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5th Grade Math

5th Grade Math Practice Test: Practice Test 3

Practice Test 3 for 5th Grade Math: real questions and explanations from the Varsity Tutors practice-test pool.

0 / 25 answered

Question 1 of 25

A baker has 3 cups of flour. Each batch of muffins needs $\tfrac{1}{4}$ cup of flour. Dividing by a unit fraction asks how many of those fractional units fit into the whole. Which claim about $3 \div \tfrac{1}{4}$ is incorrect?

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Question 1

The quotient tells how many $\tfrac{1}{4}$ -cup groups fit into 3 cups.
The quotient should be greater than 3 because each group is smaller than 1 cup.
The quotient tells how many batches can be made if each uses $\tfrac{1}{4}$ cup.
The quotient must be less than 3 because dividing always makes numbers smaller. (correct answer)

Explanation: Dividing a whole number by a unit fraction measures how many of those fractional units fit into the whole number. In the context of baking, with 3 cups of flour and each batch needing 1/4 cup, 3 ÷ 1/4 calculates the number of batches possible. You count the fractional units by seeing that each cup holds 4 units of 1/4 cup, so 3 cups hold 3 × 4 = 12 units. A bar model can represent this, with 3 bars each split into 4 quarters, totaling 12 quarters. A common misconception is that division always produces a quotient smaller than the dividend, but here 12 is larger than 3, making claim D incorrect. In general, dividing by smaller unit fractions results in larger quotients because more tiny units fit. For example, 3 ÷ 1/5 = 15, which is larger than 3 ÷ 1/4 = 12.

Question 2

The word form "six and forty-seven thousandths" written as a decimal and then compared to $6.047$ . Which statement is correct?

The word form represents $6.47$ , which is greater than $6.047$
The word form represents $6.407$ , which is greater than $6.047$
The word form represents $6.0047$ , which is less than $6.047$
The word form represents $6.047$ , which is equal to $6.047$ (correct answer)

Explanation: When you encounter word form numbers with decimal parts, you need to carefully identify which decimal place each part belongs in. The key is understanding that "thousandths" refers to the third decimal place, not just any small decimal. Let's convert "six and forty-seven thousandths" step by step. The whole number part is clearly 6. For the decimal part, "forty-seven thousandths" means 47 in the thousandths place. Since thousandths is the third decimal place, you write this as $0.047$ . Combined, this gives you $6.047$ . Now let's examine why each wrong answer contains a common mistake: Answer A suggests $6.47$ , which would be "six and forty-seven hundredths." This mistake happens when students confuse hundredths (second decimal place) with thousandths (third decimal place). Answer B gives $6.407$ , which represents "six and four hundred seven thousandths." This error occurs when students incorrectly place the digits, putting the 4 in the tenths place instead of keeping 47 together in the thousandths places. Answer C shows $6.0047$ , representing "six and forty-seven ten-thousandths." This mistake happens when students add an extra zero, pushing the number into the fourth decimal place instead of the third. Answer D correctly shows $6.047$ , which matches our conversion exactly. Remember this strategy: when converting decimal word forms, always count the decimal places carefully. Tenths = 1 place, hundredths = 2 places, thousandths = 3 places. Write the number in those exact positions, adding zeros as placeholders when needed.

Question 3

A recipe uses the same-sized whole cup as the unit. Maya pours $\tfrac{2}{3}$ cup of milk and then adds $\tfrac{1}{4}$ cup more. Because the fractions must refer to the same-sized whole cup, she rewrites them as equivalent fractions with a common denominator: $\tfrac{2}{3}=\tfrac{8}{12}$ and $\tfrac{1}{4}=\tfrac{3}{12}$ . What is the total amount of milk she poured?

$\tfrac{3}{7}$ cup
$\tfrac{9}{12}$ cup
$\tfrac{11}{12}$ cup (correct answer)
$\tfrac{3}{12}$ cup

Explanation: To add or subtract unlike fractions, which have different denominators, we must first convert them to equivalent fractions with the same denominator to ensure they refer to parts of the same-sized whole. We find a common denominator by identifying a common multiple of the two denominators, preferably the least common multiple, such as 12 for 3 and 4 in this milk-pouring scenario. To rewrite the fractions, multiply both the numerator and denominator of each by the same number; for example, multiply 2/3 by 4/4 to get 8/12, and 1/4 by 3/3 to get 3/12. Once they have the same denominator, add the numerators while keeping the denominator the same, resulting in 11/12 cup of milk total. A common misconception is that you can simply add the numerators and denominators separately, but this doesn't account for the different part sizes. Using equivalent fractions allows us to combine or compare parts accurately by making them comparable. This method ensures that operations on fractions are meaningful and applicable in real-world measurements like recipes.

Question 4

At a book fair, a roll of stickers is 1.5 meters long. Each sticker is 10 centimeters long, and there is no space between stickers. Using the known equivalence that 1 meter = 100 centimeters (based on place value), how many full stickers can be cut from the roll?

15 stickers (correct answer)
150 stickers
1 sticker
105 stickers

Explanation: The core skill here is converting units to solve problems, such as changing meters to centimeters to determine how many items fit. The relationship between meters and centimeters is that 1 meter equals 100 centimeters, based on the metric system. To convert, multiply meters by 100, so 1.5 meters = 150 centimeters, and then divide by 10 centimeters per sticker to get 15 stickers. This conversion solves the problem by calculating the exact number of full 10-centimeter stickers from the 1.5-meter roll. One misconception is ignoring the need to convert units, which might lead to dividing meters directly by centimeters incorrectly. Unit conversion is useful in manufacturing and packaging for efficient resource allocation. It helps in retail and crafts to maximize materials without waste.

Question 5

A recipe calls for $\frac{5}{6}$ cup of flour. Emma accidentally added $\frac{1}{3}$ cup too much flour to the bowl.

If Emma wants to fix her mistake by removing the extra flour and then adding $\frac{1}{8}$ cup more than the recipe originally called for, how much flour should be in the bowl when she's finished?

$\frac{23}{24}$ cup of flour in the bowl (correct answer)
$\frac{22}{24}$ cup of flour in the bowl
$\frac{25}{24}$ cup of flour in the bowl
$\frac{21}{24}$ cup of flour in the bowl

Explanation: Emma wants the original recipe amount ( $\frac{5}{6}$ cup) plus an additional $\frac{1}{8}$ cup. Convert to common denominator 24: $\frac{5}{6} = \frac{20}{24}$ and $\frac{1}{8} = \frac{3}{24}$ . Total desired: $\frac{20}{24} + \frac{3}{24} = \frac{23}{24}$ cup. Choice B subtracts instead of adds the $\frac{1}{8}$ cup. Choice C adds the extra $\frac{1}{3}$ cup instead of removing it. Choice D represents just the original recipe amount converted incorrectly.

Question 6

Refer to the figure. A rectangular prism is built using unit cubes. If 2 more layers identical to the visible layers are added to the top, what will be the total volume?

45 cubic units
60 cubic units
75 cubic units (correct answer)
90 cubic units

Explanation: The figure shows a single layer with dimensions $5 \times 5 \times 1$ , containing $25$ unit cubes. Adding 2 more identical layers creates a prism that is $5 \times 5 \times 3$ . The total volume is $5 \times 5 \times 3 = 75$ cubic units, which can be verified as $3 \times 25 = 75$ . Choice A represents adding only 1 layer (50 total). Choice B miscounts the base area. Choice D represents adding 3 additional layers instead of 2.

Question 7

A store has a ribbon length of $900$ centimeters and writes the number sentence $900 \div 10^3$ . Powers of 10 change place value positions.

Text evidence of digit shift:

900 has digits 9, 0, 0 in the hundreds, tens, and ones places
Dividing by $10^3$ shifts each digit 3 places to the right

How does dividing by $10^3$ change the number, and what is the result?

The digits shift 3 places to the left, so $900 \div 10^3 = 900{,}000$ .
The digits shift 3 places to the right, so $900 \div 10^3 = 0.9$ . (correct answer)
You remove three zeros without shifting digits, so $900 \div 10^3 = 9$ .
Dividing by $10^3$ always makes a number bigger, so the result must be greater than $900$ .

Explanation: Powers of 10 change the place value of digits in a number. Multiplying by a power of 10 shifts digits left, making the number larger. When dividing a number like 900 by 10^3, the digits shift right by 3 places, making each digit's value 1,000 times smaller and resulting in 0.9. This shifting connects to digit positions, moving from whole numbers to decimals when shifting right multiple places. A common misconception is that dividing removes zeros without shifting, but it actually requires proper place value adjustment. Recognizing these patterns allows quick unit conversions, like from centimeters to kilometers. This efficiency helps in handling large-scale divisions in everyday math tasks.

Question 8

A thermometer shows $9.015$ degrees. Decimals can be written in multiple equivalent forms, and each digit shows place value. Which words name $9.015$ correctly?

Nine and fifteen hundredths
Nine and fifteen thousandths (correct answer)
Nine and one hundred five thousandths
Nine and one tenth five

Explanation: Decimals can be written in different forms, such as numerals, words, and expanded form. Reading decimals by place value involves grouping the decimal part, like naming 9.015 as 'nine and fifteen thousandths' to reflect the places. Writing in expanded form is summing like 9 + 0.01 + 0.005 for 9.015. Each digit connects to its value: in 9.015, the 0 is 0 tenths, 1 is 1 hundredth, and 5 is 5 thousandths. A common misconception is expanding to hundredths incorrectly, like 'nine and fifteen hundredths' for 9.15. Multiple representations are useful for interpreting readings like temperatures. They ensure accurate communication and understanding across applications.

Question 9

A teacher wants to collect data about how many minutes students spend on homework each night. Which question would give her the most useful numerical data for creating a line plot?

Do you think you spend too much time on homework each night?
How many minutes did you spend on homework last night exactly? (correct answer)
Would you prefer more homework or less homework in this class?
Is homework easy, medium, or hard for you to complete?

Explanation: For a line plot showing homework time, the teacher needs specific numerical data (exact minutes). B asks for precise numerical data that can be plotted. A asks for an opinion (yes/no), which is categorical data. C asks about preferences, which is also categorical. D asks for difficulty levels, which are categorical (easy/medium/hard), not numerical data about time.

Question 10

A recipe needs 3 cups of water. Ana only has a measuring cup marked in pints. Since 1 pint = 2 cups (a known unit equivalence), which amount should Ana measure to get exactly 3 cups?

1.5 pints (correct answer)
2 pints
3 pints
6 pints

Explanation: The core skill in this problem is converting units to solve problems, such as changing cups to pints for accurate recipe measurements. The relationship between pints and cups is that 1 pint equals 2 cups, a standard equivalence in customary liquid measurements. To convert, you divide the number of cups by 2 to get pints, so 3 cups becomes 1.5 pints. This conversion solves the problem by showing exactly how much Ana should measure using her pint-marked cup. A misconception is believing all liquid units convert the same way, like confusing pints with quarts, which could double the water needed. Unit conversion is valuable for cooking and baking to avoid errors in proportions. It also applies to broader contexts like science experiments and resource management.

Question 11

Elena is calculating how many $\frac{1}{4}$ cup servings she can get from $3\frac{1}{2}$ cups of soup. She computes $3\frac{1}{2} \div \frac{1}{4} = 14$ and concludes she can serve exactly $14$ people. Her sister argues that Elena should express the quotient as $14.0$ to show it's a division result. Who is correct about the most appropriate form?

Elena, because $14$ people is the clearest way to express the practical result
Her sister, because division problems should always show decimal answers
Both, because $14$ and $14.0$ represent the same value in this context (correct answer)
Neither, because the answer should be $14\frac{0}{4}$ to match the division format

Explanation: When you encounter problems about expressing mathematical answers, think about what form best communicates the result clearly and appropriately for the context. Elena's calculation is correct: $3\frac{1}{2} \div \frac{1}{4} = \frac{7}{2} \div \frac{1}{4} = \frac{7}{2} \times \frac{4}{1} = 14$ . The key insight here is that both $14$ and $14.0$ represent exactly the same mathematical value. In this soup-serving context, both forms accurately show that Elena can serve exactly 14 people with no soup left over. Looking at the wrong answers: Choice A suggests only $14$ is appropriate, but this ignores that $14.0$ is equally valid and clear. Choice B claims division problems should always show decimal answers, which is false—many division problems yield whole number answers that are perfectly expressed as integers. Choice D proposes $14\frac{0}{4}$ , which is mathematically correct but unnecessarily complicated since $\frac{0}{4} = 0$ . The correct answer is C because both $14$ and $14.0$ represent the same value and both clearly communicate the practical result. Whether you write a whole number as $14$ or $14.0$ doesn't change its meaning or appropriateness. Study tip: When comparing different forms of the same number (like $14$ vs. $14.0$ vs. $\frac{28}{2}$ ), remember they're all mathematically equivalent. Focus on whether each form clearly communicates the answer rather than assuming one format is always "more correct" than another.

Question 12

A 1-mile walking trail is the same-sized whole. Jada walks $\tfrac{3}{5}$ mile. She jogs $\tfrac{2}{3}$ of the distance she walked. Since fraction multiplication represents taking part of a quantity, what is the correct answer to the problem: How many miles does she jog?

$\tfrac{5}{8}$ mile
$\tfrac{6}{15}$ mile
$\tfrac{2}{5}$ mile (correct answer)
$\tfrac{3}{5}$ mile

Explanation: Fraction multiplication represents taking part of a quantity, for instance, a segment of a traveled distance. Jada walks 3/5 mile and jogs 2/3 of that walked distance. The interaction is (2/3) * (3/5) = 2/5 mile jogged. Use a bar model: divide 1 mile into 5 parts, walk 3 parts; then split those 3 into 3 equal subparts, jogging 2 subparts totals 2/5. People might mistakenly add fractions, but multiplication captures the 'of' relationship. In fitness, it calculates segments of routes for varied paces. Additionally, it's practical for travel planning, finding parts of distances covered in different modes.

Question 13

A science bottle holds 16 ounces of liquid. A student pours out $\tfrac{7}{8} \times 16$ ounces. She splits 16 ounces into 8 equal parts and takes 7 parts. What does the product represent?

It represents 2 ounces, because 16 ÷ 8 = 2 and you only use one part.
It represents 14 ounces, because each eighth is 2 ounces and 7 eighths is 7 × 2 = 14 ounces. (correct answer)
It represents 24 ounces, because multiplying must make the amount larger than 16.
It represents 16 ÷ 7 ounces, because you divide by the numerator.

Explanation: Fraction multiplication has a concrete meaning, representing taking a part of a whole amount. When multiplying a fraction like 7/8 by a whole number such as 16 ounces, you start by partitioning the 16 ounces into 8 equal parts, since the denominator is 8. The numerator 7 then tells you to take 7 of those equal parts. In this science bottle context, the product represents the ounces poured out, which is 14 ounces. A common misconception is that multiplying by a fraction always increases the value, but here it results in less than 16 since 7/8 is less than 1. Models like this help explain fraction multiplication by visually showing division into equal parts and selection of some parts. Overall, such interpretations build understanding of fractions as operators on quantities.

Question 14

A student says, “In $\frac{7}{8} \times 32$ , the product must be greater than 32.” The factors are $\frac{7}{8}$ and $32$ . Which statement correctly evaluates the student’s claim without computing the product?

The claim is correct because multiplying always makes the product larger than 32.
The claim is correct because $\frac{7}{8}$ is close to 1, so the product must be bigger than 32.
The claim is incorrect because $\frac{7}{8}$ is less than 1, and multiplying by a number less than 1 makes the product smaller than 32. (correct answer)
The claim is incorrect because you should add $\frac{7}{8}$ to 32 instead of multiplying.

Explanation: The size of a factor in multiplication directly influences the size of the product compared to the other factor. When you multiply by a number greater than 1, the product becomes larger than the original number. When you multiply by a fraction less than 1, the product becomes smaller than the original number. In the expression (\frac{7}{8} \times 32), since (\frac{7}{8}) is less than 1, the product is smaller than 32, making the student's claim incorrect. A common misconception is that fractions close to 1 will still increase the product, but any value less than 1 decreases it. By assessing the factor against 1, you can evaluate claims without performing the multiplication. This form of reasoning promotes accuracy and reduces computational effort in analysis.

Question 15

A cafeteria makes a block display from two non-overlapping right rectangular prisms. Prism A is $10\text{ cm} \times 3\text{ cm} \times 1\text{ cm}$ . Prism B is $10\text{ cm} \times 3\text{ cm} \times 2\text{ cm}$ . The boundary between them is clear, so the total volume equals the sum of the parts: $V_{\text{total}}=V_A+V_B$ . What is the total volume of the composite figure?

30 cubic centimeters
60 cubic centimeters
90 cubic centimeters (correct answer)
180 cubic centimeters

Explanation: The core idea is that the volume of a composite figure is additive when it is made up of non-overlapping parts. To find the volume, we split the figure into two separate right rectangular prisms along the clear boundary. We calculate the volume of each prism by multiplying its length, width, and height; for Prism A, that's 10 cm × 3 cm × 1 cm = 30 cubic cm, and for Prism B, 10 cm × 3 cm × 2 cm = 60 cubic cm. Then, we add these volumes together to get the total volume: 30 + 60 = 90 cubic cm. A common misconception is to average the dimensions instead of adding volumes, but that doesn't account for the actual space occupied. In general, composite volumes are found by decomposing the figure into simpler shapes like rectangular prisms. We then sum the volumes of these individual prisms to obtain the total volume, ensuring no overlaps.

Question 16

Emma measures $\frac{1}{5}$ meter of ribbon. She needs to cut this ribbon into 6 equal pieces for a craft project. After cutting, she wants to know the length of each piece in meters. What is the length of each piece?

$\frac{6}{5}$ meter per piece
$\frac{1}{11}$ meter per piece
$\frac{5}{6}$ meter per piece
$\frac{1}{30}$ meter per piece (correct answer)

Explanation: This represents $\frac{1}{5} ÷ 6 = \frac{1}{5} × \frac{1}{6} = \frac{1}{30}$ meter per piece. Choice A incorrectly multiplies instead of dividing. Choice B incorrectly adds the denominator and divisor ( $\frac{1}{5+6}$ ). Choice C incorrectly inverts the original fraction.

Question 17

Look at the number $62{,}718.043$ . The digit 4 is in the hundredths place and the digit 3 is in the thousandths place. Remember: each place is 10 times the value of the place to its right and $\tfrac{1}{10}$ of the place to its left. Which statement about these two digits is correct?

The 4 is worth 0.04 and the 3 is worth 0.003, and the 4 is more than 10 times the value of the 3. (correct answer)
The 4 is worth 0.4 and the 3 is worth 0.03, and the 4 is 10 times the value of the 3.
The 4 is worth 4 and the 3 is worth 3, and the 4 is 1 more than the 3.
The 4 is worth 0.004 and the 3 is worth 0.03, and the 4 is 10 times the value of the 3 because places get bigger to the right.

Explanation: The value of a digit in a decimal number depends on its position or place relative to the decimal point. Each place to the left of another is 10 times greater in value than the place to its right. Conversely, each place to the right is 1/10 the value of the place to its left. For example, in 62,718.043, the 4 in the hundredths place is 4 × 0.01 = 0.04, and the 3 in the thousandths place is 3 × 0.001 = 0.003, so 0.04 is more than 10 times 0.003 (actually about 13.33 times) because 4 > 3. A common misconception is that place value relationships always yield exactly 10 times between adjacent places, regardless of digit values. Understanding place value enables us to compare digits' contributions across positions, even when they differ. This helps us comprehend the structure of decimals, improving skills in comparison and arithmetic.

Question 18

A recipe calls for $2\frac{1}{3}$ cups of flour to make $16$ muffins. Lisa calculated that to make $48$ muffins, she needs $7$ cups of flour. Which estimation best checks whether her calculation is reasonable?

$48 \div 16 = 3$ , and $3 \times 2 = 6$ cups, so $7$ cups seems reasonable
$48 \div 16 = 3$ , and $3 \times 2\frac{1}{3} = 7$ cups, so $7$ cups is exactly right (correct answer)
Since $48$ is about $3$ times $16$ , she needs about $3 \times 2 = 6$ cups, so $7$ seems high
Since $2\frac{1}{3}$ is close to $2$ , and $48$ is $3$ times $16$ , she needs $6$ cups exactly

Explanation: Lisa needs 3 times the original recipe (48 ÷ 16 = 3). Three times 2⅓ cups equals exactly 7 cups, so her calculation is correct. Choice A estimates 2⅓ as 2, making the estimate less accurate. Choice C also underestimates by using 2 instead of 2⅓. Choice D makes the same error and incorrectly suggests 6 cups is exact.

Question 19

A recipe for fruit punch calls for $\frac{2}{3}$ cup of orange juice for every $\frac{1}{4}$ cup of cranberry juice. If Sarah has $\frac{5}{6}$ cup of orange juice, how much cranberry juice should she use to maintain the same ratio?

$\frac{5}{8}$ cup of cranberry juice needed
$\frac{5}{16}$ cup of cranberry juice needed (correct answer)
$\frac{5}{18}$ cup of cranberry juice needed
$\frac{5}{12}$ cup of cranberry juice needed

Explanation: The ratio is $\frac{2}{3}$ cup orange juice to $\frac{1}{4}$ cup cranberry juice. First, find how much orange juice is needed for 1 cup of cranberry juice: $\frac{2/3}{1/4} = \frac{2}{3} ÷ \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}$ . So the ratio is $\frac{8}{3}:1$ (orange:cranberry). If Sarah has $\frac{5}{6}$ cup orange juice, then cranberry needed = $\frac{5}{6} ÷ \frac{8}{3} = \frac{5}{6} \times \frac{3}{8} = \frac{15}{48} = \frac{5}{16}$ cup. Choice A uses $\frac{5}{6} \times \frac{3}{4}$ . Choice C uses $\frac{5}{6} ÷ 3$ . Choice D uses $\frac{5}{6} \times \frac{1}{2}$ .

Question 20

A recipe uses $0.630$ liters of milk. Decimals can be written in multiple equivalent forms, and each digit represents a fractional value based on its place. Which statement about the decimal is correct?

The digit 3 is in the hundredths place, so it represents $0.03$ . (correct answer)
The digit 6 is in the ones place, so it represents 6.
The digit 0 in the thousandths place changes the value of the decimal.
The digit 6 is in the tenths place, so it represents $0.06$ .

Explanation: Decimals can be written in different forms, including standard notation and expanded form to highlight place values. To read 0.630 by place value, say 'six hundred thirty thousandths,' emphasizing tenths (6 as 0.6), hundredths (3 as 0.03), and thousandths (0 as nothing). Expanded form expresses it as 0.6 + 0.03 + 0, showing each part's contribution. Each digit's value is tied to its position: the 3 in hundredths represents 3 × 0.01 = 0.03. A misconception is believing a zero in thousandths changes the value, but 0.630 equals 0.63 since trailing zeros don't add value. Multiple forms help clarify decimal concepts for calculations. They promote flexibility in thinking about fractions and decimals in everyday use.

Question 21

A recipe uses $1.250$ liters of water. What is $1.250$ rounded to the nearest whole number? Remember: rounding depends on the value of the next digit (the tenths digit).

1 (correct answer)
2
1.2
1.25

Explanation: Rounding decimals is a core skill that helps estimate values by converting to whole numbers when appropriate. To round 1.250 to the nearest whole number, identify the units place, which is the 1 before the decimal. Then, check the next digit to the right, which is the tenths place, the 2. Since 2 is less than 5, keep the units digit as 1, resulting in 1. A common misconception is that any decimal part means rounding up, but it depends on the tenths being 5 or more. Rounding simplifies quantities in recipes for easier following. It provides practical estimates for cooking or resource planning.

Question 22

A recipe uses $\tfrac{1}{2}$ cup of sugar, but you want to split that amount equally into 2 small bowls for a class activity. Think of a cup model: first show $\tfrac{1}{2}$ cup, then partition that half into 2 equal smaller parts. Dividing a fraction by a whole number creates smaller equal parts. What is the result of $\tfrac{1}{2} \div 2$ ?

$\tfrac{1}{4}$ cup (correct answer)
$1$ cup
$\tfrac{2}{2}$ cup
$\tfrac{1}{2}$ cup

Explanation: Unit fractions, which have a numerator of 1, can be divided by whole numbers to find smaller equal shares. Splitting (\frac{1}{2}) cup of sugar equally into 2 bowls means dividing that half into 2 equal amounts. This partitioning further divides the (\frac{1}{2}) into 2 smaller parts, resulting in (\frac{1}{4}) cup per bowl. A cup model shows a half-cup measure, then imagined as divided into two quarter-cups. Some might think this remains (\frac{1}{2}), but that's confusing with multiplication. Generally, dividing unit fractions decreases their value. The fraction size shrinks as you increase the number of shares.

Question 23

Marcus is solving $120 \div 10^2$ and gets 1.2. He uses this to find $120 \div 10^4$ by recognizing that $10^4 = 10^2 \times 10^2$ . What should his final answer be?

0.012 (correct answer)
0.12
1.2
12

Explanation: Since $120 \div 10^2 = 1.2$ , and $10^4 = 10^2 \times 10^2$ , dividing by $10^4$ is the same as dividing by $10^2$ twice. So $120 \div 10^4 = (120 \div 10^2) \div 10^2 = 1.2 \div 10^2 = 0.012$ . Choice B represents dividing 1.2 by $10^1$ instead of $10^2$ . Choice C is the intermediate result, not the final answer. Choice D would result from multiplying 1.2 by $10^1$ .

Question 24

A gardener has 6 square meters of soil to cover with mulch. Each bag covers $\tfrac{1}{3}$ square meter. Dividing by a unit fraction asks how many $\tfrac{1}{3}$ -square-meter groups fit into 6 square meters. Which value shows how many bags are needed to cover all 6 square meters?

2 bags
3 bags
18 bags (correct answer)
6 bags

Explanation: Dividing a whole number by a unit fraction measures how many of those fractional parts can fit into the whole number. The gardener with 6 square meters using bags covering 1/3 each computes 6 ÷ 1/3 for bags needed. We count by multiplying 6 by 3, equaling 18 bags. Connect to a model of an area grid dividing 6 units into thirds, showing 18 sections. A misconception is thinking fewer bags for smaller coverage, but smaller fractions require more. In general, tinier unit fractions produce larger quotients. For example, dividing 6 by 1/4 gives 24, larger than dividing by 1/3 which gives 18.

Question 25

A teacher asks students to find a common denominator for $\frac{5}{6}$ and $\frac{7}{8}$ without using the LCM method. Which of these alternative common denominators would work?

14, because it equals 6 + 8
30, because it equals 6 × 5
22, because it equals (6 + 8) + (6 + 2)
48, because it equals 6 × 8 (correct answer)

Explanation: When you need to find a common denominator for two fractions, you're looking for a number that both original denominators can divide into evenly. This means the common denominator must be a multiple of both denominators. For $\frac{5}{6}$ and $\frac{7}{8}$ , any common denominator must be divisible by both 6 and 8. Let's test each option by checking if both denominators divide evenly into it. Option D (48) works because 48 ÷ 6 = 8 and 48 ÷ 8 = 6. Both divisions give whole numbers, so 48 is indeed a common denominator. You can rewrite the fractions as $\frac{40}{48}$ and $\frac{42}{48}$ . Option A (14) fails because while 14 ÷ 6 = 2.33..., which isn't a whole number. Adding the denominators (6 + 8 = 14) doesn't create a valid common denominator. Option B (30) doesn't work either. While 30 ÷ 6 = 5, we get 30 ÷ 8 = 3.75, which isn't whole. Multiplying one denominator by the numerator of the other fraction doesn't guarantee a common denominator. Option C (22) also fails. You get 22 ÷ 6 = 3.67... and 22 ÷ 8 = 2.75. This random calculation doesn't produce a multiple of both denominators. Remember: A common denominator must be divisible by both original denominators. While multiplying the denominators together (like 6 × 8 = 48) always works, it's not always the smallest option—but it's always correct.