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5th Grade Math

5th Grade Math Practice Test: Practice Test 5

Practice Test 5 for 5th Grade Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

In a science notebook, two patterns describe how two plants grow each week.

Plant A pattern rule: start at 2 and add 5. Terms: 2, 7, 12, 17, 22. Plant B pattern rule: start at 2 and add 3. Terms: 2, 5, 8, 11, 14.

The paired terms are graphed as ordered pairs (A,B)(A, B)(A,B) in the first quadrant.

Which statement best explains why one pattern grows faster than the other?

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Question 1

In a science notebook, two patterns describe how two plants grow each week.

Plant A pattern rule: start at 2 and add 5. Terms: 2, 7, 12, 17, 22. Plant B pattern rule: start at 2 and add 3. Terms: 2, 5, 8, 11, 14.

The paired terms are graphed as ordered pairs (A,B)(A, B)(A,B) in the first quadrant.

Which statement best explains why one pattern grows faster than the other?

  1. Plant B grows faster because it has the larger starting value.
  2. Plant A grows faster because it adds 5 each time, which is more than adding 3 each time. (correct answer)
  3. Plant B grows faster because its points will always be above Plant A’s points on the graph.
  4. They grow at the same rate because both patterns start at 2.

Explanation: Number patterns can be compared and graphed by pairing their corresponding terms to visualize relationships. To generate terms in a pattern, start with the initial value and repeatedly add the given increment, such as starting at 2 and adding 5 to get 2, 7, 12, 17, 22 or starting at 2 and adding 3 to get 2, 5, 8, 11, 14. Forming ordered pairs involves matching Plant A's nth term with Plant B's, like (2,2), (7,5), showing growth differences. The graph shows the relationship by diverging lines, with Plant A pulling ahead due to faster addition. A common misconception is that equal starts mean equal growth, but the addition rate determines speed. Graphs help visualize pattern relationships by comparing slopes of growth. Overall, such graphs allow us to explain why one pattern overtakes another over time.

Question 2

A right rectangular prism is built from unit cubes in layers. One layer has 9 cubes across and 2 cubes deep, and the prism is 3 cubes tall (3 layers). The cubes pack the prism with no gaps or overlaps, so counting cubes is the same as multiplying the three whole-number dimensions. What is the volume of the prism?

  1. 18 cubic units
  2. 54 cubic units (correct answer)
  3. 27 cubic units
  4. 14 cubic units

Explanation: The volume of a rectangular prism can be found by packing it with unit cubes and counting how many fit inside without gaps or overlaps. In this prism, one layer has 9 cubes across and 2 cubes deep, making 18 cubes per layer, and there are 3 layers to form the height of 3 cubes tall. This means you can multiply the number of cubes in one layer by the number of layers: 18 × 3 = 54, which is the same as multiplying length × width × height. Having no gaps or overlaps ensures that every part of the prism is accounted for exactly once, giving an accurate volume measurement. A common misconception is equating 'across' with length and ignoring depth, but both define the base area. Packing with unit cubes visually demonstrates how the volume is structured as layers of area stacked to a certain height. This method helps understand that the volume formula V = l × w × h comes from the number of unit cubes along each dimension.

Question 3

A measuring cup has 13\tfrac{1}{3}31​ cup of juice. The juice is poured equally into 3 small cups. Picture the 13\tfrac{1}{3}31​ cup amount being partitioned into 3 equal parts (dividing a fraction by a whole number creates smaller equal parts). What does the quotient 13÷3\tfrac{1}{3} \div 331​÷3 represent?

  1. Each small cup gets 19\tfrac{1}{9}91​ cup of juice. (correct answer)
  2. Each small cup gets 111 full cup of juice.
  3. Each small cup gets 16\tfrac{1}{6}61​ cup of juice because you divide only the denominator.
  4. Each small cup gets 13\tfrac{1}{3}31​ cup of juice because the amount does not change when you share it.

Explanation: Unit fractions, which are fractions with a numerator of 1, can be divided by whole numbers to find smaller equal shares. In this scenario, 1/3 cup of juice is being poured equally into 3 small cups, which means dividing the fraction by 3. To do this, the 1/3 cup amount is partitioned into 3 equal smaller parts, resulting in each cup getting 1/9 cup of juice. Visually, if you imagine a cup divided into 3 equal parts with one part filled, then split that filled part into 3 equal sections, each section is 1/9 of the whole cup. A common misconception is that you only divide the denominator, giving 1/6, but actually, you divide the entire fraction, making smaller pieces. Generally, dividing a unit fraction by a whole number makes the pieces smaller, as you're splitting an already small amount into more parts. The larger the divisor, the smaller each share becomes, so 1/3 ÷ 3 = 1/9, which is one-third the size of 1/3.

Question 4

A science class has 15 grams of clay. They do two experiments:

  • Experiment X: 15×6515 \times \frac{6}{5}15×56​ grams
  • Experiment Y: 15×4515 \times \frac{4}{5}15×54​ grams Which statement correctly compares the effects and explains them using the fact that the fraction’s size compared to 1 determines whether the product is larger or smaller than the original number 15?
  1. The original number is 15; both products are larger because 6 and 4 are both greater than 1.
  2. The original number is 15; Experiment X makes a smaller product because dividing by 5 always makes things smaller, and Experiment Y makes a larger product because 4 is close to 5.
  3. The original number is 15; Experiment X makes a larger product because 65>1\frac{6}{5}>156​>1, and Experiment Y makes a smaller product because 45<1\frac{4}{5}<154​<1. (correct answer)
  4. The original number is 15; both products are smaller than 15 because multiplying by a fraction always makes the original number smaller.

Explanation: The central principle is that a fraction's magnitude relative to 1 dictates whether multiplication enlarges or reduces the original number. For fractions above 1, such as 6/5, the product grows larger than the starting value. For those below 1, like 4/5, the product becomes smaller. With 15 grams of clay, Experiment X (6/5) increases it, while Y (4/5) decreases it, as explained in choice C. It's a mistake to think denominators alone control size without considering numerators. This insight helps in scientific measurements and adjustments. It enhances critical thinking in experimental design and analysis.

Question 5

Look at the pattern of dots arranged in the figure. Ana wants to create a mathematical representation that will help her predict how many dots will be in the 10th figure of this sequence. Which approach would BEST communicate her prediction strategy?

  1. Draw all figures from 1 to 10 and count the dots in each one to find the pattern
  2. Create a table showing figure numbers and dot counts, then identify the mathematical relationship between position and dots (correct answer)
  3. Write the sequence of dot numbers and look for a repeating pattern in the differences between consecutive terms
  4. Use trial and error to guess the formula, then check if it works for the given figures

Explanation: Choice B best communicates a systematic prediction strategy by organizing data in a table format that reveals patterns and allows for mathematical analysis of the relationship between figure position and dot count. This method can be clearly explained and verified by others. Choice A is inefficient and doesn't demonstrate mathematical thinking. Choice C looks at differences but doesn't organize the approach as systematically. Choice D relies on guessing rather than mathematical reasoning and doesn't provide a clear communication strategy.

Question 6

A garden plot covers 16\frac{1}{6}61​ of a yard. If 4 different vegetables are planted in equal sections of this plot, what fraction of the entire yard will the tomato section occupy?

  1. 12\frac{1}{2}21​ of the yard, found by calculating 16÷4=46÷4=16\frac{1}{6} \div 4 = \frac{4}{6} \div 4 = \frac{1}{6}61​÷4=64​÷4=61​
  2. 46\frac{4}{6}64​ of the yard, found by calculating 16×4=46\frac{1}{6} \times 4 = \frac{4}{6}61​×4=64​
  3. 110\frac{1}{10}101​ of the yard, found by calculating 16÷4=16+4=110\frac{1}{6} \div 4 = \frac{1}{6+4} = \frac{1}{10}61​÷4=6+41​=101​
  4. 124\frac{1}{24}241​ of the yard, found by calculating 16÷4=16×14\frac{1}{6} \div 4 = \frac{1}{6} \times \frac{1}{4}61​÷4=61​×41​ (correct answer)

Explanation: When you see a problem about dividing parts of parts, you need to think step-by-step about what fraction you're working with at each level. The garden plot covers 16\frac{1}{6}61​ of the yard, and this plot is divided equally among 4 vegetables. To find what fraction of the entire yard each vegetable section occupies, you need to divide the plot's fraction by 4. This means calculating 16÷4\frac{1}{6} \div 461​÷4. When dividing by a whole number, you multiply by its reciprocal: 16÷4=16×14=124\frac{1}{6} \div 4 = \frac{1}{6} \times \frac{1}{4} = \frac{1}{24}61​÷4=61​×41​=241​. So the tomato section occupies 124\frac{1}{24}241​ of the entire yard, making D correct. Let's examine why the other answers are wrong. Choice A incorrectly changes 16\frac{1}{6}61​ to 46\frac{4}{6}64​ before dividing, which doesn't match the problem setup. Choice B multiplies instead of divides (16×4\frac{1}{6} \times 461​×4), which would give you the area of 4 plots, not one section of a single plot. Choice C uses the wrong division method by adding the numbers in the denominator (16+4\frac{1}{6+4}6+41​), but division of fractions doesn't work this way. Remember: when you're finding a fraction of a fraction, you multiply the fractions together. Since dividing by 4 is the same as multiplying by 14\frac{1}{4}41​, problems like "What's 16\frac{1}{6}61​ divided by 4?" become multiplication: 16×14\frac{1}{6} \times \frac{1}{4}61​×41​.

Question 7

A teacher has 14\tfrac{1}{4}41​ of a pan of brownies left. She shares that leftover amount equally among 2 students. Imagine the 14\tfrac{1}{4}41​ piece is split into 2 equal smaller pieces (dividing a fraction by a whole number creates smaller equal parts). What is the result of 14÷2\tfrac{1}{4} \div 241​÷2?

  1. 12\tfrac{1}{2}21​ of a pan
  2. 18\tfrac{1}{8}81​ of a pan (correct answer)
  3. 12\tfrac{1}{2}21​ of 14\tfrac{1}{4}41​ of a pan
  4. 14\tfrac{1}{4}41​ of a pan

Explanation: Unit fractions, which are fractions with a numerator of 1, can be divided by whole numbers to find smaller equal shares. In this scenario, the teacher is sharing 1/4 of a pan of brownies equally among 2 students, which means dividing the fraction by 2. To do this, the 1/4 piece is partitioned into 2 equal smaller pieces, resulting in each student getting 1/8 of the pan. Visually, if you draw a pan divided into 4 equal parts and shade one, then split that shaded part into 2 equal halves, each small part is 1/8 of the whole pan. A common misconception is that dividing 1/4 by 2 gives 1/2, but that would only be true if starting with a whole pan instead of a fraction. Generally, dividing a unit fraction by a whole number makes the pieces smaller, as you're splitting an already small amount into more parts. The larger the divisor, the smaller each share becomes, so 1/4 ÷ 2 = 1/8, which is half the size of 1/4.

Question 8

A coach has 864864864 water bottles to pack equally into boxes that each hold 242424 bottles. Using place value reasoning, you can think 24×30=72024\times 30=72024×30=720 and 24×6=14424\times 6=14424×6=144, and 720+144=864720+144=864720+144=864. Which statement gives the correct quotient and shows how multiplication can check the division 864÷24864\div 24864÷24?

  1. The quotient is 360360360 because 242424 goes into 864864864 about 363636 times and then you add a zero.
  2. The quotient is 363636 because 24×36=86424\times 36=86424×36=864, so multiplication checks the division. (correct answer)
  3. The quotient is 343434 because 24×34=81624\times 34=81624×34=816 and that is close enough to 864864864.
  4. The quotient is 3,4563{,}4563,456 because 864×24=3,456864\times 24=3{,}456864×24=3,456.

Explanation: Division with two-digit divisors uses place value to break down the dividend into manageable parts, like hundreds and tens. Estimating the quotient involves finding how many times the divisor fits into larger place values, such as seeing that 24 goes into 720 thirty times since 24×30=720. Using multiplication to check means multiplying the estimated quotient by the divisor to verify if it equals the dividend, confirming 24×36=864. This connects to the partial quotients strategy, where you add quotients from each place value, like 30+6=36. A common misconception is adding an extra zero to the quotient, but that ignores the actual place value relationships. Reasoning with place value helps ensure the quotient is accurate by aligning the multiplication back to the original dividend. Overall, this approach builds confidence in division by linking it to familiar multiplication facts.

Question 9

Two solids are built from unit cubes. In both solids, the cubes fill the space completely with no gaps or overlaps. Volume is measured in cubic units.

Solid A has 10 unit cubes. Solid B has 12 unit cubes.

Which statement about their volumes is correct?

  1. Solid A has the greater volume because it might look taller.
  2. Solid B has the greater volume because 12 unit cubes fill more space than 10 unit cubes. (correct answer)
  3. Both solids have the same volume because they are made of unit cubes.
  4. You cannot compare their volumes unless you count the outside faces.

Explanation: Volume counts the number of cubic units that make up a solid figure. The unit cubes fill the space of the solid completely without any gaps or overlaps. Each unit cube adds one cubic unit to the total volume, so Solid A with 10 cubes has 10 cubic units and Solid B with 12 has 12 cubic units, making B larger. To count the volume, simply total the number of unit cubes in each solid. A common misconception is that shape or height affects volume more than the total cubes, but it's solely the number of cubes that matters. Cubic units measure the three-dimensional space occupied by any solid. This method allows us to compare volumes of different figures accurately.

Question 10

Two number patterns are paired and then graphed as ordered pairs (x,y)(x, y)(x,y) where xxx is from Pattern X and yyy is from Pattern Y.

Pattern X rule: start at 1 and add 4. Terms: 1, 5, 9, 13, 17. Pattern Y rule: start at 2 and add 3. Terms: 2, 5, 8, 11, 14.

Which ordered pair matches the 4th terms of the two patterns?​

  1. (13,11)(13, 11)(13,11) (correct answer)
  2. (11,13)(11, 13)(11,13)
  3. (9,14)(9, 14)(9,14)
  4. (17,8)(17, 8)(17,8)

Explanation: Number patterns can be compared and graphed by pairing their corresponding terms to visualize relationships. To generate terms in a pattern, start with the initial value and repeatedly add the given increment, such as starting at 1 and adding 4 to get 1, 5, 9, 13, 17 or starting at 2 and adding 3 to get 2, 5, 8, 11, 14. Forming ordered pairs involves matching the nth term of the first pattern as x and the nth term of the second as y, so the 4th pair is (13,11). The graph shows the relationship by plotting these points, illustrating how y grows relative to x based on their rules. A common misconception is mixing up which pattern is x or y, but here x is from Pattern X and y from Pattern Y. Graphs help visualize pattern relationships by connecting points to show linear trends. Overall, such graphs allow us to identify specific pairs like the 4th terms and understand coordinated growth.

Question 11

A rectangle is 23\tfrac{2}{3}32​ yard by 34\tfrac{3}{4}43​ yard. It is partitioned into unit fraction squares that are 13\tfrac{1}{3}31​ yard by 14\tfrac{1}{4}41​ yard, making 2 squares by 3 squares. Which statement is the incorrect claim about the area, based on measuring in square units?

  1. The area is 12\tfrac{1}{2}21​ square yard because there are 6 unit fraction squares and each is 112\tfrac{1}{12}121​ square yard.
  2. The area is 1712\tfrac{17}{12}1217​ square yard because 23+34=1712\tfrac{2}{3}+\tfrac{3}{4}=\tfrac{17}{12}32​+43​=1217​. (correct answer)
  3. The area is 12\tfrac{1}{2}21​ square yard because 23×34=612\tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{6}{12}32​×43​=126​.
  4. Multiplying the side lengths gives the same area as counting the small squares in the partition.

Explanation: The core skill is finding rectangular areas with fractional sides, like 2/3 yard by 3/4 yard, via fraction multiplication. Partitioning into unit fraction squares of 1/3 yard by 1/4 yard creates a 2 by 3 grid of 6 small squares. The multiplication link is 2/3 times 3/4 equaling 1/2 square yard, matching 6 times 1/12 square yard. Square units, such as square yards, denote the area coverage. One misconception is adding fractions for area, like 2/3 + 3/4 = 17/12, which wrongly confuses it with linear addition. Such tiling models bolster the area formula by contrasting correct multiplication with errors. Overall, they generalize the support for formulas, showing how visuals prevent misconceptions in fractional geometry.

Question 12

A recipe uses 0.350.350.35 liters of milk. The cook writes the number sentence 0.35×1010.35 \times 10^10.35×101 to convert to a different unit. Powers of 10 affect place value positions by shifting digits into new places. Which statement correctly describes the pattern and the result?

  1. Each digit shifts 1 place value position to the left, so the result is 3.5. (correct answer)
  2. Each digit shifts 1 place value position to the right, so the result is 0.035.
  3. You add one zero to the end no matter what, so the result is 0.350.
  4. You add 10 one time to 0.35, so the result is 10.35.

Explanation: Powers of 10 change place value by altering digit positions, as in converting 0.35 × 10^1 for a recipe. Multiplying by 10^1 shifts digits one place left, turning 0.35 into 3.5. Dividing by powers of 10 shifts digits right, decreasing the value. This links to digit positions, moving a tenths digit to the ones place with a left shift. A misconception is always adding a zero at the end for ×10, but with decimals, it's about moving the decimal point right. These patterns allow for fast unit conversions and calculations. They support efficient math in practical situations like cooking or measuring.

Question 13

A baker has 7 identical muffins and packs them equally into 2 boxes. The fraction 72\frac{7}{2}27​ represents the result of the division 7÷27 \div 27÷2. The numerator is 7 (muffins) and the denominator is 2 (boxes). Which statement explains what 72\frac{7}{2}27​ means in this situation, showing equal sharing and that fractions can represent division?

  1. Each box gets 27\frac{2}{7}72​ of a muffin because 2 muffins are shared equally among 7 boxes.
  2. Each box gets 72\frac{7}{2}27​ muffins because 7 muffins are shared equally among 2 boxes. (correct answer)
  3. The fraction 72\frac{7}{2}27​ means 7 muffins and 2 boxes, but it does not represent division or equal sharing.
  4. Since 7 cannot be divided by 2, each box must get exactly 3 muffins and the 1 extra muffin is thrown away.

Explanation: Fractions can represent division, like dividing the total items by the number of groups to determine the amount per group. Equal sharing ensures each group gets an identical portion, which can be expressed as a fraction when the division isn't even. In this case, the numerator 7 is the number of muffins, and the denominator 2 is the number of boxes, yielding 7/2 muffins per box. This result, 7/2, is an improper fraction equal to 3.5, meaning more than 3 but less than 4 muffins per box. A misconception is that you can't divide odd numbers evenly, but fractions allow for exact equal sharing without discarding remainders. Overall, fractions show division results by quantifying shares precisely, regardless of whether they're whole or partial. This approach applies broadly, making division useful in packing and distribution scenarios.

Question 14

Sam is comparing 910\frac{9}{10}109​ and 1112\frac{11}{12}1211​ using the least common multiple method. After converting to the common denominator, which statement about the numerators is correct?

  1. The numerators are 54 and 55, with a difference of 1 (correct answer)
  2. The numerators are 108 and 110, with a difference of 2
  3. The numerators are 27 and 33, with a difference of 6
  4. The numerators are 45 and 44, with a difference of 1

Explanation: When comparing fractions using the least common multiple method, you need to find a common denominator that both original denominators divide into evenly, then convert both fractions to equivalent fractions with that denominator. To find the least common multiple (LCM) of 10 and 12, list their multiples: 10, 20, 30, 40, 50, 60... and 12, 24, 36, 48, 60... The LCM is 60, since it's the smallest number that appears in both lists. Now convert each fraction to have denominator 60. For 910\frac{9}{10}109​: since 10×6=6010 \times 6 = 6010×6=60, multiply both numerator and denominator by 6: 9×610×6=5460\frac{9 \times 6}{10 \times 6} = \frac{54}{60}10×69×6​=6054​. For 1112\frac{11}{12}1211​: since 12×5=6012 \times 5 = 6012×5=60, multiply both parts by 5: 11×512×5=5560\frac{11 \times 5}{12 \times 5} = \frac{55}{60}12×511×5​=6055​. The numerators are 54 and 55, with a difference of 1. Choice A correctly identifies these numerators and their difference. Choice B shows numerators 108 and 110 – these would result from using 120 as a common denominator instead of the LCM of 60. Choice C gives 27 and 33, which don't correspond to any valid common denominator conversion. Choice D shows 45 and 44, which also don't match any correct conversion of these fractions. Study tip: Always find the LCM first, not just any common multiple. Double-check your multiplication when converting fractions – multiply both the numerator and denominator by the same factor.

Question 15

A recipe calls for 56\frac{5}{6}65​ cup of flour and 78\frac{7}{8}87​ cup of sugar. To compare these amounts, Lisa needs to find a common denominator. What is the smallest common denominator she can use?

  1. 14
  2. 48
  3. 24 (correct answer)
  4. 16

Explanation: To find LCM(6,8): 6 = 2 × 3 and 8 = 2³. The LCM = 2³ × 3 = 8 × 3 = 24. This gives us 5/6 = 20/24 and 7/8 = 21/24 for comparison. Choice A (14) is the sum of numerators. Choice B (48) is the product 6 × 8. Choice D (16) is missing the factor of 3.

Question 16

When 47.8 is multiplied by a power of 10, the result has exactly 2 zeros between the decimal point and the first non-zero digit. Which power of 10 was used?

  1. 10110^1101
  2. 10−210^{-2}10−2
  3. 10−310^{-3}10−3 (correct answer)
  4. 10−410^{-4}10−4

Explanation: Having exactly 2 zeros between the decimal point and the first non-zero digit means the result looks like 0.0478. To get from 47.8 to 0.0478, the decimal point moved 3 places left, which happens when multiplying by 10−310^{-3}10−3. Choice A would give 478. Choice B would give 0.478 (0 zeros between decimal and first non-zero digit). Choice D would give 0.00478 (3 zeros between decimal and first non-zero digit).

Question 17

Look at the number line. Point PPP represents 4.564.564.56 and point QQQ represents 45.645.645.6. The digit 444 appears in both numbers. How many times greater is the value of the digit 444 at point QQQ compared to the value of the digit 444 at point PPP?

  1. 101010 times greater because QQQ is ten units to the right
  2. 100100100 times greater because the 444 moves from ones to tens place
  3. 101010 times greater because the 444 shifted exactly one place value position (correct answer)
  4. 919191 times greater because 45.6−4.56=41.0445.6 - 4.56 = 41.0445.6−4.56=41.04 and 41.04÷0.45≈9141.04 \div 0.45 \approx 9141.04÷0.45≈91

Explanation: In 4.56, the digit 4 is in the ones place (value = 4). In 45.6, the digit 4 is in the tens place (value = 40). Since 40 = 10 × 4, the digit 4 at point Q represents a value 10 times greater than at point P. Moving one place to the left multiplies the digit's value by 10.

Question 18

A student correctly found that 712=3560\frac{7}{12} = \frac{35}{60}127​=6035​ and 34=4560\frac{3}{4} = \frac{45}{60}43​=6045​. However, the teacher said there was a more efficient common denominator to use. What should the student have used?

  1. 24
  2. 48
  3. 12 (correct answer)
  4. 36

Explanation: LCM(12,4) = 12, not 60. The student used 60, which works but isn't the least common multiple. Using 12: 7/12 = 7/12 and 3/4 = 9/12. Choice A is double the LCM. Choice B is four times the LCM. Choice D is three times the LCM.

Question 19

Carlos finds three decimal numbers in his math book: 15.84715.84715.847, 15.85115.85115.851, and 15.85515.85515.855. When he rounds each number to the nearest tenth, how many of the rounded results are the same?

  1. All three results are identical
  2. Exactly two results are identical (correct answer)
  3. Only the first two are identical
  4. All three results are different

Explanation: Round each to the nearest tenth: 15.84715.84715.847 rounds to 15.815.815.8 (hundredths digit 4 < 5), 15.85115.85115.851 rounds to 15.915.915.9 (hundredths digit 5 ≥ 5), and 15.85515.85515.855 rounds to 15.915.915.9 (hundredths digit 5 ≥ 5). So 15.85115.85115.851 and 15.85515.85515.855 both round to 15.915.915.9, making exactly two results identical. Choice A incorrectly assumes all round the same way. Choice C misidentifies which numbers round identically. Choice D assumes each rounds differently.

Question 20

A student compares the expressions 5×455 \times \tfrac{4}{5}5×54​ and 5×655 \times \tfrac{6}{5}5×56​. The factors are 5 and 45\tfrac{4}{5}54​ (less than 1) in the first expression, and 5 and 65\tfrac{6}{5}56​ (greater than 1) in the second. Factor size affects product size. Which statement is correct (without calculating the exact products)?

  1. Both products are less than 5 because both expressions use fractions.
  2. Both products are greater than 5 because multiplication always increases a number.
  3. The product of 5×455 \times \tfrac{4}{5}5×54​ is less than 5, and the product of 5×655 \times \tfrac{6}{5}5×56​ is greater than 5. (correct answer)
  4. The two products are equal because the first factor is 5 in both expressions.

Explanation: The size of a factor in multiplication directly affects the size of the product compared to the other factor. When you multiply a number by a factor greater than 1, the product becomes larger than that original number. Conversely, multiplying by a fraction less than 1 results in a product that is smaller than the original number. In the expressions 5 × 4/5 and 5 × 6/5, the first product is less than 5 because 4/5 < 1, while the second is greater than 5 because 6/5 > 1. A common misconception is that shared factors make products equal, but the varying fractions determine the size differences. By reasoning about factor sizes, you can compare products without calculating exact values. This approach enhances efficiency and strengthens understanding of fraction multiplication.

Question 21

A recipe uses 10 cups of flour as the original amount. Two changes are suggested:

• Change 1: 10×6510 \times \frac{6}{5}10×56​ • Change 2: 10×2510 \times \frac{2}{5}10×52​

Which explanation correctly matches how each fraction affects the product compared to the original number 10? (Original number: 10; products: 10×6510\times\frac{6}{5}10×56​ and 10×2510\times\frac{2}{5}10×52​.)

  1. Both products are smaller than 10 because fractions always make a product smaller than the original number.
  2. Change 1 makes the product larger than 10 because 65\frac{6}{5}56​ is greater than 1, and Change 2 makes the product smaller than 10 because 25\frac{2}{5}52​ is less than 1. (correct answer)
  3. Change 1 makes the product smaller than 10 because you are dividing into fifths, and Change 2 makes the product larger than 10 because multiplying always increases.
  4. Both products are larger than 10 because multiplying by a fraction is the same as adding the number again and again.

Explanation: The core idea in fraction multiplication is that the fraction's size relative to 1 affects whether the product is bigger or smaller than the starting number. Fractions greater than 1, such as 6/5, make the product larger because they represent more than a whole unit. Fractions less than 1, like 2/5, make the product smaller because they represent only a portion of the whole. Imagine a bar model where 10 units are divided and regrouped: multiplying by 6/5 adds extra parts, exceeding 10, while 2/5 takes less than half, falling below 10. One misconception is that multiplication always increases a number, but with fractions less than 1, it actually decreases it. Recognizing a fraction's relation to 1 allows for quick comparisons without full computation. This skill supports logical thinking in problems involving scaling, like adjusting recipes or budgets.

Question 22

Sarah's book has 240 pages. She wants to finish reading it in 12 days by reading the same number of pages each day. However, on the first day she only reads 15 pages. How should Sarah adjust her daily reading plan for the remaining 11 days?

  1. Read 20 pages per day for the remaining days since 240 ÷ 12 = 20
  2. Read 25 pages per day for the remaining days since 225 ÷ 9 = 25
  3. Read about 20.5 pages per day for the remaining days since 225 ÷ 11 = 20.45 (correct answer)
  4. Read about 18.75 pages per day for the remaining days since 225 ÷ 12 = 18.75

Explanation: When you encounter a multi-step word problem like this, you need to carefully track what changes after each step and adjust your plan accordingly. Let's work through Sarah's situation step by step. She starts with 240 pages and wants to finish in 12 days. On day 1, she reads only 15 pages instead of her planned amount. Now you need to figure out her new daily plan. After day 1, Sarah has 240−15=225240 - 15 = 225240−15=225 pages remaining. She also has 12−1=1112 - 1 = 1112−1=11 days left to read. To find her new daily reading amount, divide the remaining pages by the remaining days: 225÷11=20.45225 ÷ 11 = 20.45225÷11=20.45 pages per day. This matches answer choice C. Now let's see why the other answers miss the mark. Answer A uses the original plan (240 ÷ 12 = 20) but ignores that Sarah already read 15 pages on day 1, so this won't help her catch up. Answer B incorrectly divides 225 by 9 instead of 11 - there's no reason to use 9 days in this problem. Answer D divides 225 by 12, but Sarah only has 11 days remaining, not 12. The key strategy here is to always recalculate based on what's left after each change. When a plan gets disrupted, update both your numerator (pages remaining) and denominator (days remaining) before dividing. Don't get trapped by using the original numbers when the situation has changed.

Question 23

Error detection: A student writes, “Because a square is a kind of rectangle, squares do not need to have 4 right angles.” The class chart is Quadrilateral ⟶ Rectangle ⟶ Square.

  • Quadrilateral: 4 sides
  • Rectangle: 4 right angles
  • Square: 4 equal sides

Which statement best fixes the student’s error?

  1. Squares can choose which rectangle attributes they want to keep.
  2. Squares inherit rectangle attributes, so squares must have 4 right angles. (correct answer)
  3. Rectangles inherit square attributes, so all rectangles must have 4 equal sides.
  4. Subcategories do not share any attributes with their categories.

Explanation: Shape categories have shared attributes that define how different shapes relate to each other in a hierarchy. A subcategory is a more specific group within a larger category that inherits all the properties of the parent category while adding its own unique traits. This means that attributes from the main category automatically apply to all items in the subcategory, ensuring consistency across the hierarchy. For example, squares inherit 4 right angles from rectangles, so they must have them despite the student's claim. A common misconception is that being a subcategory means optional inheritance, but inheritance is mandatory for all attributes. Hierarchies help classify shapes by organizing them based on increasingly specific attributes, making it easier to understand relationships. This system corrects the error by showing that squares must have 4 right angles, as in statement B.

Question 24

Carmen is making a rectangular quilt patch. She cuts fabric pieces that are each 16\frac{1}{6}61​ yard by 18\frac{1}{8}81​ yard. If she arranges them in a 4-by-3 pattern (4 pieces along length, 3 pieces along width), what is the total area of her quilt patch?

  1. 714\frac{7}{14}147​ square yards
  2. 1248\frac{12}{48}4812​ square yards
  3. 414\frac{4}{14}144​ square yards
  4. 14\frac{1}{4}41​ square yards (correct answer)

Explanation: Total dimensions: length = 4×16=46=234 × \frac{1}{6} = \frac{4}{6} = \frac{2}{3}4×61​=64​=32​ yard, width = 3×18=383 × \frac{1}{8} = \frac{3}{8}3×81​=83​ yard. Total area = 23×38=624=14\frac{2}{3} × \frac{3}{8} = \frac{6}{24} = \frac{1}{4}32​×83​=246​=41​ square yard. Choice A incorrectly adds 46+38\frac{4}{6} + \frac{3}{8}64​+83​. Choice B represents 12×14812 × \frac{1}{48}12×481​ from multiplying piece count by wrong unit area. Choice C uses incorrect fraction arithmetic.

Question 25

A coach has 13\tfrac{1}{3}31​ of a water bottle left and splits it equally among 2 players. Picture a rectangle model: the bottle is divided into 3 equal parts, then one part is divided into 2 equal smaller parts. Dividing a fraction by a whole number creates smaller equal parts. Which model description matches 13÷2\tfrac{1}{3} \div 231​÷2?

  1. Split the whole into 2 equal parts, then split one part into 3 equal parts.
  2. Split the whole into 3 equal parts, then split one of those parts into 2 equal parts. (correct answer)
  3. Split the whole into 3 equal parts and take 2 of the parts.
  4. Split the whole into 2 equal parts and take 1 part.

Explanation: Unit fractions, which have a numerator of 1, can be divided by whole numbers to find smaller equal shares. Splitting (\frac{1}{3}) of a water bottle equally among 2 players means dividing that third into 2 equal portions. This partitioning further divides the (\frac{1}{3}) into 2 smaller parts, each being (\frac{1}{6}). A rectangle model shows the bottle divided into 3 equal sections, with one section then split into 2 equal subsections. A misconception is modeling it as splitting into 2 then 3, but the order is important: first into 3, then one into 2. In general, this division reduces fraction size. Bigger divisors create proportionally smaller fractions.