Practice Test: Elementary Mathematics (68)

Correct Response: A. Base 3 uses the digits 0, 1, and 2 and has place value positions defined by powers of 3. The rightmost placeholder position represents 3⁰, 3 to the zero power the position to the left is 3¹, 3 to the first power the position to the left of that is 3², 3 squared and so on.

The number 10101 1 0 1 0 1  in base 3 can be written in expanded form as shown:
1(3⁴) + 0(3³) + 1(3²) + 0(3¹) + 1(3⁰) 1 times 3 to the fourth power plus 0 times 3 cubed plus 1 times 3 squared plus 0 times 3 to the first power plus 1 times 3 to the zero power  = 81 + 9 + 1 = 91.

Incorrect Response: B. This response may be the result of misinterpreting 3⁰ 3 to the zero power  to be equal to 3 in the expression 81 + 9 + 1(3⁰ 1 times 3 to the zero power ).

Incorrect Response: C. This response may be the result of misinterpreting the rightmost place value position as 3¹ 3 to the first power  and also misinterpreting the value of 1(3¹) 1 times 3 to the first power  to be 1, which would result in a computation of 1(3⁵) + 1(3³) + 1(3¹) 1 times 3 to the fifth power plus 1 times 3 cubed plus 1 times 3 to the first power  = 243 + 27 + 1 = 271.

Incorrect Response: D.This response may be the result of misinterpreting the rightmost place value position as 3¹, 3 to the first power which would result in a computation of 1(3⁵) + 1(3³) + 1(3¹) 1 times 3 to the fifth power plus 1 times 3 cubed plus 1 times 3 to the first power  = 243 + 27 + 3 = 273.

Correct Response: A. The absolute value of a number represents its magnitude, or its distance from 0. When comparing negative numbers, the number with the least absolute value is closest to 0 and greatest in value. Of the numbers given, − negative 2.1 has the least absolute value: | −2.1 | Absolute value of negative 2.1  = 2.1.

Incorrect Response: B. | −2.4 | Absolute value of negative 2.4  = 2.4, which is greater than | −2.1 | Absolute value of negative 2.1  = 2.1. This means that − negative 2.4 is located farther to the left on a number line than − negative 2.1, which means − negative 2.4 < is less than  − negative 2.1.

Incorrect Response: C. This response is equivalent to − negative 2.2, which has a greater absolute value than − negative 2.1. This implies that − negative 2.2 <  is less than  − negative 2.1.

Incorrect Response: D. This response is equivalent to − negative 2.8, which has the greatest absolute value of all numbers in the set. Since the number is negative, this means that − negative 2.8 is the farthest to the left of zero on a number line of all numbers in the responses, which means it has the least value.

Correct Response: D. The side lengths of the cube are all equal, and the volume of the cube is the product of its side lengths. The equation V = s³ s cubed  relates the volume of the cube, V, to the length of a side, s. The inverse of this equation—the equation that describes how to determine s when V is known—can be found by taking the cube root of both sides of the equation:
³√ V = ³√ s³ → s = √ V.   The cube root of v equals the cube root of s cubed, which implies that s is equal to the cube root of V. 

Incorrect Response: A. This statement is true, but it does not lead to a direct comparison of how finding an edge length of a cube from its volume is related to finding a volume of a cube from its edge length.

Incorrect Response: B. This statement is true, but the edge length of a cube and its volume are related by iterative multiplication (i.e., exponentiation). This statement lacks the depth of the detail that is captured in the correct response.

Incorrect Response: C. This statement is true but does not lead to a direct comparison of how finding an edge length of a cube from its volume is related to finding a volume of a cube from its edge length.

Correct Response: B. The 35 chairs may be arranged into 5 rows, with 7 chairs per row. This arrangement fulfills the criteria provided.

Incorrect Response: A. There are multiple ways to arrange 30 chairs such that all the provided criteria are met. For example, the chairs could be arranged into 2 rows, with 15 chairs per row, or 3 rows, with 10 chairs per row.

Incorrect Response: C. There are multiple ways to arrange 36 chairs such that all the provided criteria are met. For example, the chairs could be arranged into 2 rows, with 18 chairs per row, or 3 rows, with 12 chairs per row.

Incorrect Response: D. Prime numbers such as 37 can only be factored into a product of 1 and the number itself. In the context of this problem, there is only one way that 37 chairs can be arranged with an equal number of chairs per row, and that is 1 row of 37 chairs. This means that the last criterion cannot be met.

Correct Response: D. √ 12  radical 12  is an irrational number since it cannot be written as a fraction in the form ^p / _q  P divided by q  where p and q are integers. The value of √ 12  radical 12  is between 3 and 4 since √ 9< √ 12< √ 16 → 3< √ 12<4.   Radical 9 is less than radical 12 is less than radical 16, which implies 3 is less than radical 12 is less than 4.  This means that √ 12  radical 12  could represent point X on the number line shown.

Incorrect Response: A. A rational number can be written as a fraction in the form ^p / _q ,  P divided by q where p and q are integers. Since ³⁸ / ₁₁  38 elevenths  meets these criteria, it is a rational number.

Incorrect Response: B. √ 7  Radical 7  is an irrational number since it cannot be written as a fraction in the form ^p / _q ,  P divided by q where p and q are integers. However, the value of √ 7  Radical 7  is between 2 and 3 since √ 4< √ 7< √ 9 → 2< √ 7<3.  Radical 4 is less than radical 7 is less than radical 9, which implies 2 is less than radical 7 is less than 3.  For this reason, √ 7  Radical 7  cannot represent point X on the number line shown.

Incorrect Response: C. A rational number can be written as a fraction in the form ^p / _q ,  P divided by q where p and q are integers. Since 3.46 = 3 ⁴⁶ / ₁₀₀ = ³⁴⁶ / ₁₀₀,  3 and 46 hundredths which equals 346 over 100 this response is a rational number.

Correct Response: D. The area of the large rectangle can be obtained from its dimensions or by adding together the areas of the two smaller rectangles that compose it. The dimensions of the large rectangle are 5 × times  14, where the 14 represents the sum of the 8 and the 6, and its area is 70. The area of the 5 × times  8 rectangle is 40 and the area of the 5 × times  6 rectangle is 30. Together, these add to 70. This model is a useful representation for the distributive property of multiplication because it illustrates why the operation of multiplication distributes over addition. That is, it explains why 5 × (8 + 6) = 5 × 8 + 5 × 6. 5 times the quantity 8 plus 6 equals 5 times 8 plus 5 times 6

Incorrect Response: A. Student B is also correct since 5 × times  8 + 5 × times  6 = 5 × times  14.

Incorrect Response: B. Student A is also correct since 5 × times  8 + 5 × times  6 = 5 × times  14.

Incorrect Response: C. A repeated addition interpretation of multiplication should refer to a repeated quantity, a number of repetitions, and an overall result. None of these quantities are mentioned here.

Correct Response: B. The sum of a number and its additive inverse is 0. This equation shows that 3 and − negative 3 are additive inverses because their values add to 0.

Incorrect Response: A. This equation is a result of the identity property of addition, where the 4 plus 0 returns a value of 4. The identity property of addition can also be used in conjunction with the property of an additive inverse to write 4 − 0 = 4 + (−0) 4 minus 0 is equal to 4 plus negative 0  = 4, but this is not the best equation to use to illustrate the concept of an additive inverse.

Incorrect Response: C. This equation illustrates the commutative property of addition using integers, but it does not illustrate the concept of an additive inverse.

Incorrect Response: D. This equation illustrates the commutative property of addition using a whole number and its reciprocal, but it does not illustrate the concept of an additive inverse.

Correct Response: C. One-to-one correspondence is a mathematical concept found in counting and, more broadly, in set theory. It is based upon a relationship where each element from one set (e.g., names of numbers, numerals, a collection of textbooks) is uniquely matched to an element from another set (e.g., objects being counted, sections of a spinner, a collection of crayons). This idea is clearly illustrated in this response, since each book is associated with exactly one crayon.

Incorrect Response: A. This response associates one object (the textbook) with multiple other objects (crayons), so it does not illustrate the concept of one-to-one correspondence. The relationship shown here is similar to using the number 1 to count each object in a set of 3 objects.

Incorrect Response: B. This response associates multiple objects (books) with exactly one object (the crayon), so it does not illustrate the concept of one-to-one correspondence. The relationship shown here is similar to using the numbers 1, 2, and 3 to count a single object.

Incorrect Response: D. This response illustrates a complex relationship between two sets that does not follow the definition of a one-to-one correspondence. For example, following the arrows between the sets, the top left text book is associated with the right crayon and, through its subsequent associations, to every other object in both sets.

Correct Response: C. The correct response is the result of dividing the following expressions as shown: ^{3.6 × 107} / _{2.4 × 10⁵} = ^3.6 / _2.4 × ¹⁰⁷ / _10⁵ = 1.5 × 10² = 1.5 × 100 = 150.  3.6 times 10 to the seventh power divided by 2.4 times 10 to the fifth power equals 3.6 divided by 2.4 times 10 to the seventh power divided by 10 to the fifth power, which equals 1.5 times 10 squared, which equals 1.5 times 100, which equals 150

Incorrect Response: A. This response may be the result of dividing the rational numbers 3.6 and 2.4 to get a quotient of 1.5, which is correct, but then multiplying that result by 10. This may be due to an error in dividing 10⁷ 10 to the seventh power  by 10⁵ 10 to the fifth power  or a misconception about how to manipulate the decimal point.

Incorrect Response: B. This response may be the result of a misconception about how to apply exponent rules to evaluate the expression ¹⁰⁷ / _10⁵ .  10 to the seventh power divided by 10 to the fifth power

Incorrect Response: D. This response may be the result of a misconception about how to apply exponent rules to evaluate the expression ¹⁰⁷ / _10⁵ .  10 to the seventh power divided by 10 to the fifth power

Correct Response: D. The total number of stickers is 1,125, so x × times  y × times  z = 1,125. The prime factorization of 1,125 is 3² × 5³. 3 squared times 5 cubed These factors may be grouped in different ways to form the numbers x, y, and z (e.g., x = 15, y = 5, z = 15; x = 25, y = 9, z = 5). For every grouping, each number must be either 3, 5, or a product of powers of those numbers, which leads to the result that all three numbers are odd (i.e., 3 is odd, 5 is odd, and any product formed from odd factors like 3 × 5² 3 times 5 squared  is also odd).

Incorrect Response: A. The prime factorization of 1,125 is 3² × 5³. 3 squared times 5 cubed Since there are no factors of 2 in the prime factorization, 1,125 cannot have any even factors.

Incorrect Response: B. The prime factorization of 1,125 is 3² × 5³. 3 squared times 5 cubed Since there are no factors of 2 in the prime factorization, 1,125 cannot have any even factors.

Incorrect Response: C. The prime factorization of 1,125 is 3² × 5³. 3 squared times 5 cubed Since there are no factors of 2 in the prime factorization, 1,125 cannot have any even factors.

Correct Response: D. In this response, a = − negative 2, b = − negative 1, c = − negative 2. The first condition (a ≤ b)  a is less than or equal to b  is satisfied since −2 ≤ −1,  negative 2 is less than or equal to negative 1 and the second condition (b > c)  b is greater than c  is satisfied since −1 > −2.  negative 1 is greater than negative 2

Incorrect Response: A. In this response, a = 1, b = 1, c = 2. While the first condition (a ≤ b)  a is less than or equal to b  is satisfied since 1 ≤ 1,  1 is less than or equal to 1 the second condition (b > c)  b is greater than c  is not: 1 < 2.  1 is less than 2

Incorrect Response: B. In this response, a = − negative 3, b = − negative 2, c = 1. While the first condition (a ≤ b)  a is less than or equal to b  is satisfied since −3 ≤ −2,  negative 3 is less than or equal to negative 2 the second condition (b > c)  b is greater than c  is not: −2 < 1.  negative 2 is less than 1

Incorrect Response: C. In this response, a = − negative 1, b = − negative 2, c = 3. The first condition (a ≤ b)  a is less than or equal to b  is not satisfied by these values: −1 ≥ −2.  negative 1 is greater than or equal to negative 2 The second condition is also false: −2 > 3.  negative 2 is greater than 3

Correct Response: B. Point G represents the third value when the list is written in ascending order: 0.250, 0.252, 0.255, 0.259. The value of 0.255 corresponds to Sheet 2 in the table.

Incorrect Response: A. This response represents point E on the number line, since 0.25 = 0.250.

Incorrect Response: C. This response represents point H since 0.259 is the value closest to 0.260.

Incorrect Response: D. This response represents point F since 0.252 is the second lowest value in the table.

Correct Response: C. Let the thousandths place digit be A, A × ¹ / ₁₀₀₀ = A × 10⁻³,  A times 1 over 1000 equals a times 10 to the negative third power and the hundredths place digit be B, B × ¹ / ₁₀₀ = B × 10⁻² .  B times 1 over 100 equals b times 10 to the negative second power The product of these numbers is (A × 10⁻³)(B × 10⁻²) = (A × B) × 10⁻⁵.  A times 10 to the negative third power times b times 10 to the negative second power is equal to a times b times 10 to the negative fifth power If the product of A and B is less than 10, then the smallest place value that can contain a nonzero digit in the product is the hundred-thousandths place.

Incorrect Response: A. This response could be chosen if incorrect powers of 10 were used to describe the smallest place value for each decimal.

Incorrect Response: B. This response could be chosen if incorrect powers of 10 were used to describe the smallest place value for each decimal.

Incorrect Response: D. This response could be chosen if incorrect powers of 10 were used to describe the smallest place value for each decimal.

Correct Response: B. The distributive property was used to divide each addend by 20: (400 + 20 + 6) ÷ 20 = ¹ / ₂₀(400 + 20 + 6) = ⁴⁰⁰ / ₂₀ + ²⁰ / ₂₀ + ⁶ / ₂₀ = 20 + 1 + ³ / ₁₀ .  The quantity 400 plus 20 plus 6 divided by 20 equals 1 twentieth times the quantity 400 plus 20 plus 6 which equals 400 divided by 20 plus 20 divided by 20 plus 6 divided by 20 which equals 20 plus 1 plus 3 tenths

Incorrect Response: A. The associative property of addition is used to rewrite 426 into expanded form.

Incorrect Response: C. The commutative property is used to change the order in which expressions are added or multiplied.

Incorrect Response: D. The multiplicative identity property is used to reduce the fraction ⁶ / ₂₀  6 twentieths : ⁶ / ₂₀ = ^{3 × 2} / _{10 × 2} = ³ / ₁₀ × ( ² / ₂) = ³ / ₁₀ .  6 twentieths is equal to the quantity 3 times 2 divided by the quantity 10 times 2, which is equal to 3 tenths times 2 over 2, which is equal to 3 tenths

Correct Response: A. Let the regular price of a shirt be x. The regular price of two shirts is 2x, but the customer only pays a price of 1.5x for these shirts because they buy the first one at price x and the second one at price 0.5x. This represents a discount of 25% percent  since ^0.5x / _2.0x  0.5 x divided by 2.0 x  = 0.25.

Incorrect Response: B. This response may be chosen by comparing the discount to the sale price and then rounding: ^0.5x / _1.5x = 0.3 ≈ 33%.  0.5 x divided by 1.5 x equals 0.3 repeating or approximately 33 percent 

Incorrect Response: C. This response may be chosen if the half-off sale is incorrectly applied to both items.

Incorrect Response: D. This response represents the percent of the regular price that the customer pays for the two shirts.

Correct Response: D. The correct response can be found by calculating the concrete use as shown: 0.05 + ¹ / ₅ = 0.05 + 0.20 = 0.25.  0.05 plus one fifth equals 0.05 plus 0.20 which equals 0.25

Incorrect Response: A. This response may be the result of misinterpreting the value of ¹ / ₅  1 fifth  to be 0.02, which would lead to the equation 0.05 + 0.02 = 0.07.

Incorrect Response: B. This response could be the result of misinterpreting 5% percent  to be equal to 0.5 and multiplying instead of adding: 0.5 × times  0.2 = 0.10.

Incorrect Response: C. This response represents the percent of the original supply of concrete that was used on Tuesday: ¹ / ₅  1 fifth  = 0.20.

Correct Response: B. The left model shows ⁵⁶ / ₁₀₀  56 over 100  = 0.56 and the right model shows ² / ₁₀  2 tenths  = 0.20. The difference of these two values is 0.36.

Incorrect Response: A. This response could be the result of taking the difference between the area of the white region on the left square, 0.44, and the area of the gray region on the right square, 0.2.

Incorrect Response: C. This response could be the result of misinterpreting the difference to be 0.44 − minus  0.02.

Incorrect Response: D. This response could be the result of misinterpreting the difference to be 0.56 − minus  0.02.

Correct Response: C. The number line has tick marks that divide the distance between 0 and 1 into quarters. Point P is closer to the second quarter than the first quarter (0.50) and point Q is very close to the third quarter (0.75). The values P = 0.42 and Q = 0.77 are consistent with this information in the number line.

Incorrect Response: A. The number line has tick marks that divide the distance between 0 and 1 into quarters. Point P is between the first and second quarter (0.25 < is less than  P < is less than  0.50) and point Q is between the third and fourth quarters (0.75 < is less than  Q < is less than  1).

Incorrect Response: B. Point P could represent the value 0.40, but Q cannot represent 0.70 because it must be greater than 0.75.

Incorrect Response: D. The location for point P in the diagram is plausible, but the location of point Q is much too close to 0.75 to accurately represent the value 0.92.

Correct Response: D. ¹ / ₁₁ = 0.09.  1 eleventh equals 0 point 0 9 repeating Multiplying this unit fraction by any whole number from 1 to 10 produces a decimal that repeats in a similar fashion using multiples of 9 e.g., ² / ₁₁ = 0.18, ³ / ₁₁ = 0.27, …, ¹⁰ / ₁₁ = 0.90.  2 elevenths equals 0 point 18 repeating, 3 elevenths equals 0 point 2 7 repeating, 10 elevenths equals 0 point 9 0 repeating 

Incorrect Response: A. This claim can be disproven by considering n = 4. ⁴ / ₁₁ = 0.36.  4 elevenths equals 0 point 3 6 repeating

Incorrect Response: B. This claim can be disproven by considering n = 4. ⁴ / ₁₁ = 0.36.  4 elevenths equals 0 point 3 6 repeating

Incorrect Response: C. This claim can be disproven by considering n = 1. ¹ / ₁₁ = 0.09.  1 elevenths equals 0 point 0 9 repeating

Correct Response: A. Any simplified fraction ^p / _q  p over q  may be written as p × ¹ / _q ,  p times the fraction 1 over q and the decimal equivalent of ^p / _q  p over q  will only terminate if ¹ / _q  1 over q  terminates. This happens when q has no prime factors other than 2 and 5, because only powers of these values or their products (e.g., 2, 5² 5 squared  = 25, 2² · 5³ 2 squared times 5 cubed  = 500) divide 1 without producing a repeating decimal. (If ^p / _q  p over q  is a terminating decimal, then ^p / _q × 10^x  P over q times 10 to the power of x  will produce an integer for some whole number x.)

Incorrect Response: B. Consider counterexamples, such as ³ / ₅  3 fifths  and ⁷ / ₈ .  7 eighths These numerators are divisible by 3 and 7, respectively, and the fractions are equivalent to the terminating decimals 0.60 and 0.875, respectively.

Incorrect Response: C. Consider a counterexample, such as ⁴ / ₅ ,  4 fifths which is equal to 0.80.

Incorrect Response: D. Consider a counterexample, such as ³ / ₄ ,  3 fourths which is equal to 0.75.

Correct Response: C. The regular price of the refrigerator is $755,  755 dollars and the sale price is $755 × 0.8.  755 dollars times 0.8 When the shopper rounds the price up, the estimated sale price becomes $800 × 0.8.  800 dollars times 0.8 The difference between the estimate and the sale price is $800 × 0.8 − $755 × 0.8 = ($800 − $755) × 0.8 = $45 × 0.8 = $36.00.  800 dollars times 0.8 minus 755 dollars times 0.8 equals the quantity 800 dollars minus 755 dollars times 0.8 equals 45 dollars times 0.8 equals 36 dollars 

Incorrect Response: A. This response represents the difference between the actual and estimated discount: (800 × 0.2) − (755 × 0.2) = 160 − 151 = 9.  The quantity 800 times 0.2 minus the quantity 755 times 0.2 equals 160 minus 151 equals 9

Incorrect Response: B. This response represents the result of finding 20% percent  of the difference between the list price of the refrigerator and the estimate rounded down to the nearest hundred: (755 − 700) × 0.2 = 55 × 0.2   The quantity 755 minus 700 times 0.2 equals 55 times 0.2  = 11.

Incorrect Response: D. This response represents the difference between the actual and estimated discount after rounding the value of $755  755 dollars  down to $700  700 dollars : (755 − 700) × 0.8   The quantity 755 minus 700 times 0.8  = 44.

Correct Response: A. The operation of subtraction can be used to determine how much greater one quantity is than another. The difference between the length of the humerus and the length of the human hand is 36.46 − minus  18.9 = 17.56.

Incorrect Response: B. This response could be the result of subtracting 36.46 − minus  18.09.

Incorrect Response: C. This response could be the result of a regrouping error where the 3 (representing 30 in 36.46) is incorrectly grouped as 3 tens and 10 ones (rather than 2 tens and 10 ones), as follows:

The standard subtraction algorithm is shown for the problem 36.46 minus 18.90, in which a regrouping error is shown. The 6 in 36 is rewritten as a 15 and the 4 in 36.46 is rewritten as 14. The product in this case is shown to be 27.56.

Incorrect Response: D. This response applies the regrouping error described in the rationale for Incorrect Response C to both the digits 3 and 6 in 36.46.

The standard subtraction algorithm is shown for the problem 36.46 minus 18.90. The 6 in 36 is rewritten as a 16 to produce the difference 28.56.

Correct Response: C. 42% percent  of the pancake batter is flour. This percentage does not change when the person doubles the recipe to 600 grams: 0.42(600) 0.42 times 600  = 252 grams.

Incorrect Response: A. This response may be the result of thinking that 42 percent is to be doubled to 84 and taken as the answer.

Incorrect Response: B. This response represents the amount of flour used in 300 grams of pancake batter: 300 × times  0.42 = 126.

Incorrect Response: D. This response may be the result of thinking that doubling the pancake batter implies that the percentage of flour in the mix is also doubled: 2(0.42) × 600 = 0.84 × 600 2 times 0.42 times 600 equals 0.84 times 600  = 504.

Correct Response: B. If r is regular price, 0.6r is sale price during the first week. Each subsequent week, the clothing is discounted by 10%, which can be done by multiplying the expression 0.6r by 0.9 for each week of the sale. The price during week 3 can be calculated using the expression 0.6r(0.9)² = 0.486r. 0.6 r times 0.9 squared equals 0.486 r

Incorrect Response: A. This expression is the result of misinterpreting the initial sale price of the clothing and multiplying 0.4r times 0.9². 0.9 squared

Incorrect Response: C. This response represents the total amount that the original price has been discounted, not the price itself: r − (0.6r × 0.9²) = r − 0.486r R minus the quantity 0.6 r times 0.9 squared equals r minus 0.486 r  = 0.514r.

Incorrect Response: D. This response may be the result of misinterpreting the original sale price as 0.40r and then misrepresenting the sale price of the clothing on subsequent weeks as being 10% percent  of that price: r − (0.4r × 0.1²) = 0.996r R minus the quantity 0.4 r times 0.1 squared equals 0.996 r 

Correct Response: C. This response uses equal-sized rectangles to represent each box when it is full of bars. It also shows that the removal of 21 honey bars and 32 raisin bars leaves the honey box with twice as many bars compared to the raisin box.

Incorrect Response: A. This response does not make any reference to the number of bars that were eaten or to the number of bars in each of the full boxes. Instead, it represents the number of eaten honey bars, 21, as being equal to half of the remaining number of honey bars and equal to all the remaining number of raisin bars. This is not consistent with this situation.

Incorrect Response: B. This response reverses the relationship between the ratio of the leftover bars, and it does not account for the bars that have already been eaten.

Incorrect Response: D. This response reverses the relationship between the ratio of the leftover bars.

Correct Response: B. The division 11 ÷ divided by  4 can be interpreted to mean the number of groups of 4 that are needed to make 11. Two whole groups of 4 makes 8; another 3 objects ( ³ / ₄  3 fourths of a full group) are still needed to make 11. Another way to identify the significance of the remainder is to represent the division using fractions: ¹¹ / ₄ = ⁴ / ₄ + ⁴ / ₄ + ³ / ₄ = 2 ³ / ₄.  11 fourths is equal to four fourths plus four fourths plus 3 fourths which is equal to 2 and 3 fourths

Incorrect Response: A. For the division algorithm, a remainder can be represented as a fraction of the form ^remainder / _divisor .  Remainder divided by divisor This response reflects the misconception that the remainder can be represented as ^remainder / _quotient .  Remainder divided by quotient

Incorrect Response: C. For the division algorithm, a remainder can be represented as a fraction of the form ^remainder / _divisor .  Remainder divided by divisor This response reflects the misconception that the remainder can be represented as ^quotient / _remainder .  Quotient divided by remainder

Incorrect Response: D. For the division algorithm, a remainder can be represented as a fraction of the form ^remainder / _divisor .  Remainder divided by divisor This response reflects the misconception that the remainder can be represented as ^divisor / _remainder .  Devisor divided by remainder

Correct Response: C. Since the two fractions are equal, the product can be found using a proportion: ^x / ₄ = ⁶ / _y → 4 ^x / ₄ = ⁶ / _y4 → x = ²⁴ / _y → xy = 24.  X over 4 equals 6 over y implies 4 times x over 4 equals 6 over y times 4 implies x equals 24 over y, which implies the product of x and y is 24

Incorrect Response: A. This response may be the result of a misconception that the proportion ^x / ₄ = ⁶ / _y  X over 4 equals 6 over y  has a property that the sums of opposite numerators and denominators are equal (e.g., that an equivalent of cross products exists with addition).

Incorrect Response: B. This response represents the least common multiple of 4 and 6.

Incorrect Response: D. This response could be the result of the misconception that fractions are equivalent if their numerator and denominator differ by the same amount. If that difference is taken to be 1, then the fractions are supposed to be ⁵ / ₄ and ⁶ / ₅.  5 fourths and 6 fifths

Correct Response: A. The expression ³⁰ / ₈  30 divided by 8  represents the number of pizzas required to feed 30 students if each student has 1 slice and each pizza is cut into 8 slices (3.75 pizzas total). The expression ^{(30 × 2)} / ₈  The quantity 30 times 2 divided by 8  represents the number of pizzas required to feed 30 students if they each have 2 slices of pizza (7.5). Since one third of students want cheese pizza and the remaining students want pepperoni pizza, it follows that two thirds of the pizzas need to be pepperoni pizzas. The number of pepperoni pizzas needed is ² / ₃  2 thirds  of the total number of pizzas needed: ^{(30 × 2)} / ₈ × ² / ₃ .  The quantity 30 times 2 divided by 8 times the fraction 2 thirds

Incorrect Response: B. This response may be the result of the misconception that the fraction representing the total number of pizzas needed, ^{(30 × 2)} / ₈ ,  The quantity 30 times 2 divided by 8 can be divided by ¹ / ₃  1 third  to find the result. (Since dividing by a fraction is the same as multiplying by the reciprocal, dividing by ¹ / ₃  1 third  triples the number of pizzas needed.)

Incorrect Response: C. This response represents the number of cheese pizzas that are needed, not the number of pepperoni pizzas.

Incorrect Response: D. This response may be the result of the misconception that the fraction representing the total number of pizzas needed, ^{(30 × 2)} / ₈ ,  The quantity 30 times 2 divided by 8 can be divided by ² / ₃  2 thirds  to find the result. (Since dividing by a fraction is the same as multiplying by the reciprocal, dividing by ² / ₃  2 thirds  increases the total number of pizzas needed by a factor of one half.)

Correct Response: A. The combined weight of the art pieces can be found by evaluating the expression

9 plus 5 sixths times 9 plus 4 fifths (5 in denominator is crossed out) times 5 sixths (5 in numerator is crossed out) times 9 is equal to 9 plus 45 sixths plus 2 thirds times 9 is equal to 9 plus 7 and one half plus 6 which is equal to 22 and 1 half

Incorrect Response: B. This response may be the result of misinterpreting the combined weight of the art pieces to 9 + ⁵ / ₆9 + ⁴ / ₅ 9.  9 plus 5 sixths times 9 plus 4 fifths times 9

Incorrect Response: C. This response may represent a misinterpretation of the combined weight of the art pieces as well as a miscalculation in the sum 9 + ⁵ / ₆ + 9 + ⁴ / ₅ + 9  9 plus 5 sixths plus 9 plus 4 fifths plus 9  (adding the numerators and denominators without finding a common denominator).

Incorrect Response: D. This response may represent a misinterpretation of the combined weight of the art pieces as 9 + ⁵ / ₆ + 9 + ⁴ / ₅ + 9.  9 plus 5 sixths plus 9 plus 4 fifths plus 9

Correct Response: D. In the image shown, x, y, and z represent the missing values in the table.

The table from the stem is shown with an x representing the digit in the tens place for the payment, y representing the digit in the tens place for the number of monthly payments, and z representing the digit in the ones place for the number of monthly payments.

Since 5y = 150, y must equal 30. If y = 30, then xy x y  = 600 implies that x = ⁶⁰⁰ / ₃₀  600 divided by 30  = 20. 20z 20 z  must equal 120, which implies that z = ¹²⁰ / ₂₀  120 divided by 20  = 6. Since y = 30 and z = 6, the total number of monthly payments is equal to y + z or 36.

Incorrect Response: A. This value represents the amount of each monthly payment, not the number of monthly payments.

Incorrect Response: B. This response is the result of adding two of the missing values diagonally in the table (x and z in the image shown in the rationale for Correct Response D).

Incorrect Response: C. This response is the result of adding two of the values diagonally in the table (y and 5 in the image shown in the rationale for Correct Response D).

Correct Response: D. This response represents the product of 50 and 240, which is the result of correctly rounding each value to the nearest tens place: 50 × times  240 = 12,000.

Incorrect Response: A. This response is the result of rounding the values down to the nearest ten in the estimate: 40 × times  230 = 9,200.

Incorrect Response: B. This response may be the result of incorrectly rounding the value of 46 to 40 in the estimate: 40 × times  240 = 9,600.

Incorrect Response: C. This response may be the result of incorrectly rounding the value of 237 to 230 in the estimate: 50 × times  230 = 11,500.

Correct Response: C. The estimated value is the sum of three values that are rounded up to the nearest tenth and one value that is rounded down. The difference between the estimated sum and the actual sum is maximized when the maximum amount of money is rounded up and the minimum amount of money is rounded down. This occurs when three piles of money are rounded up by 5 cents and the remaining pile of money is rounded down by 0 cents.

Incorrect Response: A. This value represents the three piles being rounded up by 5 cents and one pile being rounded down by 5 cents. It is not the greatest possible error.

Incorrect Response: B. This value represents the three piles being rounded up by 5 cents and one pile being rounded down by 4 cents. It is not the greatest possible error.

Incorrect Response: D. This value could represent the sum of rounding four piles of coins up by 4.5 cents, or rounding three piles of coins up by 9 cents and one pile down by 9 cents. Neither case happens in this problem.

Correct Response: B. The total volume of the fudge in the tray is equal to 12 × times  15 × times  1 = 180 cubic inches. Since 20 cubic inches of fudge equals one pound, the total weight of the fudge in the tray is ¹⁸⁰ / ₂₀  180 divided by 20  = 9 lbs. The number of pieces of fudge can be found by dividing the total weight of the fudge in the tray by the weight of each individual piece: 9 ÷ ¹ / ₁₆ = 9 × 16 = 144.  9 divided by 1 sixteenth equals 9 times 16 which equals 144

Incorrect Response: A. This response may be the result of misinterpreting the information and using the calculation (16 × 20) − (12 × 15)  The quantity 16 times 20 minus the quantity 12 times 15  = 140.

Incorrect Response: C. This response represents the total volume of the fudge in the tray in cubic inches, not the number of pieces that can be formed using that volume.

Incorrect Response: D. This response may be the result of misinterpreting the information and using the calculation 12 × 15 ÷ ¹ / ₂₀ ÷ 16 = 225.  The quantity 12 times 15 divided by 1 twentieth all divided by 16 is equal to 225

Correct Response: B. The order of operations (PEMDAS P E M D A S ) implies that first priority when evaluating this expression is simplifying the exponential term (since there are no operations to complete within parentheses): (−5)² = −5 × −5 = 25 → − ³ / ₂ ÷ 25.  The quantity negative 5 squared equals negative 5 times negative 5 which equals 25, which implies negative 3 halves divided by 25 The second operation that appears in the expression is division, and since dividing by a fraction is the same as multiplying by the reciprocal, − ³ / ₂ ÷ 25 = − ³ / ₂ × ¹ / ₂₅ = − ³ / ₅₀.  Negative 3 halves divided by 25 equals negative 3 halves times 1 twenty-fifth which equals negative 3 fiftieths

Incorrect Response: A. This response may be the result of misinterpreting the order of operations and simplifying the expression − ³ / ₂ × 5² = − ³ / ₁₀²  The quantity negative 3 halves times 5 squared is equal to the quantity negative 3 tenths squared and then making an error with the sign when squaring the fraction.

Incorrect Response: C. This response may be the result of a sign error when applying the order of operations (interpreting the value of − ³ / ₂  negative 3 halves  as positive at some point in the process), or misinterpreting the value of (−5)².  the quantity negative 5 squared

Incorrect Response: D. This response may be the result of incorrectly multiplying the denominator of the fraction − ³ / ₂  negative 3 halves  by the whole number 5 and simplifying the expression as shown: − ³ / ₂ × 5² = − ³ / ₁₀² = ⁹ / ₁₀₀.  The quantity negative 3 halves times 5 squared equals the quantity negative 3 tenths squared equals 9 hundredths

Correct Response: B. The length of segment d is the sum of the distances that a and b are from zero on the number line. | b |  The absolute value of b  gives the distance that b is from zero on the number line, and | a |  The absolute value of a  gives the distance that a is from zero on the number line, so the sum | a | + | b |  The absolute value of a plus the absolute value of b  gives the total length of segment d.

Incorrect Response: A. This response is the result of a misconception involving absolute value. If a is positive and b is negative, then the sum has a value that is less than the total distance between a and b on the number line.

Incorrect Response: C. Since b is negative and a is positive, the sum a + b will produce a value that is less than the total distance between the endpoints.

Incorrect Response: D. The length of segment d is the combined distance from b to 0 and from 0 to a. These distances are combined with addition and not subtraction.

Correct Response: A. Finding one fourth of a quantity is the same as dividing it by 4 or multiplying it by one fourth. The amount of paint that was used to paint the wall can be found by multiplying the initial amount of paint, 2 ¹ / ₂ ,  2 and 1 half by ¹ / ₄ .  1 fourth

Incorrect Response: B. The cyclist travels 2 ¹ / ₂  2 and 1 half  miles in one quarter of an hour, or ^{2 1 / ₂ miles} / _{¹ / 4 hour} .  2 and 1 half miles divided by one fourth of an hour Then the speed of the cyclist, in miles per hour, is four times that quantity: 2 ¹ / ₂ ÷ ¹ / ₄ = 2 ¹ / ₂ × 4 .  2 and 1 half divided by 1 fourth equals 2 and 1 half times 4

Incorrect Response: C. Consider a rectangle with side lengths a and b. The area of the rectangle is ab, a times b  which is equal to 2 ¹ / ₂  2 and 1 half  in this context. If one dimension, a, is equal to ¹ / ₄ ,  1 fourth then ¹ / ₄ × b = 2 ¹ / ₂ → b = 2 ¹ / ₂ ÷ ¹ / ₄.  1 fourth times b equals 2 and 1 half which implies b equals 2 and 1 half divided by 1 fourth

Incorrect Response: D. Like response C, this problem implies division. The total number of quarter cup portions in 2 ¹ / ₂  2 and 1 half  cups of chocolate chips is equal to 2 ¹ / ₂ ÷ ¹ / ₄.  2 and 1 half divided by one fourth 

Correct Response: A. This equation rewrites the original expression using mathematical concepts, the first of which is the fact that a fraction like 2 ³ / ₄  and 3 fourths  is equal to 2 + ³ / ₄ .  3 fourths The commutative and associative properties of addition are then applied to add the whole number values first and the fractional values second.

Incorrect Response: B. This response represents a misconception related to the distributive property of multiplication for the expression 2 + ³ / ₄ × 3 + ¹ / ₃.  The quantity 2 plus 3 fourths times the quantity 3 plus one third

Incorrect Response: C. Consider the following property of division for c + d ≠ is not equal to  0: ^{a + b} / _{c + d} = ^a / _{c + d} + ^b / _{c + d} .  The quantity a plus b divided by the quantity c plus d is equal to a divided by the quantity c plus d plus b divided by the quantity c plus d To apply this property correctly to the expression in this response, let a = 2 and b = ³ / ₄ ,  3 fourths divide both a and b by 3 ¹ / ₃ ,  and 1 third and then add their quotients together. This response is incorrect because it shows only a being divided by 3 and b being divided by 4√ x.  4 radical x

Incorrect Response: D. This response represents a misconception (sign error) with using the distributive property for subtraction. The following equation corrects the error: 2 + ³ / ₄ − 3 + ¹ / ₃ = 2 + ³ / ₄ + −3 − ¹ / ₃ = (2 − 3) + ³ / ₄ − ¹ / ₃ .  The quantity 2 plus 3 fourths minus the quantity 3 plus 1 third equals the quantity 2 plus 3 fourths plus the quantity negative 3 minus 1 third equals the quantity 2 minus 3 plus the quantity 3 fourths minus 1 third

Correct Response: D. The fence is a square enclosure with an area of x square feet. Therefore, the side length of the fence is √ x  Radical x  and its total perimeter is 4√ x.  4 Radical x Because the information states that the perimeter of the fence is equal to y, y = 4√ x.  4 Radical x

Incorrect Response: A. The equation y = 2x describes a linear equation with a y-intercept of 0 and a slope of 2. This line passes through points such as (1, 2), (2, 4), and (3, 6). The graph in the stem is not linear, so this equation is incorrect.

Incorrect Response: B. The equation y = √ x  Radical x  + 6 describes a square root function that increases in a way that is similar to the graph shown (in that the correct response is a square root equation) but has a y-intercept at the point (0, 6), which is inconsistent with the y-intercept shown in the graph.

Incorrect Response: C. This response can be eliminated by considring that it predicts a perimeter of − negative 7 for an area of 1, which is inconsistent with the values shown in the graph and makes no sense for this context.

Correct Response: D. This function generates the correct values since each output, f(x), f of x is equal to the reciprocal of the square of the input, x:

Incorrect Response: A. This function would generate the following values:

Incorrect Response: B. This function would generate the following values:

Incorrect Response: C. This function would generate the following values:

Correct Response: A. The function machine applies two operations to produce the output (subtract the input by 2, then square that value). These rules produce a positive output for an input of 0, an output of 0 for an input for 2, and positive outputs for inputs greater than 2. This graph is consistent with that information.

Incorrect Response: B. This graph always grows more slowly as input values increase, which is not consistent with the squaring operation. Consider that the given rules of the function machine match the input values of 2, 12, and 22 to the output values of 0, 100, and 400, respectively.

Incorrect Response: C. This graph shows output values always increasing with increasing input values. This is not consistent with the rules for the function machine for input values less than 2: the input value 0 has an output value of 4 and the input value of 1 has an output value of 1. The output values only begin to increase when the input values exceed 2.

Incorrect Response: D. This graph lacks the properties described in the correct response. Consider that the given rules of the function machine match the input values of 2, 12, and 22 to the output values of 0, 100, and 400, respectively.

Correct Response: C. The given information states that the mass needed to balance a 30-gram mass is inversely proportional to its distance from the fulcrum. This means that mx m x  = k, where k is a constant. Substituting the values given in the problem yields 30 × 60 = 1800 30 times 60 = 1,800  = k. If x is changed to 20, then m(20) = 1800 → m   M times 20 equals 1,800, which means m  = 90.

Incorrect Response: A. This response is the result of the misconception that mass is proportional to the distance from the fulcrum, rather than inversely proportional.

Incorrect Response: B. This is the result of misinterpreting the question and identifying the distance between the fulcrum and m, rather than the mass needed to balance the board.

Incorrect Response: D. This response is the correct mass needed to balance the board when x is 15 centimeters (one fourth the starting distance) from the fulcrum.

Correct Response: D. The total number of triangles in any figure of the pattern can be represented by the expression 4n², 4 n squared where n is the figure number. Since one of the four triangles in each square is shaded, there are a total of ¹ / ₄4n² = n²  One fourth times 4 n squared equals n squared  shaded squares in each figure. The number of unshaded squares can be determined by finding the difference: 4n² − n² = 3n².  4 n squared minus n squared equals 3 n squared The seventh figure in the pattern has a total of 3(7)²  3 times 7 squared  = 147 unshaded triangles.

Incorrect Response: A. This response may represent the misconception that the number of unshaded triangles in a figure is equal to n²  n squared : 7²  7 squared  = 49.

Incorrect Response: B. This response may represent the misconception that the number of unshaded triangles is equal to 3⁴  3 to the fourth power  = 81.

Incorrect Response: C. This represents the number of unshaded triangles in the sixth figure in the pattern: 3(6)²  3 times 6 squared  = 108.

Correct Response: B. The poured water increases ever more slowly in height as it fills the container from its base to its widest point. Where the container narrows in its middle region, the height of the poured water increases ever more quickly until it reaches the top section of the container. That final section has a short height and a constant, narrow width, so the water fills relatively quickly and at a constant rate. The graph in this response best represents these changes.

Incorrect Response: A. This response shows the height of the water in the container to be increasing at an increasing rate, which is not true for the initial shape of the container.

Incorrect Response: C. The horizontal stretch in the graph suggests that water is being poured in without changing its height in the container. This will not happen with the container shown.

Incorrect Response: D. The horizontal stretch in the graph suggests that water is being poured in without changing its height in the container. This will not happen with the container shown.

Correct Response: C. This scenario is correct because it describes the cyclist's movement in a way that is consistent with the graph:

• "1 mile in the first 10 minutes" implies that the graph passes through the point (10, 1).
• "An additional mile on a rough dirt road during the next 20 minutes" implies a lesser rate of change, passing through the point (30, 2).
• "The cyclist stops for 30 minutes…" implies that the graph is horizontal from the point (30, 2) to (60, 2).
• "…before continuing 3 miles down a paved road to a river" implies a greater rate of change up to a y-value of 5.
• "The cyclist rides home on a direct route on a paved road, completing the trip in 2 hours" implies a constant negative rate of change to the x-axis (at x = 120).

Incorrect Response: A. The first sentence of this scenario implies that the graph should pass through the point (20, 3) since the bicyclist has traveled a total of 3 miles in 20 minutes. The graph shown does not pass through this point.

Incorrect Response: B. This scenario is consistent with the graph shown from the origin to x = 30, but at that point, the graph shows a horizontal line (which implies that the bicyclist has stopped). This scenario describes the cyclist traveling along a side road at that point, which would mean that the cyclist's distance from home would not be constant in that interval.

Incorrect Response: D. This scenario states that the bicyclist has traveled 3 miles in 30 minutes, which implies the graph should pass through the point (30, 3). The graph shown does not pass through this point, so the scenario is not accurate.

Correct Response: C. Evaluating the expression for x = 2 requires careful interpretation of the negative signs and the exponents: 3(−2)⁻¹ − (−2)³ = 3− ¹ / ₂ − (−8) = − ³ / ₂ + 8 = −1 ¹ / ₂ + 8 = 6 ¹ / ₂.  3 times the quantity negative 2 to the negative first power minus the quantity negative 2 cubed equals 3 times the quantity negative one half minus the quantity negative 8 equals negative 3 halves plus 8 equals negative 1 and 1 half plus 8 equals 6 and 1 half. 

Incorrect Response: A. An error related to the sign for the value of (−2)³  The quantity negative 2 cubed  could lead to this result. Interpreting the value to be 8 leads to the expression − ³ / ₂ − 8 = −9 ¹ / ₂.  Negative 3 halves minus 8 = negative 9 and 1 half

Incorrect Response: B. This response may be the result of misinterpreting the value of 3(−2)⁻¹  3 times the quantity negative 2 to the negative first power  to be − ¹ / ₆  negative 1 sixth  and making a sign error when interpreting (−2)³  The quantity negative 2 cubed : − ¹ / ₆ − 8 = −8 ⁵ / ₆.  negative 1 sixth minus 8 equals negative 8 and 5 sixths

Incorrect Response: D. This response may be the result of misinterpreting the value of 3(−2)⁻¹  3 times the quantity negative 2 to the negative first power  to be − ¹ / ₆  negative 1 sixth : − ¹ / ₆ + 8 = 7 ⁵ / ₆.  negative 1 sixth plus 8 equals 7 and 5 sixths

Correct Response: B. In the equation Qt  Q times t  = V, Q is directly proportional to V (e.g., doubling flow rate Q for a given amount of time will double volume V) and inversely proportional to t (e.g., doubling the flow rate Q and halving the time allows the same volume of water to pass through the pipe). In general, two variables are directly proportional when they are in a proportional relationship and inversely proportional when their product is constant.

Incorrect Response: A. This equation shows that Q is inversely proportional to V since Q = ^t / _V .  t divided by v (This implies that the value of Q decreases as the value of V increases.)

Incorrect Response: C. This equation shows that Q is directly proportional to both t and V (because Q = Vt  v times t ).

Incorrect Response: D. This equation shows that Q is inversely proportional to both t and V (because QVt  Q times v times t  = 1).

Correct Response: D. Applying the distributive property and then combining like terms produces the following result: 8x − 5(x − 1) + 3x = 8x − 5x + 5 + 3x.  8 x minus 5 times the quantity x minus 1 plus 3 x equals 8 x minus 5 x plus 5 plus 3 x This further simplifies to 3x + 5 + 3x = 6x + 5.  3 x plus 5 plus 3 x equals 6 x plus 5

Incorrect Response: A. This response is the result of sign errors with the distributive property, as well as errors with distributing the factor of − negative 5 to the − negative 1 in the quantity (x − 1): 8x + 5x − 1 + 3x = 10x − 1.  X minus 1: 8 x plus 5 x minus 1 plus 3 x equals 10 x minus 1

Incorrect Response: B. This response is the result of forgetting about the factor 3x at the end of the expression: 8x − 5x + 5 = 3x + 5.  8 x minus 5 x plus 5 equals 3 x plus 5

Incorrect Response: C. This response is the result of not distributing the factor of − negative 5 to the − negative 1 in the quantity (x − 1): 8x − 5x − 1 + 3x = 3x − 1 + 3x = 6x − 1.  X minus 1: 8 x minus 5 x minus 1 plus 3 x equals 3 x minus 1 plus 3 x equals 6 x minus 1

Correct Response: C. Solving the equation for n yields the following result: y = x + dn − d → y = x + d(n − 1) → y − x = d(n − 1) → ^{y − x} / _d = n − 1 → n = 1 + ^{y − x} / _d  Y equals x plus d n minus d implies y equals x plus d times the quantity n minus 1 implies y minus x equals d times the quantity n minus 1 implies the quantity y minus x divided by d equals n minus 1 implies n equals 1 plus the quantity y minus x divided by d. 

Incorrect Response: A. This response may be the result of sign errors along with a misconception that the two values of d in the initial equation can be combined to form 2d.

Incorrect Response: B. This response represents misconceptions about what it means to "move" a variable from one side of the equal sign to another. (This may be the result of moving the variables without regard to the addition property of equality.)

Incorrect Response: D. This response may be the result of a sign error when moving one value of d to the left with y, and then dividing both sides by (x − d), x minus d which could be the result of a misinterpretation of the order of operations in the problem.

Correct Response: A. The double shaded region in the graph below describes all the possible combinations of Cruisers and Racers that the company may develop without exceeding 24 total bicycles (i.e., where x + y ≤  is less than or equal to  24) and without exceeding 160 total hours of labor (i.e., where 8x + 4y ≤  is less than or equal to  160).

A graph is shown in the first quadrant of the coordinate plane such that the horizontal axis represents the number of racers and the vertical axis represents the number of cruisers. The graph shows the shaded region defined by the inequalities x plus y is less than or equal to 24 and 8 x plus 4 y is less than or equal to 160.

Incorrect Response: B. This response shows the solution to the system of inequalities x + y ≤  is less than or equal to  24 and y ≤  is less than or equal to  20. The inequality y ≤  is less than or equal to  20 may be the result of dividing 160 by 8.

Incorrect Response: C. This response shows the solution to the system of inequalities x + y ≤  is less than or equal to  24 and x ≤  is less than or equal to  12. The inequality x ≤  is less than or equal to  12 may be the result of adding 4 and 8.

Incorrect Response: D. This response is the graph of the inequality x + y ≤  is less than or equal to  24, which is only one of the inequalities used to solve the system.

Correct Response: D. The ratio of the investments shown in the table X: to Y: to Z is equal to 750: to 500: to 250 or 3: to 2: to 1. The total investment is equal to $2100, 2,100 dollars so we can use the equation 3x + 2x + x = 2100  3 x plus 2 x plus x equals 2,100  to determine the investment in Company Z: 3x + 2x + x = 2100 → 6x = 2100 → x = 350  3 x + 2 x + x = 2,100 implies 6 x = 2,100 implies x = 350  This means the investor may invest $350  350 dollars  in Company Z, twice that amount (2 × 350 = $700)  2 times 350 = 700 dollars  in Company Y, and triple that amount (3 × 350 = $1100)  3 times 350 = 1,100 dollars  in Company X.

Incorrect Response: A. This response maintains the same ratio between the companies as the table, but the sum of the investments is not equal to $2100.  2,100 dollars

Incorrect Response: B. In this response, the second investor spends a total of $2100  2,100 dollars  by investing $200  200 dollars  more than what the first investor allocated for each of the companies. This addition does not keep the ratios equal. For example, ^{$750 + $200} / _{$500 + $200} = ^$950 / _$700 ≠ ^$750 / _$500.  The sum 750 dollars plus 200 dollars divided by the sum 500 dollars plus 200 dollars equals 950 dollars divided by 700 dollars which is not equal to 750 dollars divided by 500 dollars.

Incorrect Response: C. In this response, the ratio comparing the investment in Company X to the investment in Company Z is ^$1050 / _$700 = ³ / ₂.  1,050 dollars divided by 700 dollars is equal to 3 divided by 2 However, the table shows that the ratio of these investments is actually ^$750 / _$250 = ³ / ₁.  750 dollars divided by 250 dollars is equal to 3 divided by 1

Correct Response: B. The unit rate (i.e., the cost for one muffin) can be found by the equation ^13.5 / ₃  13.5 divided by 3  = 4.50. The missing value in the table corresponds to the price for 5 muffins, which is 5 times the unit rate: 4.50 ×  times  5 = 22.50.

Incorrect Response: A. This response is the result of interpreting 5 to be ² / ₅  2 fifths  of the distance from 3 to 8, which may lead to the calculation 13.50 + 13.50( ² / ₅  2 fifths ) = 13.50 + 5.40 = 18.90.

Incorrect Response: C. This response is the result of finding the average of the costs shown in the table: ^{13.50 + 36.00} / ₂ = ^49.50 / ₂ = 24.75.  The sum of 13.50 and 36.00 divided by 2 is equal to 49.50 divided by 2 which is equal to 24.75 This would imply that the price for each muffin is $4.95, 4 dollars and 95 cents which is not equal to ^13.5 / ₃  13.5 divided by 3  = 4.50.

Incorrect Response: D. This response may be the result of misinterpreting the unit cost for a muffin to be $1.00,  1 dollar then subtracting $3.00  3 dollars  from the value of $36.00.  36 dollars This is inconsistent with the unit price of ^13.5 / ₃  13.5 divided by 3  = 4.50 per muffin.

Correct Response: C. If the 12 kittens represent ² / ₃  2 thirds  of the cats, then the total number of cats can be found using the proportion ² / ₃ = ¹² / _x → x = 18 .  2 thirds equals 12 divided by x which implies x equals 18 Since 25% percent  of the animals at the shelter are cats, the total number of animals can be found using the percent proportion ²⁵ / ₁₀₀ = ¹⁸ / _a.  25 divided by 100 equals 18 divided by a

Incorrect Response: A. This response is the result of misinterpreting the total number of cats to be 8 ( ² / ₃  2 thirds  of 12). This contradicts the given information that there are 12 kittens at the animal shelter.

Incorrect Response: B. This response is the result of misinterpreting the total number of cats to be 8 ( ² / ₃  2 thirds  of 12) and then setting up the percent proportion incorrectly. This gives the result that there are only 2 animals at the shelter.

Incorrect Response: D. This is the result of correctly identifying the number of cats at the shelter but then setting up the final percent proportion incorrectly. This response gives a total of 4.5 animals at the shelter.

Correct Response: D. Using the similarity statement provided, we can conclude that PQ ∼ MN  P Q is similar to M N  and RQ ∼ ON.  R Q is similar to O N We can calculate PQ by setting up a proportion where the ratios compare the lengths of the segments within each triangle: ^PQ / _RQ = ^MN / _ON → ^PQ / ₁₀₈ = ¹⁹ / ₂₇.  P Q divided by R Q is equal to M N divided by O N implies P Q divided by 108 is equal to 19 divided by 27

Incorrect Response: A. This response is likely the result of misinterpreting the similarity statement shown in the stem. This could lead to the construction of an invalid proportion.

Incorrect Response: B. This response is likely the result of misinterpreting the similarity statement shown in the stem. This could lead to the construction of an invalid proportion.

Incorrect Response: C. This response is likely the result of misinterpreting the similarity statement shown in the stem. This could lead to the construction of an invalid proportion.

Correct Response: D. When two variables vary directly, the ratio between them always stays the same. In this equation, the dependent variable (y) is always 7 times the independent variable (x).

Incorrect Response: A. In this equation, x and y are inversely related. The ratio between x and y does not stay the same for different inputs for this equation. (Consider the input/output pairs x = 1, y = 2 and x = 8, y = ¹ / ₄ .)  1 fourth

Incorrect Response: B. This describes a linear equation with a constant rate of change and a y-intercept of 9, but a linear equation only describes direct variation when the y-intercept is 0.

Incorrect Response: C. This equation is equal to y = ^{x + 1} / _x ,  the expression x plus 1 all divided by x which describes a rational function. The values of x and y do not maintain the same ratio. (Consider the input/output pairs x = 1, y = 2 and x = 4, y = ⁵ / ₄ .)  5 fourths

Correct Response: B. In a proportion, cross products are equal:

R divided by s equals t divided by u implies u times r divided by s equals t divided by u (u in denominator is crossed out) times u (which is crossed out) implies u r divided by s equals t implies s (which is crossed out) times u r divided by s (s in denominator is crossed out) equals s t implies u r equals s t.

Incorrect Response: A. This response may be the result of the misconception that diagonal sums are equal in a proportion. (For a counterexample, consider ¹ / ₂ = ⁵ / ₁₀  1 half = 5 tenths : 1 + 10 ≠  is not equal to  5 + 2).

Incorrect Response: C. This response represents the misconception that the sum of the numerators must be equal to the sum of the denominators in a proportion. (For a counterexample, consider ¹ / ₂ = ⁵ / ₁₀  1 half = 5 tenths : 1 + 5 ≠  is not equal to  2 + 10.)

Incorrect Response: D. This response represents the misconception that the product of the numerators must be equal to the product of the denominators in a proportion. (For a counterexample, consider ¹ / ₄ = ²⁵ / ₁₀₀: 1 × 25 ≠ 4 × 100.)  1 fourth equals 25 hundredths, but 1 times 25 is not equal to 4 times 100 

Correct Response: A. The 260-foot parcel is 1.3 times bigger in area than the 200-foot parcel (because ^{260 × 200} / _{200 × 200} = ¹³ / ₁₀  The product 260 times 200 divided by the product 200 times 200 is equal to 13 tenths ). The price of this parcel must also be 1.3 times greater than the smaller parcel because it is directly proportional to the area. Therefore, the lot costs $24,000 × 1.3 = $31,200.  24,000 dollars times 1.3 = 31,200 dollars

Incorrect Response: B. This value represents the area, in square feet, of the 200-foot by 200-foot parcel.

Incorrect Response: C. This value may be the result of using a 260-foot by 260-foot parcel in the calculations: ^{260 × 260} / _x = ^{200 × 200} / _24,000 → 40,000x = 1,622,400,000 → x = 40,560.  The product of 260 times 260 divided by x is equal to the product 200 times 200 divided by 24 thousand, which implies 40 thousand x is equal to 1 billion 6 hundred 22 million 4 hundred thousand, which implies x is equal to 40 thousand 5 hundred 60

Incorrect Response: D. This value represents the area, in square feet, of the 200-foot by 260-foot parcel.

Correct Response: A. Assuming that the walkway is moving at a constant speed, we can find the rate of change using any two data points (x₁, y₁) and (x₂, y₂)  x 1 y 1 and x 2 y 2  in the table: ^{y₂ − y₁} / _{x2 − x1} .  Y 2 minus y 1 divided by x 2 minus x 1 Using the points (0, 500) and (380, 1), the rate of change is equal to ^{500 − 380} / _{0 − 1} = −120.  500 minus 380 divided by 0 minus 1 is equal to negative 120 This means that the distance from the traveler to end of the walkway is decreasing by 120 feet per minute, which implies that the walkway is moving the traveler at a rate of 120 feet per minute.

Incorrect Response: B. This response is the result using the data point (2, 260) to create the ratio ²⁶⁰ / ₂ = ¹³⁰ / ₁.  260 divided by 2 equals 130 divided by 1 Since the context refers to the distance from the end of the 500 ft. foot  walkway, the correct ratio is ^{500 − 260} / _{0 − 2} = ²⁴⁰ / ₋₂ = −120.  500 minus 260 divided by 0 minus 2 equals 240 divided by negative 2 equals negative 120

Incorrect Response: C. This response may be the result of finding the average for the distance values in the table: ^{500 + 380 + 260 + 140} / ₄ = ¹²⁸⁰ / ₄ = 320.  500 plus 380 plus 260 plus 140 divided by 4 equals 1,280 divided by 4 equals 320

Incorrect Response: D. This may be the result of misinterpreting the data point (1, 380) to represent the rate of change for the scenario. This would be true only if the data described the traveler's total distance traveled per minute, rather than their distance from the end of the walkway.

Correct Response: A. The initial rate for the machine is ³⁶ / ₂  36 divided by 2  = 18 pies per hour. The new rate is ²⁴⁰ / ₈  240 divided by 8  = 30 pies per hour. The difference in the rates is equal to 30 −  minus  18 = 12.

Incorrect Response: B. This response is the initial rate of production for the machine in the factory, not the difference in the rate of production.

Incorrect Response: C. This response is the rate of production for the machine after the maintenance, not the difference in the rate of production.

Incorrect Response: D. This response is the result of using the ratio ^{240 − 36} / _{8 − 2} = ²⁰⁴ / ₆  240 minus 36 divided by 8 minus 2 equals 204 divided by 6  = 34. This answer is not accurate because the scenario involves two different, constant rates of change.

Correct Response: C. The x-intercept for a line occurs when y = 0. To find the x-intercept for the linear equation shown, substitute 0 for y and solve for x: 3x + 2(0) = 12 → 3x = 12 → x = 4.  3 x plus 2 times 0 equals 12 implies 3 x equals 12 implies x equals 4

Incorrect Response: A. This response may be the result of interpreting the coefficient for y in the given equation as the x-intercept for the line.

Incorrect Response: B. This response may be the result of interpreting the coefficient for x in the given equation as the x-intercept for the line.

Incorrect Response: D. This response represents the y-intercept for the given line: 3(0) + 2y = 12 → 2y = 12 → y = 6.  3 times 0 plus 2 y equals 12 implies 2 y equals 12 implies y equals 6

Correct Response: D. The solution to this problem can be found by replacing p with (p + 0.50) in the equation provided: d = −17,500(p + 0.50) + 78,750 → d = −17,500p − 8,750 + 78,750 → d = −17,500p + 70,000.  D = negative 17,500 times the quantity p plus 0.50 plus 78,750 implies d equals negative 17,500 p minus 8750 plus 78,750 implies d equals negative 17,500 p plus 70,000 The new equation is 8,750 less than the original equation, which corresponds to a decrease in demand by 8,750 units.

Incorrect Response: A. This response may be the result of misinterpreting the new equation d = −17,500 + 70,000.  D = negative 17,500 plus 70,000 The misconception may involve interpreting the y-intercept of the new equation as the change in demand.

Incorrect Response: B. This response may be the result of a sign error when combining the constant terms in the equation d = −17,500(p + 0.50) + 78,750.   D equals negative 17,500 times the quantity p plus 0.50 plus 78,750 (Accidentally using a minus sign instead of a plus sign after combining − negative8,750 and 78,750 may lead to the conclusion that 70,000 implies a decrease in demand by 70,000 units.)

Incorrect Response: C. This response may be the result of misinterpreting the given equation and the new equation d = −17,500p + 70,000.  D = negative 17,500 p plus 70,000 If d = −17,500p + 78,750   D equals negative 17,500 p plus 78,750  is considered the new equation, it would appear to correspond to an increase in demand by 8,750 units.

Correct Response: B. The slope of each line represents the cost per mile for renting each truck. This information would allow someone to determine which truck is more reasonable for a given distance.

Incorrect Response: A. The x-intercepts for each line would represent the number of miles each truck can be driven for $0, 0 dollars so they are not in the domain for the functions in this context.

Incorrect Response: C. The y-intercepts show the initial cost to rent each truck, but they do not provide information on the cost per mile to rent each truck.

Incorrect Response: D. The intersection shows the number of miles for which the total cost of renting each truck is the same, but it does not provide information about the specific rate of change for each line.

Correct Response: B. The x-intercept for a line occurs when y = 0. To find the x-intercept for the given equation, substitute 0 for y and solve for x: 48x = 12(0) − 17 − 3x → 48x = −17 − 3x → 51x = −17 → x = − ¹ / ₃.  48 x equals 12 times 0 minus 17 minus 3 x implies 48 x equals negative 17 minus 3 x implies 51 x equals negative 17 implies x equals negative 1 third

Incorrect Response: A. This may be the result of combining the like x terms to get 51x  51 x  and then dividing the coefficient by 17.

Incorrect Response: C. This may be the result of an error when solving the equation for x, as well as an error when combining the like terms for x: 48x − 3x = 45x.  48 x minus 3 x = 45 x 

Incorrect Response: D. This response represents the y-intercept for the given equation: 48(0) = 12y − 17 − 3(0) → 0 = 12y − 17 → 17 = 12y → y = ¹⁷ / ₁₂.  48 times 0 equals 12 y minus 17 minus 3 times 0 implies 0 equals 12 y minus 17 implies 17 equals 12 y implies y equals 17 twelfths 

Correct Response: B. The volume of a cube, V, is equal to a side length, s, cubed: V = s³.  x cubed Since the volume of each cube is 8, (8) = s³ → ³√ 8 = ³√ s³ → 2 = s .  8 equals s cubed implies the cube root of 8 equals the cube root of s cubed implies s equals 2 Exactly one face of each cube is touching the ground, and the area of one face of the cube is equal to length ×  times  width, or 2 ×  times  2, which is equal to 4 ft.².  4 square feet Since there are 25 boxes that each occupy 4 ft.²  4 square feet  of floor space, the total floor space for the configuration is 25 × 4 = 100 ft.².  25 times 4 or 100 square feet

Incorrect Response: A. This response may be the result of misinterpreting the side length of each box to be 1 ft. foot : 1² × 25  1 squared times 25  = 25.

Incorrect Response: C. This response may be the result of misinterpreting the side length of each box to be 4 ft. feet : 4² × 25  4 squared times 25  = 400.

Incorrect Response: D. This response may be the result of misinterpreting the side length of each box to be 8 ft. feet : 8² × 25 = 1600.  8 squared times 25 = 1,600

Correct Response: C. Consider the following diagram of equilateral triangle ABC:

Equilateral triangle A B C is shown. The midpoints of each side form a smaller inscribed equilateral triangle D E F such that D is the midpoint of A B, E is the midpoint of B C and F is the midpoint of A C. The larger equilateral triangle is divided by the inscribed triangle into four congruent triangles. The three bottom triangles are shaded gray.

It follows from the information provided that D is the midpoint of AB  A B  and E is the midpoint of BC,  B C which implies AD ≅ DB and BE ≅ EC.  A D is congruent to D B and B E is congruent to E C By using properties of special right triangles ( 30°–60°–90°  30 60 90 degree  formed by drawing an altitude in each of the smaller triangles), and properties of parallel lines ( DE ∥ AC  D E is parallel to A C ), we can conclude that the four smaller triangles are congruent ( ΔADF ≅ ΔDBE ≅ ΔDEF ≅ ΔFEC  Triangle A D F is congruent to triangle D B E is congruent to triangle D E F is congruent to triangle F E C ). The ratio of unshaded triangles to shaded triangles is 1: to 3, which means the amount of paint necessary to finish the task is ¹ / ₃  1 third  of what has already been used.

Incorrect Response: A. This response may be the result of misinterpreting the triangles in the diagram shown in the rationale for Correct Response C by comparing the top triangle to the halves of the shaded equilateral triangles. This could lead to a ratio of 1: to 6.

Incorrect Response: B. This response may be the result of misinterpreting the number of smaller triangles in the figure (i.e., each smaller triangle in the figure represents ¹ / ₄  1 fourth  the area of the large triangle).

Incorrect Response: D. This response may stem from the misconception that the painted area represents half of the triangle since the horizontal line segment is located halfway up the triangle.

Correct Response: A. Because of the relationship between the area of a triangle and the area of a rectangle with the same base ¹ / ₂bh, bh,  1 half base times height, base times height  respectively it is true that Δ triangle 1 represents half of the area of rectangle EFGH.  E F G H It follows that the remaining triangles must sum to half the area of that rectangle, which implies that the area of Δ triangle 1 is equal to the sum of triangles 2 and 3. Similarly, Δ triangle 5 represents half of the area of rectangle JKLM,  J K L M and its area is equal to the sum of the areas of triangles 4 and 6. Since the rectangles are congruent, the following relationships are implied for the areas of the triangles: Δ1 = (Δ2 + Δ3) = Δ5 = (Δ4 + Δ6).  Triangle 1 equals the sum of triangles 2 and 3 which equals triangle 5 which equals the sum of triangles 4 and 6

Incorrect Response: B. This response may be the result of the misconception that the two triangles (2 and 3) together represent ² / ₃  2 thirds  of the total area of the rectangle.

Incorrect Response: C. Because of the relationship between the area of a triangle and the area of a rectangle with the same base ¹ / ₂bh, bh,  1 half base times height, base times height respectively it is true that Δ triangle 1 represents half of the area of rectangle EFGH.  E F G H It follows that the remaining triangles must sum to half the area of that rectangle, which implies that the area of Δ triangle 1 is equal to the sum of triangles 2 and 3. Since triangles 2 and 3 are equal, they each represent ¹ / ₄  1 fourth  of the total area. The same is true in rectangle JKLM,  J K L M where triangles 4 and 6 each represent ¹ / ₄  1 fourth  of the total area. This response is incorrect because the sum of triangles 1 and 2 is equal to the sum of triangles 4 and 5 both sums represent ³ / ₄  3 fourths  of the total area of the rectangle.

Incorrect Response: D. The sum of the areas of triangles 1 and 2 is equal to ³ / ₄  3 fourths  of the total area of the rectangle, and the sum of the areas of triangles 4 and 6 is ¹ / ₂  1 half  of the total area of the rectangle.

Correct Response: B. Adjacent angles are supplementary, which makes this a parallelogram and not necessarily a rhombus. (A rhombus must have 4 congruent sides.)

The interior angles for all the composing shapes for trapezoid W V Y U are shown. U W Z is equal to 30 degrees, and W Z U is equal to 60 degrees. Angles X W Z and X Y Z are equal to 60 degrees, and angles W Z Y and W X Y are equal to 120 degrees. Each interior angle for triangle X V Y is equal to 60 degrees.

Incorrect Response: A. The claim that the four sides are congruent is not necessarily true.

Incorrect Response: C. A rhombus has 4 congruent sides, congruent opposite angles, and supplementary adjacent angles. The information in the figure can only confirm the last two criteria (e.g., WX  segment W X  is not necessarily congruent to WZ).  segment W Z 

Incorrect Response: D. The statement that a parallelogram must have four congruent sides is false. Quadrilateral WXYZ  W X Y Z  is a parallelogram because its opposite angles are congruent.

Correct Response: B. Since the arc shown in the diagram has a 60°  60 degree  central angle, it must represent ⁶⁰ / ₃₆₀ = ¹ / ₆  60 divided by 360 equals 1 sixth  of the total circumference for the circle. This means that the circumference of the circle is 6(4) = 24 ft.  6 times 4 = 24 feet Since the circumference of a circle is 2πr,  2 pie r we can determine the length of the radius: 2πr = 24 → r = ²⁴ / _2π = ¹² / _π.  2 pie r equals 24 implies r equals 24 divided by 2 pie equals 12 divided by pie Substituting this value for r into the equation for the area of a circle yields the area: πr² = π ¹² / _π² = π ¹⁴⁴ / _π² = ¹⁴⁴ / _π  Pie r squared equals pie times the quantity 12 divided by pie squared equals pie times 144 divided by pie squared which equals 144 divided by pie square feet.

Incorrect Response: A. This response may be the result of misinterpreting the radius to be 4 and then finding the circumference using that value: 2πr = 2π(4) = 8π.  2 pie r equals 2 times pie times 4 equals 8 pie

Incorrect Response: C. This response may be the result of calculating the area of the circle using 4 as the radius: πr² = π(4)² = 16π.  Pie r squared equals pie times 4 squared equals 16 pie

Incorrect Response: D. This response gives the area of the circle excluding the 60°  60 degree  sector shown multiplying the area of the circle by ⁵ / ₆  5 sixths  after finding the correct radius.

Correct Response: C. The net shown is for a square pyramid.

The cross-section described in this response creates an isosceles triangle where two legs are equal to the slant height of the pyramid:

The image indicates the cross section that is the result of a slice through the top vertex of the pyramid that is perpendicular to the base and parallel to one side of the base, resulting in a cross section that is the shape of an isosceles triangle.

Incorrect Response: A. This response represents the shape of the cross section when the square pyramid is cut with a plane that is parallel to the square base:

The image indicates the cross section that is the result of a slice through the pyramid that is parallel to the base, resulting in a cross section that is the shape of a square.

Incorrect Response: B. This response represents the shape of the cross section when the square pyramid is cut with a plane that is perpendicular to the base, intersects the base at a line that is parallel to one side of the square, but does not pass through Point P:

The image indicates the cross section that is the result of a slice through the pyramid that is perpendicular to the base, parallel to one side of the base, and not passing through the top vertex P, resulting in a cross section that is the shape of a trapezoid.

Incorrect Response: D. This response may be the result of the misconception that slicing through a square pyramid in a particular way will result in a distortion of the base resulting in a rectangle. Less common cross sections for a square pyramid can create variations of trapezoids (as shown in the rationale for Incorrect Response B) but not rectangles.

Correct Response: B. It follows from the diagram that the length of the diagonal for the square is 8 inches. Applying properties of special right triangles to a triangular half of the square (in a 45°–45°–90°  45 degree 45 degree 90 degree  triangle, the ratio of the legs to the hypotenuse is 1:1:√ 2  1 to 1 to the square root of 2 ), the side length of the square must be ⁸ / _√ 2 = ⁸ / _√ 2 × ^√ 2 / _√ 2 = ^8√ 2 / _√ 2 = 4√ 2  8 divided by the square root of 2 equals 8 divided by the square root of 2 times the square root of 2 divided by the square root of 2 which equals 8 radical 2 divided by 2 which equals 4 radical 2  inches. This implies that the area of the square is (4√ 2)² = 16 × 2 = 32  4 radical 2 the quantity squared is equal to 16 times 2 or 32  square inches. Since the circle has a radius of 4 inches, the area of the circle can be approximated as follows, approximating pi pie  as a value just slightly greater than 3: πr² = π(4)² = 16π ≈ 50  Pie times r squared equals pi times 4 squared which equals 16 pie or approximately 50  square inches. The area of the shaded region is the difference between these two values: 50 −  minus  32 = 18 square inches.

Incorrect Response: A. This response may be the result of the calculation 8π − 16,  8 pie minus 16 in which the area of the square was interpreted to be 16, and it was subtracted from the circumference of the circle.

Incorrect Response: C. This response may be the result of the calculation 16π − 16√ 2,  16 pie minus 16 radical 2 in which the perimeter of the square was subtracted from the area of the circle.

Incorrect Response: D. This response may be the result of the calculation 16π − 32,  16 pie minus 32 in which a side length of the square was interpreted to be 8 inches, and the perimeter was used instead of the area.

Correct Response: B. The length of cable needed on one side of the tree can be calculated by using the Pythagorean Theorem with side lengths 6, (8 −  minus  2), and 8: a² + b² = c² → √ 6² + 8² = c → √ 100 = c → c = 10 feet.  A squared plus b squared equals c squared implies the square root of the quantity 6 squared plus 8 squared equals c which implies the square root of 100 equals c which implies c equals 10 feet. Since an additional 1.5 ft. feet  of cable is needed for each side, the total length of cable needed for one side of the tree is 11.5 ft. feet  Doubling this value produces the correct response, 23 feet.

Incorrect Response: A. This response may be the result of calculating the length of 10 feet, then doubling the value without regard for the additional cable needed on each side.

Incorrect Response: C. This response may be the result of tripling the value of 8 ft. feet  in the problem without regard for the information about the additional cable.

Incorrect Response: D. This response may be the result of adding the leg lengths for the triangles on both sides of the tree, then adding 3 for the additional cable: 2(8 + 6)  2 times the quantity 8 plus 6  + 3 = 28 + 3 = 31.

Correct Response: C. Applying the Pythagorean Theorem with a perpendicular line from the center of the base to the vertex yields the slant height of the pyramid: 30² + 40² = x² → x = √ 30² + 40² → x = √ 2500 → x = 50 feet.  30 squared plus 40 squared equals x squared implies x equals the square root of the quantity 30 squared plus 40 squared which implies x equals the square root of 2,500 which implies x equals 50 feet  The area of one triangular face of the pyramid is ¹ / ₂bh = ¹ / ₂(60)(50) = 1500  1 half b times h equals 1 half 60 times 50 which equals 1,500  square feet. Since there are 4 congruent triangular faces, the area of all triangular faces is 4(1500) = 6000  4 times 1,500 = 6,000  square feet.

Incorrect Response: A. This response represents the area of a single triangular face of the pyramid.

Incorrect Response: B. This response may be the result of interpreting the slant height to be equal to the overall height of the pyramid: 4( ¹ / ₂ × 40 × 60) = 4800.  4 times the quantity 1 half times 40 times 60 equals 4,800

Incorrect Response: D. This response represents the overall surface area of the pyramid (6000  6,000  for the four triangular faces plus 3600  3,600  for the square base).

Correct Response: A. To create an isosceles triangle with AB  A B  as the base, point C must be placed such that it is equidistant from A and B. That is, AC  A C  and AB  A B  must have the same length. Since the point (3, 1) is the midpoint of AB,  A B placing a point anywhere along the vertical line x = 3 will ensure that both the vertical and horizontal distance is equal for AC  A C  and CB, which will result in congruent side lengths for the segments.

Incorrect Response: B. This response may be the result of finding the midpoint between 0 and 8.

Incorrect Response: C. This is a possibility but not a necessary condition.

Incorrect Response: D. This is a possibility but not a necessary condition.

Correct Response: B. The definition of a trapezoid is a quadrilateral with at least one set of parallel sides. This figure meets that condition since AB ∥ DC.  A B is parallel to D C

Incorrect Response: A. Since ABCD  A B C D  is a quadrilateral with at least one pair of opposite sides parallel ( AB ∥ DC  A B is parallel to D C ), it meets the criteria for a trapezoid.

Incorrect Response: C. The shape is a trapezoid, but this response incorrectly relates the sides in the figure. AD  A D  is not parallel to BC  B C  since they have different slopes.

Incorrect Response: D. While one pair of opposite sides is parallel ( AB ∥ DC  A B is parallel to D C ), this is not true for AD  A D  and BC  B C  since they have different slopes.

Correct Response: A. The length of segment XZ X Y  is 7 (found by counting horizontal units). The length of segment XY and YZ Y Z  can be found using the distance formula:
 the square root of the sum of the two expressions x 2 minus x 1 the quantity squared and y 2 minus y 1 the quantity squared. 
XY: d = √ ((−2) − 2)² + (0 − 4)² = √ 16 + 16 = √ 32 ≈ 5.7  the square root of the quantity negative 2 minus 2 squared plus the quantity 0 minus 4 squared equals the square root of 16 plus 16, which equals the square root of 32, which is approximately 5.7 
YZ Y Z : d = √ (2 − 5)² + (4 − 0)² = √ 9 + 16 = √ 25 = 5  the square root of the quantity 2 minus 5 squared plus the quantity 4 minus 0 squared equals the square root of 9 plus 16, which equals the square root of 25 which is equal to 5. 
Since all three sides have a different length, the triangle is scalene.

Incorrect Response: B. This response may be the result of a miscalculation using the distance formula which resulted in same length sides XY and YZ Y Z .

Incorrect Response: C. This response may be the result of the misconception that the length of YZ Y Z  is equal to the sum differences in the x and y values: 3 + 4 = 7.

Incorrect Response: D. This may be the result of using a visual assessment to determine that the side lengths are equal. The calculated values are all distinct. (See the rationale for Correct Response A.)

Correct Response: D. Consider the diagram below:

The image from the stem is shown with labels for all interior angles in the diagram. The two base angles in the isosceles triangle are 5 and 6, the third interior angle for the isosceles triangle is 4, the vertical angle to 4 is 1, the angle opposite the 130-degree angle is 2, and the final interior angle in the parallelogram is 3.

Since the 130° degree  angle is one angle in a parallelogram, we can deduce that the opposite angle, ∠ angle 2 is also equal to 130°. degrees The sum of the interior angles in a parallelogram is 360°, degrees and opposite angles in a parallelogram are congruent, so both ∠ angle 1 and ∠ angle 3 are equal to ¹ / ₂(360 − 2(130)) = ¹ / ₂(100) = 50° .  1 half the quantity 360 minus 2 times 130 equals 1 half times 100 equals 50 degrees. Since ∠ angle 1 and ∠ angle 4 are vertical angles, the measure of ∠ angle 4 is also equal to 50°. degrees The triangle in the image is isosceles, where ∠ angle 5 and ∠ angle 6 are the base angles of the triangle. This means that they both have the same measure, which is ¹ / ₂(180 − 50) = 65°.  1 half the quantity 180 minus 50 equals 65 degrees.

The image from the stem is shown with measures for all interior angles in the diagram. The two base angles in the isosceles triangle are 65 degrees, the third interior angle for the isosceles triangle is 50 degrees, the vertical angle is 50 degrees, the angle opposite the 130-degree angle is also 130 degrees, and the final interior angle in the parallelogram is 50 degrees.

Incorrect Response: A. This response represents the measure of angles 1 and 4 in the diagram, shown in the rationale for Correct Response D, not angle Q.

Incorrect Response: B. Consider the diagram shown in the rationale for Correct Response D. This response cannot be true since the measure of angle 4 must be 50°, degrees the measure of angles 5 and 6 must be equal, and the sum of the measures of angles 4, 5, and 6 must be 180°. degrees

Incorrect Response: C. This response may be the result of misinterpreting the triangle in the image to be an equilateral triangle.

Correct Response: B. The midpoint of segment AB  A B  is equidistant from both points. To find it, we can average the x and y values for the coordinates (−4, 4) and (8, −2)  Negative 4, 4 and 8, negative 2 : ( ^{−4 + 8} / ₂, ^{4 + −2} / ₂) → ( ⁴ / ₂, ² / ₂) → (2, 1) .  Half of negative 4 plus 8, half of 4 plus negative 2 implies 4 halves, 2 halves implies 2, 1

Incorrect Response: A. A point is only equidistant from A and B if it is a point on the perpendicular bisector of AB.  A B The slope of AB  A B  is ^{y₂ − y₁} / _{x2 − x1} = ^{4 − (−2)} / _{−4 − 8} = ⁶ / ₋₁₂ = − ¹ / ₂,  A B is y 2 minus y 1 divided by x 2 minus x 1 equals 4 minus negative 2 divided by negative 4 minus 8 equals 6 divided by negative 12 equals negative 1 half. and the midpoint is (2, 1). (See the reasoning provided in the rationale for Correct Response B). The equation for the perpendicular bisector for AB  A B  must have a slope of 2 (the opposite reciprocal of − ¹ / ₂  Negative 1 half ) and pass through the point (2, 1). Substituting this information into a linear equation in slope-intercept form yields the equation of the perpendicular bisector: y = 2x + b → (1) = 2(2) + b → b = −3 → y = 2x − 3.  Y equals 2 x plus b implies 1 equals 2 times 2 plus b implies b equals negative 3 implies y equals 2 x minus 3 This response is not correct because it is not a point on the line—that is, the x and y coordinates do not satisfy the linear equation: (1) ≠ 2(1) − 3.  1 is not equal to 2 times 1 minus 3 

Incorrect Response: C. This response is not correct because it is not a point on the perpendicular bisector of AB  A B  (see the rationale for Incorrect Response A)—that is, the x and y coordinates do not satisfy the linear equation: (−1) ≠ 2(4) − 3.  Negative 1 is not equal to 2 times 4 minus 3

Incorrect Response: D. This response is not correct because it is not a point on the perpendicular bisector of AB  A B  (see the rationale for Incorrect Response A)—that is, the x and y coordinates do not satisfy the linear equation: (4) ≠ 2(8) − 3.  4 is not equal to 2 times 8 minus 3

Correct Response: B. Consider the following image, which shows triangle ABC  A B C  after a 90-degree rotation clockwise about the origin.

The image shows a clockwise rotation of triangle A B C about the origin, followed by a translation that moves triangle A B C into the same position as triangle A prime B prime C prime.

Note that the location of the rotated image (gray) is close to correct, as is the orientation. A translation (or slide) will move the triangle into the correct location.

Incorrect Response: A. This set of transformations places Triangle ABC  A B C  in the following location:

This image shows a counterclockwise rotation of triangle A B C about point A followed by a reflection over the y axis.

Incorrect Response: C. See the rationale shown for Correct Response B to verify that a rotation is necessary at some point to orient the triangle in the correct position.

Incorrect Response: D. See the rationale shown for Correct Response B to verify that a rotation is necessary at some point to orient the triangle in the correct position.

Correct Response: A. Consider the following diagram, which labels three angles that can be used to determine the measure of angle n:

The image from the stem is labeled such that the triangle formed between line a, line c, and line d has interior angle labels 1, 2, and 3. 1 is a vertical angle to the 57 degree angle. 2 is supplementary to the 127 degree angle, and 3 is supplementary to n.

Angle 1 represents the vertical angle that is opposite the 57° degree  angle, so its measure is also 57° degrees. Angle 2 must be supplementary with the 127° degree  angle since they are a linear pair, which means the measure of angle 2 is (180 − minus  127) = 53° degrees. Angles 1, 2, and 3 form the interior angles of a triangle, so the sum of their measures must equal 180° degrees, which implies the measure of angle 3 is equal to 180 − minus  57 − minus  53 = 70° degrees. Finally, angle n must be supplementary to angle 3 since they form a linear pair. This means the measure of angle n is equal to 180 − minus  70 = 110° degrees.

Incorrect Response: B. This response may stem from the misconception that angle n is twice the measure of the angle labeled 57° degrees.

Incorrect Response: C. This is the measure of an angle that is supplementary to 57° degrees, but it is not equal to the measure of angle n. (See the rationale for Correct Response A.)

Incorrect Response: D. This response may be the result of thinking that angle n is congruent to the angle labeled 127° degrees, which is not the case. (See the rationale for Correct Response A.)

Correct Response: C. Two triangles are similar if two of their corresponding angles are congruent. One pair of congruent angles is given in this response and the other comes from both triangles having right angles (because all right angles are congruent).

Incorrect Response: A. One corresponding angle and one congruent corresponding leg is not enough information to establish similarity. Consider the counterexample shown:

Incorrect Response: B. This information is insufficient to establish similarity. As a counterexample, consider a 30°–60°–90°  30 degree 60 degree 90 degree  triangle and a 45°–45°–90°  45 degree 45 degree 90 degree  triangle. These satisfy the condition (the sums of the non-right angles are equal), but they are not similar.

Incorrect Response: D. This information is insufficient to establish similarity. As a counterexample, consider a 30°–60°–90°  30 degree 60 degree 90 degree  triangle and a 45°–45°–90°  45 degree 45 degree 90 degree  triangle. It is possible that these triangles have the same hypotenuse length but are not similar.

Correct Response: C. Quadrilateral ABCD  A B C D  is shown in the coordinate plane below:

Quadrilateral A B C D is shown graphed using the coordinates from the question. A horizontal line segment connects the vertices A and C.

To find the area, consider a horizontal line connecting points A and C and calculate the area of each triangle separately. The area of the upper triangle is ¹ / ₂(10)(5)  1 half times 10 times 5  = 25 and the area of the lower triangle is ¹ / ₂(10)(4)  1 half times 10 times 4  = 20, so the total area of the shape is 45 square units.

Incorrect Response: A. This response may be the result of applying the same logic as what is shown in the rationale for Correct Response C but interpreting the value of the base for each triangle to be 6: (|−8 + 2| = 6).  The absolute value of negative 8 plus 2 equals 6 

Incorrect Response: B. This response may be the result of finding the product of the lengths of AB  A B  and BC  B C  after using the distance formula: (2√ 13)(4√ 2) = 8√ 26.  2 radical 13 times 4 radical 2 equals 8 radical 26  This is only valid if the quadrilateral is a rectangle.

Incorrect Response: D. This response may be the result of finding the product of the lengths of AB  A B  and AD  A D  after using the distance formula: (√ 61)(4√ 2) = 4√ 122.  Radical 61 times 4 radical 2 equals 4 radical 122  This is only valid if the quadrilateral is a rectangle.

Correct Response: C. Consider the graph shown.

The graph shows a reflection of the original triangle over the line y = x, followed by a reflection over the line x = 1. The final location of A after those transformations is negative 4, 4.

Reflecting point A (4, 2) across the line y = x creates the point A′,  A prime  located at (2, 4). (Note that this transformation can also be performed algebraically by transposing the x and y coordinates in the original point A.) The second transformation is a reflection across the line x = −  negative 1, which transforms point A′  A prime  to point A″,  A double prime  located at (−4, 4).  negative 4, 4  (This can also be performed algebraically by considering the horizontal distance point A′  A prime  is away from the line of reflection.)

Incorrect Response: A. This response may be the result of reflecting point A over the x-axis, and then reflecting the point over the line x = − negative 1.

Incorrect Response: B. This response represents the final location of a different vertex for the original triangle (the vertex initially located at (6, 0)).  6 0 

Incorrect Response: D. This response represents the location of point A after a reflection across the line y = x and a reflection across the x-axis.

Correct Response: B. The prefix "deci" refers to "tenth," so there are 10 decimeters in a meter. 18 decimeters is equal to 1.8 meters, which is equivalent to approximately 6 feet.

Incorrect Response: A. This response is the result of a misconception about the prefix "centi." There are 100 centimeters in one meter. (18 centimeters is equivalent to approximately 7 inches.)

Incorrect Response: C. This response is the result of misconceptions related to the length of a meter. One meter is approximately 3.2 feet, so 18 meters is approximately 58 feet.

Incorrect Response: D. This response is the result of a misconception about the prefix "kilo." There are 1000 meters in a kilometer. 18 kilometers is 1,800 meters or approximately 5,900 feet.

Correct Response: B. The total cost for the lunch order is equal to $42.75.  42 dollars and 75 cents  The change due for a $50  50 dollars  bill is 50 −  minus  42.75 = 7.25. The fewest number of bills that can be used is 3: 2 one-dollar bills and 1 five-dollar bill.

Incorrect Response: A. This response may be the result of misinterpreting the lunch order calculation, specifically misinterpreting the cost of the drinks to be 4 times $2.50  2 dollars and 50 cents  or $10,  10 dollars  and the cost of the 3 sandwiches as $19.  19 dollars  This would result in a total cost of $44,  44 dollars  which implies $6  6 dollars  of change. This can be accomplished with 2 bills.

Incorrect Response: C. This response may be the result of using the correct computation for the lunch order but misinterpreting the question. If the question asked for the fewest number of bills and coins needed to make the change, the answer would be 4.

Incorrect Response: D. This response may be the result of a correct computation of the lunch order but the misinterpretation that the change must be made in one-dollar bills.

Correct Response: A. The relationship between degrees Fahrenheit and degrees Celsius is C = ⁵ / ₉(F − 32).  C equals 5 ninths times the quantity f minus 32  When F = 98.6, C = ⁵ / ₉((98.6) − 32) = ⁵ / ₉(66.6).  C equals 5 ninths times the quantity 98.6 minus 32 which equals 5 ninths times 66.6  For the purposes of estimation, we can round 66.6 to the nearest multiple of 9, 63, to get

5 ninths times 63 is equal to 5 ninths (9 in the denominator is crossed out) times 9 times 7 over 1 (9 in the numerator is crossed out) equals 5 times 7 or 35 degrees Celsius.

Incorrect Response: B. This response may be the result of using an incorrect formula (C = ⁵ / ₉(F + 32))  C equals 5 ninths times the quantity f plus 32  to find the temperature in degrees Celsius.

Incorrect Response: C. This response may be the result of using an incorrect formula (C = ⁵ / ₉(F − 32))  C equals 5 ninths times the quantity f minus 32  to find the temperature in degrees Celsius.

Incorrect Response: D. This response may be the result of using an incorrect formula (C = ⁵ / ₉(F + 32))  C equals 5 ninths times the quantity f plus 32  to find the temperature in degrees Celsius.

Correct Response: D. The ratio ^width / _height  Width divided by height  for the model and the actual building will be equal since the model is to scale. This means that the proportion ⁶ / ₁₂ = ⁴⁵ / _h  6 divided by 12 equals 45 divided by h  can be used to calculate the height of the building: ⁶ / ₁₂ = ⁴⁵ / _h → 6h = 540 → h = 90.  6 divided by 12 equals 45 divided by h implies 6 h equals 540 implies h equals 90 

Incorrect Response: A. This proportion does not describe an accurate relationship between the dimensions of the model and the dimensions of the building. This proportion results in an answer of 22.5 feet for the height of the building.

Incorrect Response: B. This proportion does not describe an accurate relationship between the dimensions of the model and the dimensions of the building. This proportion results in an answer of 22.5 feet for the height of the building.

Incorrect Response: C. This proportion does not describe an accurate relationship between the dimensions of the model and the dimensions of the building. This proportion results in an answer of 1.6 feet for the height of the building.

Correct Response: B. The figure is composed of two similar triangles. The triangles are similar because their corresponding angles are congruent (i.e., the bottom angle is common to both triangles and the left and right edges are transversals that form 2 pairs of congruent corresponding angles where they intersect the parallel lines). To find y, we can use the following proportion: ⁹ / ₁₂ = ¹² / _{12 + y} → 144 = 9(12 + y) → 144 = 108 + 9y → 36 = 9y → y = 4.  9 divided by 12 equals 12 divided by 12 plus y implies 144 equals 9 times the quantity 12 plus y implies 144 equals 108 plus 9 y implies 36 equals 9 y implies y equals 4.  To find x, we can use the proportion ⁹ / _x = ¹² / ₈ → 12x = 72 → x = 6.   9 divided by x is equal to 12 divided by 8 implies 12 x equals 72 implies x equals 6.  The ratio of x to y is 6: to 4 or 3: to 2.

Incorrect Response: A. This response may be the result of using an incorrect proportion or misinterpreting the value of y to be equal to 3.

Incorrect Response: C. This response may be the result of analyzing the ratio 12: to y, which gives 12: to 4 or 3: to 1.

Incorrect Response: D. This response may be the result of analyzing the ratio of 12: to 9, which is equal to 4: to 5.

Correct Response: D. The lines intersect to form the right triangle shown:

The image shows the triangular area that is bounded by the lines in the question. The triangle has a base of 6 units and a height of 6 units.

The triangle has a width of 6 units and a height of 6 units, so the area is equal to ¹ / ₂  1 half  × 6 × 6 = 18  One half times 6 times 6 equals 18  square units.

Incorrect Response: A. This response may be the result of incorrectly interpreting the graph of the lines, or by using the coefficient in the calculation for the area of a triangle.

Incorrect Response: B. This may be the result of misinterpreting the dimensions of the triangle to be 3 units tall and 7 units wide.

Incorrect Response: C. This may be the result of misinterpreting the dimensions of the triangle to be 3 units tall and 9 units wide.

Correct Response: B. Consider a triangle with base, b, and height, h. The area of this triangle is equal to ¹ / ₂bh. 1 half base times height Doubling the base and the height affects the expression for area as follows: ¹ / ₂(2b)(2h) = 2bh.  One half times 2 b times 2 h equals 2 b h  This expression is equal to 4 times ¹ / ₂bh,  1 half base times height  so doubling both dimensions causes the area to increase by a factor of 4.

Incorrect Response: A. This response may be the result of misinterpreting the value 2b2h to mean the area of the original triangle has doubled.

Incorrect Response: C. Consider the following counterexample: An isosceles triangle has a base of 6 units and a height of 4 units. The length of the two congruent sides can be found by drawing a perpendicular line from the vertex to the midpoint of the base and using the Pythagorean Theorem to get = √ 25 = 5.  The square root of 3 squared plus 4 squared equals the square root of 25 equals 5  The perimeter of the triangle is 6 + 5 + 5 = 16 units. Doubling the base and height means the new base is 12 units long, but each of the congruent sides must be equal to = √ 100 = 10.  The square root of 6 squared plus 8 squared is equal to the square root of 100 which is equal to 10.  The new perimeter is now 12 + 10 + 10 = 32 units. This process can be generalized to show that the perimeter doubles when both dimensions are doubled.

Incorrect Response: D. Consider the following counterexample: An isosceles triangle has a base of 6 units and a height of 4 units. The length of the two congruent sides can be found by drawing a perpendicular line from the vertex to the midpoint of the base and using the Pythagorean Theorem to get = √ 25 = 5.  The square root of 3 squared plus 4 squared equals the square root of 25 equals 5  The perimeter of the triangle is 6 + 5 + 5 = 16 units. Doubling the base and height means the new base is 12 units long, but each of the congruent sides must be equal to = √ 100 = 10.  The square root of 6 squared plus 8 squared is equal to the square root of 100 which is equal to 10.  The new perimeter is now 12 + 10 + 10 = 32 units. This process can be generalized to show that the perimeter doubles when both dimensions are doubled.

Correct Response: C. The sample space represents all possible outcomes for the contest. There are 6 possible outcomes for a winner from two participants A and B choosing one of three difference gifts: {(A, book), (A, gift certificate), (A, smartphone), (B, book), (B, gift certificate), (B, smartphone)}. The number of outcomes in the sample space is equal to the number of people (2) times the number of prizes (3).

Incorrect Response: A. In this context, only one person wins, and that winner then makes a choice of which gift to take. The number 3 does not represent the total number of outcomes.

Incorrect Response: B. The sum of the number of people and the number of prizes is 2 + 3 = 5, which does not represent the total number of possible outcomes for the contest. (See the reasoning in the rationale for Correct Response C.)

Incorrect Response: D. This response indicates that there are 2³  2 cubed  = 8 possible outcomes, but there are only 6 unique outcomes for the contest. (See the reasoning in the rationale for Correct Response C.)

Correct Response: C. The radius of the large circle is 4, which means the area of the large circle is equal to πr² = π(4)² = 16π  Pie times r squared equals pie times 4 squared equals 16 pie  square units, and the radius of the small circle is 2, which means the area of the small circle is equal to πr² = π(2)² = 4π  Pie times r squared equals pie times 2 squared equals 4 pie  square units. The shaded area represents one quarter of the outer ring, which represents the difference between the area of the large circle and the area of the small circle: ¹ / ₄(16π − 4π) = ¹ / ₄(12π) = 3π. 1 fourth times the quantity 16 pie minus 4 pie equals 1 fourth 12 pie which equals 3 pie This shaded region accounts for ^3π / _16π = ³ / ₁₆  3 pie divided by 16 pie equals 3 divided by 16  of the total circular region.

Incorrect Response: A. This response represents the area of the shaded region shown.

The image from the question is shown, with a shaded region indicating the area of the small circle only between the radii

The area of this shaded region is ¹ / ₄(4π) = π 1 fourth of 4 pie equals pie, and it represents ^π / ₁₆ = ¹ / ₁₆  Pie divided by 16 equals 1 sixteenth  of the total circular region.

Incorrect Response: B. This response may be the result of interpreting the shaded area to be twice as much as the area shown in the rationale for Incorrect Response A. This would result in a conclusion that the shaded region represents ^2π / ₁₆ = ² / ₁₆  2 Pie divided by 16 equals 2 sixteenths  of the circular area.

Incorrect Response: D. This response represents the probability that a beanbag lands in the shaded region shown.

The image from the question is shown with a shaded region that indicates the total area of the diagram (including the small and large circle) between the radii.

The shaded area is ¹ / ₄(16π) = 4π 1 fourth 16 pie equals 4 pie, and it represents one fourth of the total area for the circular region.

Correct Response: A. There are 4 prime numbers in each hat: 2, 3, 5, and 7. The total number of possible combinations where both numbers are prime is 4 ×  times  4 = 16. There are 10 ×  times  10 or 100 possible combinations for drawing numbered slips from the hats, so the theoretical probability that two slips are both prime is equal to ¹⁶ / ₁₀₀ .  16 hundredths In the experiment, drawing two prime numbers occurred 20 times out of 100, which is 4 more pairs than what we would expect using theoretical probability.

Incorrect Response: B. This is most likely the result of the misconception that 1 is a prime number, which would lead to a theoretical probability of ²⁵ / ₁₀₀ .  25 hundredths

Incorrect Response: C. This is the result of the misconception that only 3 of the numbers in each hat are prime, which leads to a theoretical probability of ⁹ / ₁₀₀ .  9 hundredths

Incorrect Response: D. This could be the result of reasoning that because 4 out of the 10 numbers in each hat are prime, the probability of drawing two prime numbers is 40% percent  or ⁴⁰ / ₁₀₀ .  40 hundredths (This is equal to the theoretical probability for drawing a single prime number from one hat.)

Correct Response: A. Since the letters e and a must be placed consecutively, we can think of ea e a  as a single character within the context of the problem. This means that there are 4 unique characters to use to create a word. The number of possibilities for this is equal to 4! = 4 × 3 × 2 × 1 = 24. 4 factorial equals 4 times 3 times 2 times 1 equals 24 The stem asks for how many additional arrangements can be formed (excluding "heart"), so the answer is 23.

Incorrect Response: B. This response includes the word "heart," which is excluded in the question by "how many additional arrangements …."  dot dot dot

Incorrect Response: C. This is the result of using the incorrect expression 5! (5 × 4 × 3 × 2 × 1)  5 factorial, or 5 times 4 times 3 times 2 times 1  and then subtracting 1 to compensate for the word "heart" since the stem specifies "How many additional arrangements…  dot dot dot ": 120 − minus  1 = 119.

Incorrect Response: D. This is the result of the incorrect expression 5! (5 × 4 × 3 × 2 × 1),  5 factorial, or 5 times 4 times 3 times 2 times 1 which is most likely the result of considering e and a to be separate characters when calculating the different word arrangements: 5! = 120.  5 factorial equals 120

Correct Response: C. The expression 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720  6 factorial equals 6 times 5 times 4 times 3 times 2 times 1 equals 720  can be used to find the number of different arrangements possible for the letters in the word "Boston." However, this counting technique is only the solution if each letter in the word is unique. Since there are two of the letter "o" and swapping the two letters produces the same word, the value of 720 must be halved to compensate for the duplicates: ⁷²⁰ / ₂  720 divided by 2  = 360.

Incorrect Response: A. This response may be the result of misinterpreting the value of 6!  6 factorial  to be equal to the sum of the whole numbers from 6 to 1, rather than the product: 6 + 5 + 4 + 3 + 2 + 1 = 21.

Incorrect Response: B. This response may be the result of using the expression ^6! / ₆ .  6 factorial divided by 6 (The denominator in this expression is incorrect for the reasons described in the rationale for Correct Response C.)

Incorrect Response: D. This response does not take into consideration the fact that swapping two of the same letter produces the same word rather than a unique arrangement.

Correct Response: B. Treat the selection of the 2 particular people as one possible outcome and compare it to the total number of ways that any 2 of 10 people may be paired together (i.e., the number of ways to choose 2 of 10 people when the order of their selection does not matter). There are 5 ways to choose the first person and 4 ways to choose the second person, so there are 5 ×  times  4 = 20 ways in which two people may be chosen when the order of their selection matters. However, each pair may have been chosen in 2 different orders, so the total number of pairings is actually 20 ÷  divided by  2 = 10. Therefore, the probability is ¹ / ₁₀  1 tenth  or 10%.

Incorrect Response: A. This response may be the result of interpreting the number of possible two-person groupings as 5 ×  times  4 (without dividing by two to account for different combinations of the same two employees).

Incorrect Response: C. This response may be the result of assuming there are two unique outcomes in which employees can work together. Since order does not matter, different combinations of the same two employees must all be considered the same outcome.

Incorrect Response: D. This response may be the result of using the ratio ² / ₅  2 fifths  to calculate the probability.

Correct Response: C. The left edge of the box in the box plot represents the first quartile of the data set. The first quartile is the median of the lower half of the data. This means that 75% percent  of the data in the set are greater than 68.

Incorrect Response: A. The median of the data set, represented by the middle bar in the box for the box plot, is 77. The mean test score cannot be deduced from the information shown.

Incorrect Response: B. This response may be the result of misinterpreting the relationship between the number line and the box plot. The median of the data set is 77, which is represented by the middle line in the box for the box plot.

Incorrect Response: D. Since 68 is the first quartile of the data set, 25% percent  of the data fall between 45 and 68. 77 is the median of the data set, which means that 50% percent  of the data fall between 77 and the maximum value, 100.

Correct Response: C. The average amount of tips, in dollars, for the 4 days worked is equal to ^{34 + 40 + 56 + 82} / ₄ = ²¹² / ₄ = 53. The sum 34 plus 40 plus 56 plus 82 all divided by 4 equals 212 divided by 4 which equals 53 The equation ^{34 + 40 + 56 + 82 + x} / ₅ = 58  The sum 34 plus 40 plus 56 plus 82 plus x all divided by 5 equals 58  can be used to find the amount of tips needed on Saturday to make the average $5 5 dollars  greater than the 4-day average. Solving this equation for x leads to: ^{212 + x} / ₅ = 58 → 212 + x = 290 → x = 78. The sum of 212 plus x divided by 5 equals 58 implies 212 plus x equals 290 which implies x equals 78.

Incorrect Response: A. This response is the average amount of tips for the four days that are provided: ^{34 + 40 + 56 + 82} / ₄ = ²¹² / ₄ = 53. The sum 34 plus 40 plus 56 plus 82 all divided by 4 equals 212 divided by 4 which equals 53

Incorrect Response: B. This response is the average described in the question ($5 5 dollars  greater than the 4-day average of $53 53 dollars ), but it does not represent the amount of tips that must be earned on Saturday to achieve this.

Incorrect Response: D. This response is $5 5 dollars  greater than the amount of tips earned on Friday. It does not represent the amount needed on Saturday to increase the average in the way specified in the question.

Correct Response: D. Since this sample only includes respondents with an e-mail address, or respondents who regularly check their e-mail, it may not be a representative sample of the population. The data from this survey could be skewed by the sample of respondents (sampling bias).

Incorrect Response: A. Including specific examples does not inherently affect the validity of the conclusion. The question includes an "e.g. list," which provides some specific examples but implies that there are more recreational uses for computers and cell phones than what is provided.

Incorrect Response: B. This statement is only true if the sample is biased for some reason. A random sample can be used to make valid conclusions about the population.

Incorrect Response: C. While the devices listed are not exhaustive, the conclusion is about the use of a cell phone or computer only. The conclusion does not make claims about use of recreational electronic devices beyond the scope of computer or cell phone use.

Correct Response: A. A scatter plot is an effective way to analyze two variables and determine correlation. For instance, in this context, it may be the case that the number of absences is negatively correlated with performance on standardized tests (as the number of absences increases, standardized test scores decrease).

Incorrect Response: B. Since this type of data is referencing parts (different ages of students) of a whole (students in a primary school) a circle graph would be most appropriate.

Incorrect Response: C. A line graph is most appropriate for tracking how one value changes over time (in this case, height).

Incorrect Response: D. A double bar graph is appropriate for comparing measures from two distinct sets.

Correct Response: B. The value 145 is greater than 6 of the values in the data set, which contains a total of 11 values. This means that a height of 145 corresponds to the 55th percentile: ⁶ / ₁₁ × 100 ≈ 0.55. 6 elevenths times 100 is approximately 0.55

Incorrect Response: A. This response represents the percent of students with a height greater than 145: ⁴ / ₁₁ × 100 ≈ 0.36. 4 elevenths times 100 is approximately 0.36

Incorrect Response: C. This response is the product of ¹ / ₁₁  1 eleventh  times 7 (the number of values in the data set that are less than or equal to 145): ⁷ / ₁₁ × 100 ≈ 0.64. 7 elevenths times 100 is approximately 0.64

Incorrect Response: D. This response may be the result of interpreting the percentile using the fraction ⁷ / ₁₀  7 tenths  (using the number of values less than or equal to 145 and mistaking the number of values in the set to be 10).

Correct Response: C. This statement involves the use of interpolation, and the values referenced are consistent with the trend in the scatter plot. This claim is a valid prediction that does not extend beyond the scope of the scatter plot.

Incorrect Response: A. This statement is too broad a generalization given the data provided.

Incorrect Response: B. A scatter plot allows for analysis of correlation between two variables. The data shown do not provide evidence for claims regarding different variables (factors).

Incorrect Response: D. A scatter plot allows for analysis of correlation between two variables. Additional evidence is needed to establish a causal relationship.