Exercises: Interpret and compute quotients of fractions - Common Core Math - Grade 6

A

Recall / Warm-Up

1.

Which pair of fractions are reciprocals of each other?

A.

$\frac{2}{3}$ and $\frac{3}{2}$

B.

$\frac{2}{3}$ and $\frac{2}{3}$

C.

$\frac{2}{3}$ and $-\frac{2}{3}$

D.

$\frac{2}{3}$ and $\frac{1}{3}$

2.

What is the reciprocal of $\frac{3}{5}$ ?

A.

$\frac{3}{5}$

B.

$\frac{5}{3}$

C.

$-\frac{3}{5}$

D.

$\frac{2}{5}$

$Number line from 0 to 1 divided into fourths. The interval from 0 to 3/4 is highlighted, containing three segments of length 1/4 each.$

3.

A number line from 0 to 1 is divided into fourths. The segment from 0 to $\frac{3}{4}$ contains how many $\frac{1}{4}$ -length segments?

A.

1

B.

2

C.

3

D.

4

B

Fluency Practice

1.

Which word problem matches the expression $\frac{1}{2} \div 3$ ?

A.

How many groups of 3 are in $\frac{1}{2}$ ?

B.

How many groups of $\frac{1}{2}$ are in 3?

C.

Three people share $\frac{1}{2}$ lb of cheese equally. How much does each person get?

D.

A recipe uses $\frac{1}{2}$ lb of cheese per batch. How many batches can you make with 3 lbs?

$Unit square with a teal rectangle occupying 3/4 of the width and 2/3 of the height, showing area = 1/2 sq mi. The width is labeled 3/4 mi and the height is labeled with a question mark.$

2.

Use the area model: a rectangle with area $\frac{1}{2}$ sq mi and length $\frac{3}{4}$ mi. What is the width?

A.

$\frac{3}{8}$ mi

B.

$\frac{2}{3}$ mi

C.

$\frac{1}{2}$ mi

D.

$\frac{4}{3}$ mi

3.

Compute $\frac{2}{3} \div \frac{3}{4}$ . Show the reciprocal of the divisor and the product.

4.

Compute $\frac{5}{6} \div \frac{2}{3}$ . Express your answer as a mixed number if the result is greater than 1.

5.

Compute $\frac{3}{4} \div \frac{3}{8}$ .

C

Varied Practice

1.

"How many $\frac{3}{4}$ -cup servings are in $\frac{2}{3}$ of a cup of yogurt?" Which expression correctly represents this problem?

A.

$\frac{3}{4} \div \frac{2}{3}$

B.

$\frac{2}{3} \div \frac{3}{4}$

C.

$\frac{2}{3} \times \frac{3}{4}$

D.

$\frac{3}{4} \times \frac{2}{3}$

2.

Compute $\frac{1}{2} \div \frac{3}{4}$ .

3.

Compute $\frac{4}{5} \div \frac{2}{3}$ step by step.
The divisor is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .
The reciprocal of the divisor is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .
$\frac{4}{5} \times$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $=$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ (simplify fully).

divisor:

reciprocal:

multiplication expression:

final answer:

4.

Compute $\frac{7}{8} \div \frac{7}{4}$ .

5.

Which statement correctly justifies why $\frac{2}{3} \div \frac{3}{4} = \frac{8}{9}$ ?

A.

Because $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ , and flipping gives $\frac{2}{1} = 2$ .

B.

Because $\frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3}$ — the product of the divisor and quotient equals the dividend.

C.

Because division always gives a smaller result than the dividend.

D.

Because $\frac{2 \div 3}{3 \div 4} = \frac{2}{3} \div \frac{3}{4}$ .

D

Word Problems

1.

Mia has $\frac{5}{6}$ of a yard of ribbon. She wants to cut it into pieces that are each $\frac{1}{3}$ of a yard long.

1.

Which division expression gives the number of pieces Mia can cut?

A.

$\frac{1}{3} \div \frac{5}{6}$

B.

$\frac{5}{6} \div \frac{1}{3}$

C.

$\frac{5}{6} \times \frac{1}{3}$

D.

$\frac{5}{6} + \frac{1}{3}$

2.

How many pieces can Mia cut?

2.

A recipe uses $\frac{2}{3}$ cup of sugar per batch of cookies. Priya has $\frac{5}{6}$ cup of sugar.

Complete the reasoning.
Quotient = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
Number of full batches = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

quotient:

full batches:

3.

A rectangular garden has an area of $\frac{1}{2}$ sq mi and a length of $\frac{3}{4}$ mi.

Complete the reasoning.
Width = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
Width = ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ mi

division expression:

width:

4.

Three friends share $\frac{3}{4}$ lb of trail mix equally.

How much trail mix does each person receive in pounds?

E

Error Analysis

1.

Demi computes $\frac{2}{3} \div \frac{4}{5}$ :
$\frac{2}{3} \div \frac{4}{5} = \frac{2 \div 4}{3 \div 5} = \frac{0.5}{0.6} \approx 0.83$

What error did Demi make? What is the correct answer?

A.

Demi divided numerators by numerators and denominators by denominators. Correct method: invert the divisor and multiply. $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$ .

B.

Demi's method is valid. The answer is approximately 0.83.

C.

Demi should have added the fractions: $\frac{2}{3} + \frac{4}{5} = \frac{22}{15}$ .

D.

Demi inverted the wrong fraction. She should have inverted $\frac{2}{3}$ instead.

2.

Leon computes $\frac{2}{3} \div \frac{3}{4}$ :
Step 1: Invert the first fraction (dividend): $\frac{3}{2}$
Step 2: Multiply: $\frac{3}{2} \times \frac{3}{4} = \frac{9}{8}$
Leon writes: "The answer is $\frac{9}{8}$ ."

What error did Leon make? What is the correct answer?

A.

Leon inverted the wrong fraction. He inverted the dividend ( $\frac{2}{3}$ ) instead of the divisor ( $\frac{3}{4}$ ). Correct: $\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$ .

B.

Leon's answer is correct.

C.

Leon should have inverted both fractions: $\frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2$ .

D.

Leon's method is correct, but he made an arithmetic error.

F

Challenge / Extension

1.

A strip of fabric is $\frac{5}{6}$ of a yard long. You cut it into pieces that are each $\frac{1}{4}$ of a yard long.
Number of full pieces: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
Fraction of a piece left over: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

full pieces:

fraction of piece remaining:

2.