Dividing Fractions: practice solutions, Grade 6 – download the PDF

1 Counting pieces ★★★

Each whole contains as many pieces as the denominator says.

1. \( 6 \times 2 = 12 \)
2. \( 3 \times 5 = 15 \)
3. \( 8 \times 4 = 32 \)

Check (a): \( 12 \times \dfrac{1}{2} = 6 \).

2 Reciprocals ★★★

1. Swap: \( \dfrac{5}{3} \).
2. \( 7 = \dfrac{7}{1} \), so the reciprocal is \( \dfrac{1}{7} \).
3. \( 9 \), because \( \dfrac{1}{9} \times 9 = 1 \).
4. \( 2\dfrac{1}{2} = \dfrac{5}{2} \), so the reciprocal is \( \dfrac{2}{5} \).

3 Sharing a fraction ★★★

1. \( \dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6} \)
2. \( \dfrac{3}{4} \times \dfrac{1}{2} = \dfrac{3}{8} \)
3. \( \dfrac{6}{7} \div 3 = \dfrac{2}{7} \) (6 sevenths in 3 groups gives 2 sevenths).

4 Loaves of bread ★★★

\( 2 \div \dfrac{1}{4} = 2 \times 4 = 8 \).

She can make 8 loaves. Check: \( 8 \times \dfrac{1}{4} = 2 \) pounds.

5 True or false? ★★★

1. True: \( 3 \times 6 = 18 \).
2. False: \( \dfrac{1}{3} \div 2 = \dfrac{1}{3} \times \dfrac{1}{2} = \dfrac{1}{6} \). The quotient must be smaller than \( \dfrac{1}{3} \), but \( \dfrac{2}{3} \) is larger.
3. True: \( \dfrac{4}{9} \times \dfrac{9}{4} = 1 \).

6 Using the bar model ★★★

The 4 colored parts are each \( \dfrac{1}{6} \), so \( \dfrac{2}{3} \div \dfrac{1}{6} = 4 \).

Check by the rule: \( \dfrac{2}{3} \times 6 = 4 \). Four pieces fit.

7 Garden hose breaks ★★★

\( 9 \div \dfrac{3}{4} = 9 \times \dfrac{4}{3} = \dfrac{36}{3} = 12 \).

Maya gets 12 pieces. Check: \( 12 \times \dfrac{3}{4} = 9 \) feet.

8 Fraction divided by a fraction ★★★

\( \dfrac{3}{5} \times \dfrac{10}{3} = \dfrac{30}{15} = 2 \).

Check: \( 2 \times \dfrac{3}{10} = \dfrac{6}{10} = \dfrac{3}{5} \). The answer is 2.

9 Three quotients ★★★

1. \( \dfrac{5}{6} \times \dfrac{3}{2} = \dfrac{15}{12} = \dfrac{5}{4} \)
2. \( \dfrac{7}{8} \times 4 = \dfrac{28}{8} = \dfrac{7}{2} \)
3. \( \dfrac{4}{9} \times \dfrac{3}{8} = \dfrac{12}{72} = \dfrac{1}{6} \)

10 Find the mistake ★★★

Mia multiplied by 3 instead of by the reciprocal \( \dfrac{1}{3} \). Sharing \( \dfrac{2}{5} \) into 3 parts must give less than \( \dfrac{2}{5} \), but \( \dfrac{6}{5} \) is more than 1.

Correct: \( \dfrac{2}{5} \div 3 = \dfrac{2}{5} \times \dfrac{1}{3} = \dfrac{2}{15} \).

11 Ribbon for gift bows ★★★

\( \dfrac{9}{10} \div \dfrac{3}{20} = \dfrac{9}{10} \times \dfrac{20}{3} = \dfrac{180}{30} = 6 \).

Six bows can be made. Each bow uses \( \dfrac{3}{20} \) m \( = 0.15 \) m \( = 15 \) cm.

12 Jumps on a number line ★★★

The jumps land on \( \dfrac{2}{3} \), \( \dfrac{4}{3} \) and 2, so there are 3 jumps.

Check: \( 2 \times \dfrac{3}{2} = 3 \), and \( 3 \times \dfrac{2}{3} = 2 \). So \( 2 \div \dfrac{2}{3} = 3 \).

13 Mixed numbers ★★★

1. \( \dfrac{3}{2} \times \dfrac{4}{3} = \dfrac{12}{6} = 2 \)
2. \( \dfrac{9}{4} \times \dfrac{2}{3} = \dfrac{18}{12} = \dfrac{3}{2} \)
3. \( 3 \times \dfrac{2}{3} = 2 \)

14 Missing factor ★★★

1. \( \square = 6 \div \dfrac{2}{5} = 6 \times \dfrac{5}{2} = 15 \).
2. \( \square = \dfrac{9}{10} \div \dfrac{3}{4} = \dfrac{9}{10} \times \dfrac{4}{3} = \dfrac{36}{30} = \dfrac{6}{5} \).

15 Pizza night ★★★

\( \dfrac{5}{6} \div 5 = \dfrac{5}{6} \times \dfrac{1}{5} = \dfrac{1}{6} \).

Each friend eats \( \dfrac{1}{6} \) of the pizza.

16 Hiking speed ★★★

Speed = distance \( \div \) time \( = \dfrac{3}{4} \div \dfrac{1}{6} = \dfrac{3}{4} \times 6 = \dfrac{18}{4} = \dfrac{9}{2} \).

Her speed is \( 4\dfrac{1}{2} \) mph (4.5 mph, about 7.2 km/h). That is a brisk but realistic pace.

17 Predict, then compute ★★★

1. The divisor is less than 1, so the quotient is greater than 5: \( 5 \times \dfrac{4}{3} = \dfrac{20}{3} = 6\dfrac{2}{3} \).
2. The divisor \( \dfrac{5}{4} \) is greater than 1, so the quotient is less than \( \dfrac{2}{3} \): \( \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15} \).
3. The divisor is 1, so the quotient is equal: \( \dfrac{7}{8} \).

18 Height of a fish tank ★★★

Height = volume \( \div \) base area \( = \dfrac{9}{8} \div \dfrac{3}{4} = \dfrac{9}{8} \times \dfrac{4}{3} = \dfrac{36}{24} = \dfrac{3}{2} \).

The water is \( \dfrac{3}{2} \) ft \( = 1.5 \) ft \( = 18 \) inches high. Check: \( \dfrac{3}{4} \times \dfrac{3}{2} = \dfrac{9}{8} \).

19 Friendship bracelets ★★★

1. \( \dfrac{15}{2} \div \dfrac{3}{4} = \dfrac{15}{2} \times \dfrac{4}{3} = \dfrac{60}{6} = 10 \). She can make 10 bracelets.
2. Total cord: \( \dfrac{15}{2} + \dfrac{9}{4} = \dfrac{30}{4} + \dfrac{9}{4} = \dfrac{39}{4} \) m. Then \( \dfrac{39}{4} \div \dfrac{3}{4} = 13 \). She can make 13 bracelets.

20 Write your own problem ★★★

Sample problem: a bottle holds \( \dfrac{4}{5} \) liter of paint, and each craft project uses \( \dfrac{1}{10} \) liter. How many projects can be painted?

\( \dfrac{4}{5} \div \dfrac{1}{10} = \dfrac{4}{5} \times 10 = 8 \). Check: \( 8 \times \dfrac{1}{10} = \dfrac{8}{10} = \dfrac{4}{5} \). The answer is 8 projects. Any context works as long as the question asks how many pieces of size \( \dfrac{1}{10} \) fit in \( \dfrac{4}{5} \).

21 Find the unknown number ★★★

1. \( x \div \dfrac{2}{3} = \dfrac{9}{4} \), so \( x = \dfrac{9}{4} \times \dfrac{2}{3} = \dfrac{18}{12} = \dfrac{3}{2} \). Check: \( \dfrac{3}{2} \times \dfrac{3}{2} = \dfrac{9}{4} \).
2. \( \dfrac{5}{6} \div n = \dfrac{5}{3} \), so \( n = \dfrac{5}{6} \div \dfrac{5}{3} = \dfrac{5}{6} \times \dfrac{3}{5} = \dfrac{1}{2} \). Check: \( \dfrac{5}{6} \div \dfrac{1}{2} = \dfrac{5}{3} \).