Multiplying and Dividing Fractions: practice solutions, Grade 5 – download the PDF

1 Fractions as division ★★★

A fraction is a division, so \(a \div b = \dfrac{a}{b}\).

1. \(3 \div 8 = \dfrac{3}{8}\).
2. \(7 \div 4 = \dfrac{7}{4} = 1\tfrac{3}{4}\) because \(7 = 4 + 3\).
3. \(9 \div 5 = \dfrac{9}{5} = 1\tfrac{4}{5}\) because \(9 = 5 + 4\).

2 Whole number times fraction ★★★

1. \(4 \times \dfrac{1}{5} = \dfrac{4}{5}\).
2. \(3 \times \dfrac{2}{7} = \dfrac{6}{7}\).
3. \(5 \times \dfrac{3}{8} = \dfrac{15}{8} = 1\tfrac{7}{8}\).

3 Product of two fractions ★★★

Multiply the numerators, then the denominators.

1. \(\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}\).
2. \(\dfrac{2}{3} \times \dfrac{1}{5} = \dfrac{2}{15}\).
3. \(\dfrac{3}{4} \times \dfrac{1}{2} = \dfrac{3}{8}\).

4 Sharing juice ★★★

Sharing equally is a division: \(2 \div 3 = \dfrac{2}{3}\).

Each friend gets \(\dfrac{2}{3}\) of a liter.

5 Read an area model ★★★

The shaded columns are \(\dfrac{3}{5}\) of the width and the shaded row is \(\dfrac{1}{2}\) of the height. The overlap has 3 small rectangles out of \(5 \times 2 = 10\).

\(\dfrac{1}{2} \times \dfrac{3}{5} = \dfrac{3}{10}\).

6 Unit fraction divided by a whole number ★★★

Use \(\dfrac{1}{b} \div n = \dfrac{1}{b \times n}\).

1. \(\dfrac{1}{4 \times 3} = \dfrac{1}{12}\).
2. \(\dfrac{1}{2 \times 5} = \dfrac{1}{10}\).
3. \(\dfrac{1}{6 \times 2} = \dfrac{1}{12}\).

7 Whole number divided by a unit fraction ★★★

Use \(n \div \dfrac{1}{b} = n \times b\): each whole holds \(b\) pieces.

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

8 Greater, less, or equal? ★★★

1. Less than 8, because \(\dfrac{3}{5} < 1\).
2. Equal to 8, because \(\dfrac{7}{7} = 1\).
3. Greater than 8, because \(\dfrac{9}{4} > 1\) (it is \(2\tfrac{1}{4}\)).
4. Less than 8, because \(\dfrac{11}{12} < 1\).

9 Mixed number times fraction ★★★

1. \(1\tfrac{1}{2} = \dfrac{3}{2}\), so \(\dfrac{3}{2} \times \dfrac{2}{3} = \dfrac{6}{6} = 1\).
2. \(2\tfrac{1}{4} = \dfrac{9}{4}\), so \(\dfrac{9}{4} \times \dfrac{4}{5} = \dfrac{36}{20} = \dfrac{9}{5} = 1\tfrac{4}{5}\).

10 Baking batches ★★★

Five groups of \(\dfrac{3}{4}\): \(5 \times \dfrac{3}{4} = \dfrac{15}{4} = 3\tfrac{3}{4}\).

Five batches need \(3\tfrac{3}{4}\) cups of flour.

11 Part of a garden ★★★

1. \(\dfrac{2}{3} \times \dfrac{3}{5} = \dfrac{6}{15} = \dfrac{2}{5}\) of the garden.
2. \(\dfrac{2}{5} \times 40 = \dfrac{80}{5} = 16\). Tomatoes cover 16 square yards.

12 Area of a small rectangle ★★★

Area = length \(\times\) width \(= \dfrac{3}{4} \times \dfrac{2}{5} = \dfrac{6}{20} = \dfrac{3}{10}\).

The tile has an area of \(\dfrac{3}{10}\) square foot.

13 Cutting ribbon ★★★

\(\dfrac{1}{3} \div 4 = \dfrac{1}{3 \times 4} = \dfrac{1}{12}\).

Each piece is \(\dfrac{1}{12}\) yard long.

14 Scoops of flour ★★★

Each cup holds 4 scoops, so \(6 \div \dfrac{1}{4} = 6 \times 4 = 24\).

She can fill 24 scoops.

15 Two mixed numbers ★★★

Estimate: \(2\tfrac{1}{2} \times 1\tfrac{1}{3}\) is a bit more than \(2\tfrac{1}{2}\), so it should be between 3 and 4.

\(\dfrac{5}{2} \times \dfrac{4}{3} = \dfrac{20}{6} = \dfrac{10}{3} = 3\tfrac{1}{3}\). The result is between 3 and 4, so it makes sense.

16 Area of a patio ★★★

Convert: \(2\tfrac{1}{4} = \dfrac{9}{4}\) and \(1\tfrac{2}{3} = \dfrac{5}{3}\).

\(\dfrac{9}{4} \times \dfrac{5}{3} = \dfrac{45}{12} = \dfrac{15}{4} = 3\tfrac{3}{4}\).

The area is \(3\tfrac{3}{4}\) square feet.

17 Trail mix ★★★

Share: \(\dfrac{1}{2} \div 6 = \dfrac{1}{12}\) pound per hiker.

Eaten: \(\dfrac{3}{4} \times \dfrac{1}{12} = \dfrac{3}{48} = \dfrac{1}{16}\).

Each hiker eats \(\dfrac{1}{16}\) pound.

18 Find the error ★★★

She is wrong. She treated the problem as \(4 \div 5\). But \(\dfrac{1}{5}\) is a piece, and each whole holds 5 pieces of size \(\dfrac{1}{5}\).

The correct answer is \(4 \div \dfrac{1}{5} = 4 \times 5 = 20\). Check: \(20 \times \dfrac{1}{5} = 4\).

19 Rope pieces ★★★

1. \(12 \div \dfrac{1}{3} = 12 \times 3 = 36\) pieces.
2. \(\dfrac{2}{9} \times 36 = \dfrac{72}{9} = 8\) pieces.

20 A two-part hike ★★★

Morning: \(\dfrac{15}{4} \times \dfrac{2}{5} = \dfrac{30}{20} = \dfrac{3}{2}\) miles.

Remaining: \(\dfrac{15}{4} - \dfrac{3}{2} = \dfrac{15}{4} - \dfrac{6}{4} = \dfrac{9}{4}\) miles.

After lunch: \(\dfrac{1}{3} \times \dfrac{9}{4} = \dfrac{9}{12} = \dfrac{3}{4}\).

Sam hikes \(\dfrac{3}{4}\) mile after lunch.

21 Missing numbers ★★★

1. \(\dfrac{3}{8}\) is \(\dfrac{3}{4}\) cut in half, so \(\square = \dfrac{1}{2}\). Check: \(\dfrac{1}{2} \times \dfrac{3}{4} = \dfrac{3}{8}\).
2. The denominator went from 6 to 18, so the missing factor is \(\dfrac{1}{3}\). Check: \(\dfrac{5}{6} \times \dfrac{1}{3} = \dfrac{5}{18}\).
3. \(\dfrac{1}{20} = \dfrac{1}{5 \times 4}\), so \(\square = 4\). Check: \(\dfrac{1}{5} \div 4 = \dfrac{1}{20}\).

22 Does it grow or shrink? ★★★

\(\dfrac{7}{6} > 1\), so the first product is greater than \(\dfrac{3}{5}\); \(\dfrac{5}{6} < 1\), so the second is smaller.

\(\dfrac{3}{5} \times \dfrac{7}{6} = \dfrac{21}{30} = \dfrac{7}{10}\) and \(\dfrac{7}{10} > \dfrac{6}{10}\). \(\dfrac{3}{5} \times \dfrac{5}{6} = \dfrac{15}{30} = \dfrac{1}{2}\) and \(\dfrac{1}{2} < \dfrac{3}{5}\).