Equivalent Fractions and Comparing: practice solutions, Grade 4 – download the PDF

1 Name the fraction ★★★

1. The bar has 8 equal parts, so each part is \(\dfrac{1}{8}\) of the bar.
2. 5 parts are shaded: \(\dfrac{5}{8}\).
3. The other \(8-5=3\) parts are not shaded: \(\dfrac{3}{8}\). Check: \(\dfrac{5}{8}+\dfrac{3}{8}=\dfrac{8}{8}=1\) whole bar.

2 Pizza slices ★★★

1. One slice is \(\dfrac{1}{6}\) of the pizza.
2. Maya ate \(1+2=3\) slices, which is \(\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{6}=\dfrac{3}{6}\).
3. \(6-3=3\) slices are left, which is \(\dfrac{3}{6}\). Dividing the top and bottom by 3 gives \(\dfrac{3}{6}=\dfrac{1}{2}\), so half the pizza is left.

3 Fill in the missing number ★★★

1. \(2\times 3=6\), so multiply the top by 3: \(\dfrac{1\times 3}{2\times 3}=\dfrac{3}{6}\).
2. \(3\times 4=12\), so \(\dfrac{2\times 4}{3\times 4}=\dfrac{8}{12}\).
3. \(5\times 2=10\), so \(\dfrac{3\times 2}{5\times 2}=\dfrac{6}{10}\).
4. \(4\times 25=100\), so \(\dfrac{3\times 25}{4\times 25}=\dfrac{75}{100}\).

4 Name the points ★★★

1. The line is cut into 6 equal lengths. A is 1 jump from 0: \(\dfrac{1}{6}\). B is 3 jumps: \(\dfrac{3}{6}\). C is 5 jumps: \(\dfrac{5}{6}\).
2. Divide 3 and 6 by 3: \(\dfrac{3}{6}=\dfrac{1}{2}\). Point B is halfway between 0 and 1.

5 True or false? ★★★

1. True. Divide the top and bottom of \(\dfrac{2}{4}\) by 2 to get \(\dfrac{1}{2}\).
2. False. \(\dfrac{3}{6}=\dfrac{1}{2}\), not \(\dfrac{1}{3}\). Three sixths is half of the whole.
3. True. \(\dfrac{4\div 2}{10\div 2}=\dfrac{2}{5}\).
4. False. \(\dfrac{2}{3}=\dfrac{8}{12}\) and \(\dfrac{3}{4}=\dfrac{9}{12}\), and \(8\ne 9\).

6 Same denominator or same numerator ★★★

1. Same denominator, and \(2\lt 4\): \(\dfrac{2}{5}\lt\dfrac{4}{5}\).
2. Six sixths make one whole: \(\dfrac{6}{6}=1\).
3. Same numerator. Thirds are bigger pieces than eighths, so \(\dfrac{1}{3}\gt\dfrac{1}{8}\).
4. Same denominator, and \(5\gt 3\): \(\dfrac{5}{8}\gt\dfrac{3}{8}\).

7 Above or below one half ★★★

Compare each numerator with half of its denominator.

• \(\dfrac{3}{8}\): half of 8 is 4 and \(3\lt 4\), so less than one half.
• \(\dfrac{5}{6}\): half of 6 is 3 and \(5\gt 3\), so greater.
• \(\dfrac{4}{8}\): half of 8 is 4, so equal to one half.
• \(\dfrac{2}{5}\): half of 5 is 2.5 and \(2\lt 2.5\), so less.
• \(\dfrac{7}{12}\): half of 12 is 6 and \(7\gt 6\), so greater.
• \(\dfrac{1}{10}\): half of 10 is 5 and \(1\lt 5\), so less.

Less: \(\dfrac{3}{8}\), \(\dfrac{2}{5}\), \(\dfrac{1}{10}\). Equal: \(\dfrac{4}{8}\). Greater: \(\dfrac{5}{6}\), \(\dfrac{7}{12}\).

8 Two equivalent fractions ★★★

1. \(\dfrac{3\times 2}{4\times 2}=\dfrac{6}{8}\) and \(\dfrac{3\times 3}{4\times 3}=\dfrac{9}{12}\).
2. \(\dfrac{2\times 2}{5\times 2}=\dfrac{4}{10}\) and \(\dfrac{2\times 20}{5\times 20}=\dfrac{40}{100}\).

9 Simplify ★★★

1. Divide by 2: \(\dfrac{3}{4}\).
2. Divide by 2: \(\dfrac{5}{6}\). Only 1 divides both 5 and 6.
3. Divide by 4: \(\dfrac{2}{3}\) (or divide by 2 twice).
4. Divide by 25: \(\dfrac{1}{4}\).

10 Oatmeal scoops ★★★

Write \(\dfrac{2}{3}\) in sixths: \(\dfrac{2}{3}=\dfrac{2\times 2}{3\times 2}=\dfrac{4}{6}\). That is 4 copies of \(\dfrac{1}{6}\), so Lena needs 4 scoops.

For \(\dfrac{1}{3}=\dfrac{2}{6}\) she needs 2 scoops.

11 Fractions with unlike denominators ★★★

1. Twelfths: \(\dfrac{8}{12}\) and \(\dfrac{9}{12}\). Since \(8\lt 9\): \(\dfrac{2}{3}\lt\dfrac{3}{4}\).
2. Tenths: \(\dfrac{6}{10}\) and \(\dfrac{7}{10}\). Since \(6\lt 7\): \(\dfrac{3}{5}\lt\dfrac{7}{10}\).
3. Twelfths: \(\dfrac{10}{12}\) and \(\dfrac{9}{12}\). Since \(10\gt 9\): \(\dfrac{5}{6}\gt\dfrac{3}{4}\).
4. Eighths: \(\dfrac{3}{8}\) and \(\dfrac{2}{8}\). Since \(3\gt 2\): \(\dfrac{3}{8}\gt\dfrac{1}{4}\).

12 Two number lines ★★★

1. A is \(\dfrac{1}{4}\), B is \(\dfrac{3}{4}\), C is \(\dfrac{2}{8}\), D is \(\dfrac{5}{8}\).
2. A and C are at the same place: \(\dfrac{1}{4}=\dfrac{2}{8}\).
3. B is at \(\dfrac{3}{4}=\dfrac{6}{8}\). D at \(\dfrac{5}{8}\) is to the left of B, so \(\dfrac{5}{8}\lt\dfrac{3}{4}\).

13 Who ran farther? ★★★

Use eighths: \(\dfrac{3}{4}=\dfrac{6}{8}\). Compare \(\dfrac{6}{8}\) and \(\dfrac{5}{8}\): \(6\gt 5\).

Leo ran farther, by \(\dfrac{6}{8}-\dfrac{5}{8}=\dfrac{1}{8}\) of a mile.

14 Least to greatest ★★★

Use sixths: \(\dfrac{1}{2}=\dfrac{3}{6}\), \(\dfrac{1}{3}=\dfrac{2}{6}\), \(\dfrac{1}{6}\), \(\dfrac{5}{6}\).

The numerators in order are 1, 2, 3, 5, so \(\dfrac{1}{6}\lt\dfrac{1}{3}\lt\dfrac{1}{2}\lt\dfrac{5}{6}\).

15 Find the unknown number ★★★

1. \(5\times 20=100\), so \(4\times 20=80\): \(\dfrac{80}{100}\).
2. \(12\div 3=4\), so \(9\div 3=3\): \(\dfrac{3}{4}\).
3. \(3\times 3=9\), so \(8\times 3=24\): \(\dfrac{9}{24}\).
4. \(7\times 3=21\), so \(10\times 3=30\): \(\dfrac{21}{30}\).

16 Find the mistake ★★★

Ben is wrong. Adding the same number to the top and bottom does not keep the amount. The rule needs multiplying or dividing.

A correct fraction: \(\dfrac{2\times 2}{3\times 2}=\dfrac{4}{6}\).

To compare, use fifteenths: \(\dfrac{2}{3}=\dfrac{10}{15}\) and \(\dfrac{4}{5}=\dfrac{12}{15}\). Since \(12\gt 10\), \(\dfrac{4}{5}\gt\dfrac{2}{3}\), so they are not equal.

17 Which equal one half? ★★★

A fraction equals \(\dfrac{1}{2}\) when the denominator is twice the numerator.

• \(\dfrac{5}{10}\): \(2\times 5=10\), equal.
• \(\dfrac{6}{12}\): \(2\times 6=12\), equal.
• \(\dfrac{50}{100}\): \(2\times 50=100\), equal.
• \(\dfrac{4}{9}\): \(2\times 4=8\lt 9\), so \(\dfrac{4}{9}\lt\dfrac{1}{2}\).
• \(\dfrac{3}{5}\): \(2\times 3=6\gt 5\), so \(\dfrac{3}{5}\gt\dfrac{1}{2}\).

18 Ribbons for a craft ★★★

1. Use twelfths: red \(=\dfrac{10}{12}\), blue \(=\dfrac{9}{12}\), green \(=\dfrac{7}{12}\). Since \(7\lt 9\lt 10\): green, then blue, then red. That is \(\dfrac{7}{12}\lt\dfrac{3}{4}\lt\dfrac{5}{6}\).
2. \(\dfrac{2}{3}=\dfrac{8}{12}\). Red (10) and blue (9) are greater than 8, but green (7) is not. The red and blue ribbons are longer than \(\dfrac{2}{3}\) yard.

19 Same numerator ★★★

Both fractions count 3 pieces. Fifths are bigger pieces than eighths because the whole is cut into fewer parts. Three big pieces are more than three small ones, so \(\dfrac{3}{5}\gt\dfrac{3}{8}\).

The numerator is always 7. The bigger the denominator, the smaller the pieces, so the order is \(\dfrac{7}{12}\lt\dfrac{7}{10}\lt\dfrac{7}{8}\).

20 Phone batteries ★★★

Use tenths: A is \(\dfrac{4}{10}\); B is \(\dfrac{3\times 2}{5\times 2}=\dfrac{6}{10}\); C is \(\dfrac{1\times 5}{2\times 5}=\dfrac{5}{10}\).

Since \(4\lt 5\lt 6\), the order is A, then C, then B. Phone B has the most charge.

21 Vegetable garden ★★★

1. Peppers: \(\dfrac{1}{2}=\dfrac{5}{10}\), the greatest of 4, 5 and 1 tenths.
2. \(\dfrac{4}{10}\lt\dfrac{1}{2}\), because \(\dfrac{4}{10}\lt\dfrac{5}{10}\).
3. \(1-\dfrac{5}{10}=\dfrac{5}{10}=\dfrac{1}{2}\).
4. \(\dfrac{4}{10}+\dfrac{5}{10}+\dfrac{1}{10}=\dfrac{10}{10}=1\), so the whole garden is used.

22 Fractions in between ★★★

1. \(\dfrac{1}{2}=\dfrac{4}{8}\) and \(1=\dfrac{8}{8}\). The numerators between 4 and 8 are 5, 6 and 7: \(\dfrac{5}{8}\), \(\dfrac{6}{8}\), \(\dfrac{7}{8}\).
2. \(\dfrac{1}{3}=\dfrac{4}{12}\) and \(\dfrac{2}{3}=\dfrac{8}{12}\). The numerators between 4 and 8 are 5, 6 and 7: \(\dfrac{5}{12}\), \(\dfrac{6}{12}\), \(\dfrac{7}{12}\).