6th Grade Math Quiz: Divide Fractions By Fractions

20 questions · exam conditions

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Divide Fractions By FractionsQuestion 1 of 20

Maria has $\frac{3}{4}$ of a yard of fabric. She wants to cut it into pieces that are each $\frac{1}{8}$ of a yard long to make bookmarks. After cutting all possible pieces, she realizes she made an error and each bookmark actually needs $\frac{1}{6}$ of a yard of fabric. How many complete bookmarks can she make with her original $\frac{3}{4}$ yard of fabric?

4 bookmarks because she can reuse all the pieces she already cut

4 bookmarks because

\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5

6 bookmarks because

\frac{3}{4} \div \frac{1}{8} = 6

pieces were already cut

5 bookmarks because

\frac{3}{4} \div \frac{1}{6} = \frac{18}{4} = 4.5

rounds up to 5

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6th Grade Math Quiz

6th Grade Math Quiz: Divide Fractions By Fractions

Practice Divide Fractions By Fractions in 6th Grade Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Divide Fractions By Fractions, giving you a quick way to practice the rules, question types, and explanations that matter most for 6th Grade Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

4 bookmarks because she can reuse all the pieces she already cut
4 bookmarks because $\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5$ (correct answer)
6 bookmarks because $\frac{3}{4} \div \frac{1}{8} = 6$ pieces were already cut
5 bookmarks because $\frac{3}{4} \div \frac{1}{6} = \frac{18}{4} = 4.5$ rounds up to 5

Explanation: To find how many

\frac{1}{6}

-yard bookmarks can be made from

\frac{3}{4}

yard, we calculate

\frac{3}{4} \div \frac{1}{6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5

. Since we can only make complete bookmarks, the answer is 4. Choice A incorrectly assumes the previously cut pieces can be reused. Choice C uses the wrong division (by

\frac{1}{8}

instead of

\frac{1}{6}

). Choice D incorrectly rounds up when we need complete bookmarks.

Question 2

Compute and interpret: A container holds $\tfrac{3}{4}$ liter of juice. Each small bottle holds $\tfrac{1}{2}$ liter. How many small bottles can be filled? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{2}\right)$ and verify by multiplying the quotient by $\tfrac{1}{2}$ .

$\tfrac{3}{8}$ bottle; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{3}{8}\right)=\tfrac{3}{16}$
$\tfrac{2}{3}$ bottle; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{1}{3}$
$\tfrac{3}{2}$ bottles; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{3}{4}$ (correct answer)
$\tfrac{1}{4}$ bottle; and $\left(\tfrac{1}{2}\right)\times\left(\tfrac{1}{4}\right)=\tfrac{1}{8}$

Explanation: This question tests dividing fractions

(a/b) \div (c/d)

using the reciprocal method (multiply by

d/c

), interpreting in contexts like filling bottles, and understanding via visual models and the multiplication-division relationship. Dividing fractions:

(3/4) \div (1/2) = (3/4) \times (2/1) = 6/4 = 3/2

(flip divisor to reciprocal, multiply). Verification: divisor

\times

quotient

=

dividend

((1/2) \times (3/2) = 3/4

confirms quotient correct). Context: 'how many

1/2

-liter bottles from

3/4

liter' divides:

(3/4) \div (1/2) = 3/2

(1.5 bottles). Multiplication-division relationship:

(3/4) \div (1/2) = 3/2

because

(1/2)

(3/2)

equals

(3/4)

(

(1/2) \times (3/2) = 3/4

, division is inverse of multiplication). The correct choice is C, which uses the reciprocal method to get

3/2

and verifies correctly. Common errors include multiplying instead of dividing, like getting

2/3

as in B.

Question 3

A recipe uses $\tfrac{2}{3}$ cup of yogurt. One serving size is $\tfrac{3}{4}$ cup. How many $\tfrac{3}{4}$ -cup servings can you make from $\tfrac{2}{3}$ cup? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ and choose the best interpretation.

$\tfrac{3}{8}$ serving (dividing numerators and denominators separately)
$\tfrac{8}{9}$ of a serving (less than 1 full $\tfrac{3}{4}$ -cup serving) (correct answer)
$\tfrac{2}{3}$ serving (from an arithmetic slip: $\tfrac{2}{3}\times\tfrac{4}{3}=\tfrac{6}{9}$ )
$\tfrac{1}{2}$ serving (because $\tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{1}{2}$ )

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like servings from a given amount of yogurt. To divide (2/3)÷(3/4), flip the divisor to its reciprocal 4/3 and multiply: (2/3)×(4/3)=(2×4)/(3×3)=8/9. Verification: multiply the divisor by the quotient, (3/4)×(8/9)=(3×8)/(4×9)=24/36=2/3, which matches the dividend. In context, this means you can make 8/9 of a 3/4-cup serving from 2/3 cup of yogurt, which is less than one full serving. A common error is multiplying instead of dividing, like (2/3)×(3/4)=1/2 as in choice A, or flipping the wrong fraction leading to incorrect results like 2/3 in choice C. To compute: write the division, flip the divisor, multiply numerators and denominators, simplify if needed, and verify by multiplying back. Visual models, like a bar representing 2/3 cup divided into 3/4-cup segments, show it fits 8/9 of such a segment.

Question 4

Use the multiplication-division relationship to check a quotient: If $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)=\tfrac{8}{9},$ which multiplication correctly verifies the result (divisor $\times$ quotient $=$ dividend)?

$\left(\tfrac{2}{3}\right)\times\left(\tfrac{3}{4}\right)=\tfrac{8}{9}$
$\left(\tfrac{3}{4}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{2}{3}$ (correct answer)
$\left(\tfrac{8}{9}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{3}{4}$
$\left(\tfrac{2}{3}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{3}{4}$

Explanation: This question tests dividing fractions

\frac{a}{b} \div \frac{c}{d}

using the reciprocal method, emphasizing the multiplication-division relationship for verification, and understanding via contexts and visual models. The division

\frac{2}{3} \div \frac{3}{4} = \frac{8}{9}

is verified by multiplying divisor by quotient to equal dividend:

\frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3}

. Other options like

\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}

do not verify correctly. For example, the relationship shows division is the inverse of multiplication, so if quotient is correct, divisor

\times

quotient

=

dividend. The correct verification is option B. A common error is confusing which to multiply, like quotient

\times

dividend. Remember: to check, always multiply divisor by proposed quotient and see if it equals dividend.

Question 5

Number-line reasoning: On a number line, $\tfrac{3}{4}$ is one step size. How many $\tfrac{3}{4}$ -steps are in $\tfrac{1}{3}$ ? Compute $\left(\tfrac{1}{3}\right)\div\left(\tfrac{3}{4}\right)$ and verify by multiplication.

$\tfrac{4}{9}$ step; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{4}{9}\right)=\tfrac{1}{3}$ (correct answer)
$\tfrac{9}{4}$ steps; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{9}{4}\right)=\tfrac{27}{16}$
$\tfrac{1}{4}$ step; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{1}{4}\right)=\tfrac{3}{16}$
$\tfrac{1}{9}$ step; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{1}{9}\right)=\tfrac{1}{12}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in number-line contexts like step sizes, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (1/3)÷(3/4)=(1/3)×(4/3)=4/9 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((3/4)×(4/9)=12/36=1/3 confirms quotient correct). Context: 'how many 3/4-steps in 1/3' divides: 1/3÷3/4=4/9 step (less than 1). Multiplication-division relationship: (1/3)÷(3/4)=4/9 because (3/4) of (4/9) equals (1/3) (3/4×4/9=1/3, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 4/9 and verifies correctly. Common errors include flipping the wrong fraction, leading to 9/4 as in B.

Question 6

Use a visual model idea: Imagine a bar showing $\tfrac{2}{3}$ of a whole. Each group is $\tfrac{1}{4}$ of a whole. How many $\tfrac{1}{4}$ -size groups fit into $\tfrac{2}{3}$ ? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{1}{4}\right)$ and verify by multiplication.

$\tfrac{2}{12}$ groups; and $\left(\tfrac{1}{4}\right)\times\left(\tfrac{2}{12}\right)=\tfrac{1}{24}$
$\tfrac{8}{3}$ groups; and $\left(\tfrac{1}{4}\right)\times\left(\tfrac{8}{3}\right)=\tfrac{2}{3}$ (correct answer)
$\tfrac{1}{2}$ group; and $\left(\tfrac{1}{4}\right)\times\left(\tfrac{1}{2}\right)=\tfrac{1}{8}$
$\tfrac{1}{6}$ group; and $\left(\tfrac{1}{4}\right)\times\left(\tfrac{1}{6}\right)=\tfrac{1}{24}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in visual model contexts like fitting groups into a bar, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (2/3)÷(1/4)=(2/3)×(4/1)=8/3 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((1/4)×(8/3)=8/12=2/3 confirms quotient correct). Context: 'how many 1/4-size groups in 2/3' divides: 2/3÷1/4=8/3 (more than 2 groups). Multiplication-division relationship: (2/3)÷(1/4)=8/3 because (1/4) of (8/3) equals (2/3) (1/4×8/3=2/3, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 8/3 and verifies correctly. Common errors include wrong reciprocal or simplification, like getting 1/6 as in C.

Question 7

A ribbon is $\tfrac{3}{4}$ yard long. Each bracelet needs $\tfrac{1}{2}$ yard of ribbon. The division expression is $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{2}\right)$ . How many bracelets can be made from the ribbon? (Simplify.)

$\tfrac{3}{8}$
$\tfrac{1}{4}$
$\tfrac{2}{3}$
$\tfrac{3}{2}$ (correct answer)

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like how many segments of one length fit into another, and understanding via visual models and the multiplication-division relationship. To divide (3/4)÷(1/2), flip the divisor to its reciprocal and multiply: (3/4) × (2/1) = 6/4 = 3/2, so 3/2 bracelets can be made. Verification: multiply divisor by quotient to get dividend, (1/2) × (3/2) = 3/4, which confirms. For example, a ribbon of 3/4 yard divided by 1/2 yard per bracelet: (3/4) × 2 = 3/2, meaning 1 full and half another. The correct division using the reciprocal gives simplified 3/2. Common mistakes include subtracting fractions or not simplifying 6/4. Method: write division, flip divisor, multiply, reduce fraction, verify.

Question 8

A ribbon is $\tfrac{3}{4}$ yard long. Each bow needs $\tfrac{1}{2}$ yard of ribbon. How many bows can be made? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{2}\right)$ .

$\tfrac{3}{8}$
$\tfrac{1}{2}$
$\tfrac{2}{3}$
$\tfrac{3}{2}$ (correct answer)

Explanation: This question tests dividing fractions

(a/b) \div (c/d)

using the reciprocal method (multiply by

d/c

), in context of how many items can be made from a given length. Compute

(3/4) \div (1/2) = (3/4) \times (2/1) = (3 \times 2)/(4 \times 1) = 6/4 = 3/2

bows. Verification:

(1/2) \times (3/2) = 3/4

yard, matching the total. This means 1 full bow and half another from

3/4

yard. Errors include multiplying fractions to get 3/8 as in choice A, or confusing reciprocals to get 1/2. Steps: flip divisor, multiply, simplify

6/4

3/2

, verify. Context like ribbons shows practical division application.

Question 9

A rectangle has area $\tfrac{1}{2}$ square mile and length $\tfrac{3}{4}$ mile. What is the width? Compute $\left(\tfrac{1}{2}\right)\div\left(\tfrac{3}{4}\right)$ and verify by checking that $\text{length} \times \text{width} = \text{area}$ .

$\tfrac{3}{8}$ mile; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{3}{8}\right)=\tfrac{9}{32}$
$\tfrac{1}{6}$ mile; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{1}{6}\right)=\tfrac{1}{8}$
$\tfrac{3}{2}$ mile; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{9}{8}$
$\tfrac{2}{3}$ mile; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{1}{2}$ (correct answer)

Explanation: This question tests dividing fractions

\frac{a}{b} \div \frac{c}{d}

using the reciprocal method (multiply by

\frac{d}{c}

), interpreting in contexts like rectangle area

\div

length = width, and understanding via visual models and the multiplication-division relationship. Dividing fractions:

\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}

(flip divisor to reciprocal, multiply). Verification: divisor

\times

quotient = dividend (

\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}

confirms quotient correct). Context: 'width of rectangle with area

\frac{1}{2}

sq mi and length

\frac{3}{4}

mi' divides:

\frac{1}{2} \div \frac{3}{4} = \frac{2}{3}

mi. Multiplication-division relationship:

\frac{1}{2} \div \frac{3}{4} = \frac{2}{3}

because

\frac{3}{4}

\frac{2}{3}

equals

\frac{1}{2}

(

\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}

, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get

\frac{2}{3}

and verifies correctly. Common errors include incorrect flipping or multiplication, leading to wrong quotients like

\frac{3}{2}

in B.

Question 10

Three friends share $\tfrac{1}{2}$ pound of trail mix equally. How much does each friend get? Write the division expression and compute $\left(\tfrac{1}{2}\right)\div 3$ . Verify by multiplying your answer by 3.

$\tfrac{1}{6}$ lb each; and $3\times\tfrac{1}{6}=\tfrac{1}{2}$ (correct answer)
$\tfrac{3}{2}$ lb each; and $3\times\tfrac{3}{2}=\tfrac{9}{2}$
$\tfrac{2}{3}$ lb each; and $3\times\tfrac{2}{3}=2$
$\tfrac{1}{5}$ lb each; and $3\times\tfrac{1}{5}=\tfrac{3}{5}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like sharing among friends, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (1/2)÷3=(1/2)×(1/3)=1/6 (treat 3 as 3/1, flip to reciprocal, multiply). Verification: divisor×quotient=dividend (3×(1/6)=3/6=1/2 confirms quotient correct). Context: 'how much trail mix each of 3 friends gets from 1/2 lb' divides: 1/2÷3=1/6 lb each. Multiplication-division relationship: (1/2)÷3=1/6 because 3 times (1/6) equals (1/2) (3×1/6=1/2, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 1/6 and verifies correctly. Common errors include treating division incorrectly, like getting 3/2 as in B.

Question 11

A recipe uses $\tfrac{2}{3}$ cup of yogurt. Each smoothie serving needs $\tfrac{3}{4}$ cup of yogurt. How many servings can you make from $\tfrac{2}{3}$ cup? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ and interpret the result (is it more or less than 1 full serving?). Then verify your answer by multiplying: $\left(\tfrac{3}{4}\right)\times(\text{quotient})=\tfrac{2}{3}$ .

$\tfrac{2}{3}$ of a serving; and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{2}{3}\right)=\tfrac{1}{2}$
$\tfrac{1}{2}$ serving; because $\left(\tfrac{2}{3}\right)\times\left(\tfrac{3}{4}\right)=\tfrac{1}{2}$
$\tfrac{8}{9}$ of a serving (less than 1); and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{2}{3}$ (correct answer)
$\tfrac{9}{8}$ servings (more than 1); and $\left(\tfrac{3}{4}\right)\times\left(\tfrac{9}{8}\right)=\tfrac{27}{32}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like servings, sharing, or rectangle area÷length=width, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (2/3)÷(3/4)=(2/3)×(4/3)=8/9 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((3/4)×(8/9)=24/36=2/3 confirms quotient correct). Context: 'how many 3/4-cup servings in 2/3 cup yogurt?' divides: 2/3÷3/4=8/9 (less than 1 full serving, 8/9 of a 3/4-cup portion). Multiplication-division relationship: (2/3)÷(3/4)=8/9 because (3/4) of (8/9) equals (2/3) (3/4×8/9=24/36=2/3, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 8/9 and verifies correctly. Common errors include multiplying instead of dividing ((2/3)×(3/4)=1/2 as in A), or flipping the wrong fraction leading to incorrect quotients like in D.

Question 12

A science class has $\tfrac{3}{4}$ cup of saltwater. Each test tube needs $\tfrac{2}{3}$ cup. How many test tubes can be filled? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{2}{3}\right)$ and verify by multiplication.

$\tfrac{1}{2}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{1}{2}\right)=\tfrac{1}{3}$
$\tfrac{8}{9}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{8}{9}\right)=\tfrac{16}{27}$
$\tfrac{6}{12}$ test tube; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{6}{12}\right)=\tfrac{1}{3}$
$\tfrac{9}{8}$ test tubes; and $\left(\tfrac{2}{3}\right)\times\left(\tfrac{9}{8}\right)=\tfrac{3}{4}$ (correct answer)

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like filling test tubes, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (3/4)÷(2/3)=(3/4)×(3/2)=9/8 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((2/3)×(9/8)=18/24=3/4 confirms quotient correct). Context: 'how many 2/3-cup test tubes from 3/4 cup' divides: 3/4÷2/3=9/8 (more than 1). Multiplication-division relationship: (3/4)÷(2/3)=9/8 because (2/3) of (9/8) equals (3/4) (2/3×9/8=3/4, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 9/8 and verifies correctly. Common errors include incorrect multiplication, like getting 8/9 as in A.

Question 13

Which value makes this verification true? If $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)=q$ , then $\left(\tfrac{3}{4}\right)\times q=\tfrac{2}{3}$ . What is $q$ ?

$\tfrac{8}{9}$ (correct answer)
$\tfrac{9}{8}$
$\tfrac{1}{2}$
$\tfrac{4}{9}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), verifying through the multiplication-division relationship. To find q where (3/4)×q=2/3, q=(2/3)÷(3/4)=(2/3)×(4/3)=8/9. Verification: (3/4)×(8/9)=24/36=2/3, correct. This confirms division as the inverse of multiplication. Common errors: flipping incorrectly to 9/8 as in choice D, or multiplying to 1/2. Method: solve for q using reciprocal, compute, verify by plugging back. Examples like (1/2)÷(1/4)=2 verify similarly.

Question 14

A painter has $\tfrac{2}{3}$ gallon of paint. One room needs $\tfrac{1}{3}$ gallon. How many rooms can be painted? Compute $\left(\tfrac{2}{3}\right)\div\left(\tfrac{1}{3}\right)$ and verify by multiplication.

$\tfrac{2}{9}$ room; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{2}{9}\right)=\tfrac{2}{27}$
$\tfrac{3}{2}$ rooms; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{3}{2}\right)=\tfrac{1}{2}$
$2$ rooms; and $\left(\tfrac{1}{3}\right)\times 2=\tfrac{2}{3}$ (correct answer)
$\tfrac{1}{2}$ room; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{1}{2}\right)=\tfrac{1}{6}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like painting rooms, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (2/3)÷(1/3)=(2/3)×(3/1)=2 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((1/3)×2=2/3 confirms quotient correct). Context: 'how many rooms needing 1/3 gallon from 2/3 gallon' divides: 2/3÷1/3=2 rooms. Multiplication-division relationship: (2/3)÷(1/3)=2 because (1/3) of 2 equals (2/3) (1/3×2=2/3, division is inverse of multiplication). The correct choice is B, which uses the reciprocal method to get 2 and verifies correctly. Common errors include wrong division, like getting 3/2 as in C.

Question 15

A ribbon is $\tfrac{3}{4}$ meter long. You cut pieces that are each $\tfrac{1}{3}$ meter long. How many pieces can you cut? Compute $\left(\tfrac{3}{4}\right)\div\left(\tfrac{1}{3}\right)$ and verify by multiplication.

$\tfrac{1}{4}$ piece; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{1}{4}\right)=\tfrac{1}{12}$
$\tfrac{3}{12}$ pieces; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{3}{12}\right)=\tfrac{1}{12}$
$\tfrac{9}{4}$ pieces; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{9}{4}\right)=\tfrac{3}{4}$ (correct answer)
$\tfrac{4}{9}$ piece; and $\left(\tfrac{1}{3}\right)\times\left(\tfrac{4}{9}\right)=\tfrac{4}{27}$

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreting in contexts like cutting ribbons into pieces, and understanding via visual models and the multiplication-division relationship. Dividing fractions: (3/4)÷(1/3)=(3/4)×(3/1)=9/4 (flip divisor to reciprocal, multiply). Verification: divisor×quotient=dividend ((1/3)×(9/4)=9/12=3/4 confirms quotient correct). Context: 'how many 1/3-meter pieces in 3/4 meter ribbon?' divides: 3/4÷1/3=9/4 (more than 2 full pieces). Multiplication-division relationship: (3/4)÷(1/3)=9/4 because (1/3) of (9/4) equals (3/4) (1/3×9/4=3/4, division is inverse of multiplication). The correct choice is A, which uses the reciprocal method to get 9/4 and verifies correctly. Common errors include using the wrong reciprocal or arithmetic mistakes, like getting 4/9 as in B.

Question 16

On a number line, $\tfrac{2}{3}$ is a point to the right of 0. If you take steps of size $\tfrac{3}{4}$ , how many steps reach $\tfrac{2}{3}$ ? This is $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ . Which value is correct?

$\tfrac{1}{2}$
$\tfrac{3}{8}$
$\tfrac{9}{8}$
$\tfrac{8}{9}$ (correct answer)

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method (multiply by d/c), interpreted on a number line as steps to reach a point. Compute (2/3)÷(3/4)=(2/3)×(4/3)=8/9 steps. Verification: (3/4)×(8/9)=24/36=2/3, reaching the point. This is less than 1 step since 2/3 < 3/4. Mistakes include inverting the operation to get 9/8 as in choice B, or multiplying to get 1/2. Method: division to multiplication by reciprocal, compute 8/9, verify. Visual: number line with steps of 3/4 up to 2/3 fits 8/9 of a step.

Question 17

A recipe for trail mix calls for $\frac{5}{8}$ cup of nuts. Sarah wants to make multiple smaller portions, where each portion contains $\frac{3}{4}$ of the nuts called for in the original recipe. If she has $2\frac{1}{8}$ cups of nuts available, and after making as many complete smaller portions as possible she uses the remaining nuts to make one final portion at $\frac{1}{2}$ the original recipe size, will she have enough nuts?

No, because she can make 4 smaller portions using $2\frac{1}{16}$ cups total
Yes, because 4 smaller portions use $1\frac{7}{8}$ cups, leaving $\frac{1}{4}$ cup for the final portion
No, because 4 smaller portions use $1\frac{7}{8}$ cups, but the final portion needs $\frac{5}{16}$ cup (correct answer)
Yes, because she can make exactly 4 smaller portions with $\frac{1}{8}$ cup remaining for other uses

Explanation: Each smaller portion needs

\frac{3}{4} \times \frac{5}{8} = \frac{15}{32}

cup. Number of portions from

2\frac{1}{8} = \frac{17}{8}

cups:

\frac{17}{8} \div \frac{15}{32} = \frac{17}{8} \times \frac{32}{15} = \frac{544}{120} = \frac{68}{15} \approx 4.53

, so 4 complete portions. These use

4 \times \frac{15}{32} = \frac{60}{32} = \frac{15}{8} = 1\frac{7}{8}

cups. Remaining:

2\frac{1}{8} - 1\frac{7}{8} = \frac{17}{8} - \frac{15}{8} = \frac{2}{8} = \frac{1}{4}

cup. The final portion at

\frac{1}{2}

original size needs

\frac{1}{2} \times \frac{5}{8} = \frac{5}{16}

cup. Since

\frac{1}{4} = \frac{4}{16} < \frac{5}{16}

, she doesn't have enough. Choice A has wrong calculation. Choice B incorrectly concludes she has enough. Choice D ignores the final portion requirement.

Question 18

A road segment is $\tfrac{1}{2}$ mile long. A student walks at a pace of $\tfrac{1}{4}$ mile per minute. The time (in minutes) is $\left(\tfrac{1}{2}\right)\div\left(\tfrac{1}{4}\right)$ . How many minutes does it take? (Simplify.)

$\tfrac{3}{4}$
$\tfrac{1}{2}$
$\tfrac{1}{8}$
$2$ (correct answer)

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like time taken to cover distance at a given speed, and understanding via visual models and the multiplication-division relationship. To divide (1/2)÷(1/4), flip the divisor to its reciprocal and multiply: (1/2) × (4/1) = 4/2 = 2, so it takes 2 minutes. Verification: multiply divisor by quotient to get dividend, (1/4) × 2 = 1/2, which confirms. For example, distance 1/2 mile at 1/4 mile per minute: how many quarters in a half is 2. The correct division using the reciprocal gives 2. A common mistake is adding fractions or using wrong reciprocal. Method: write division, flip divisor, multiply, simplify, verify with multiplication.

Question 19

A rectangular garden plot has an area of $\frac{5}{6}$ square meters and a length of $\frac{5}{4}$ meters. If the garden is divided into smaller rectangular sections, each with a width equal to the original garden's width, and each section has a length of $\frac{1}{3}$ meter, how many complete sections can be formed?

2 sections because the original width is $\frac{2}{3}$ meter and $\frac{5}{4} \div \frac{1}{3} = \frac{15}{4}$
2 sections because the original width is $\frac{2}{3}$ meter and $\frac{2}{3} \div \frac{1}{3} = 2$
3 sections because the original width is $\frac{2}{3}$ meter and $\frac{5}{4} \div \frac{1}{3} = 3.75$ (correct answer)
4 sections because the original length $\frac{5}{4}$ divided by new length $\frac{1}{3}$ gives $\frac{15}{4} = 3.75$

Explanation: First find the original width:

\text{width} = \frac{\text{area}}{\text{length}} = \frac{5/6}{5/4} = \frac{5}{6} \times \frac{4}{5} = \frac{2}{3}

meter. The number of sections is determined by how many

\frac{1}{3}

-meter lengths fit in the original

\frac{5}{4}

-meter length:

\frac{5}{4} \div \frac{1}{3} = \frac{5}{4} \times 3 = \frac{15}{4} = 3.75

, so 3 complete sections. Choice A miscalculates the division. Choice B incorrectly divides width by new length. Choice D rounds 3.75 up instead of down for complete sections.

Question 20

A recipe needs $\tfrac{2}{3}$ cup of yogurt. You have a scoop that holds $\tfrac{3}{4}$ cup. The division expression is $\left(\tfrac{2}{3}\right)\div\left(\tfrac{3}{4}\right)$ . How many full scoops of size $\tfrac{3}{4}$ cup fit into $\tfrac{2}{3}$ cup of yogurt? (Choose the simplified value.)

$\tfrac{9}{8}$
$\tfrac{2}{3}$
$\tfrac{1}{2}$
$\tfrac{8}{9}$ (correct answer)

Explanation: This question tests dividing fractions (a/b)÷(c/d) using the reciprocal method, interpreting in contexts like servings or how many scoops of one size fit into another amount, and understanding via visual models and the multiplication-division relationship. To divide (2/3)÷(3/4), flip the divisor to its reciprocal and multiply: (2/3) × (4/3) = 8/9, meaning 8/9 of a full 3/4-cup scoop fits into 2/3 cup of yogurt. Verification: multiply divisor by quotient to get dividend, (3/4) × (8/9) = 24/36 = 2/3, which confirms the result. For example, imagine a bar representing 2/3 cup; dividing it into 3/4-cup segments shows it fits 8/9 of one segment. The correct method is to use the reciprocal for division, yielding 8/9 as the simplified value. A common mistake is multiplying without flipping, like (2/3) × (3/4) = 1/2, or flipping the wrong fraction. Remember the steps: write the division, flip the divisor, multiply, simplify, and verify with multiplication.