Fractions: add, subtract, multiply and divide – Year 6 maths worksheet answers to download as PDF

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1. Answer to Question 1

   1. The highest common factors are 6, 20, 12 and 15. So the answers are \(\frac{3}{4}\), \(\frac{2}{5}\), \(\frac{3}{4}\) and \(\frac{3}{4}\).
   2. \(\frac{2}{3}=\frac{8}{12}\) (multiply by 4). \(\frac{3}{5}=\frac{18}{30}\) (multiply by 6). \(\frac{7}{8}=\frac{35}{40}\) (multiply by 5). The missing numbers are 8, 30 and 35.
   3. Change to a common denominator of 24: \(\frac{5}{8}=\frac{15}{24}\) and \(\frac{2}{3}=\frac{16}{24}\). So \(\frac{2}{3}\) is bigger, by \(\frac{1}{24}\).

2. Answer to Question 2

   1. The common denominator is 12: \(\frac{3}{12}+\frac{8}{12}=\) \(\frac{11}{12}\).
   2. The common denominator is 12: \(\frac{10}{12}-\frac{3}{12}=\) \(\frac{7}{12}\).
   3. The common denominator is 10: \(\frac{6}{10}+\frac{7}{10}=\frac{13}{10}=\) \(1\frac{3}{10}\).
   4. The common denominator is 8: \(\frac{7}{8}-\frac{6}{8}=\) \(\frac{1}{8}\).
   5. The common denominator is 18: \(\frac{4}{18}+\frac{3}{18}=\) \(\frac{7}{18}\).

3. Answer to Question 3

   1. Add the wholes: 2 + 1 = 3. Add the fractions with denominator 4: \(\frac{2}{4}+\frac{3}{4}=\frac{5}{4}=1\frac{1}{4}\). Total: \(3+1\frac{1}{4}=\) \(4\frac{1}{4}\).
   2. Wholes: 3 + 1 = 4. Tenths: \(\frac{4}{10}+\frac{7}{10}=\frac{11}{10}=1\frac{1}{10}\). Total \(5\frac{1}{10}\).
   3. Wholes: 1 + 2 = 3. Twelfths: \(\frac{8}{12}+\frac{9}{12}=\frac{17}{12}=1\frac{5}{12}\). Total \(4\frac{5}{12}\).
   4. Wholes: 4 + 2 = 6. Sixths: \(\frac{5}{6}+\frac{2}{6}=\frac{7}{6}=1\frac{1}{6}\). Total \(7\frac{1}{6}\).

4. Answer to Question 4

   1. Common denominator 6: \(3\frac{2}{6}-1\frac{5}{6}\). Write \(3\frac{2}{6}\) as \(2\frac{8}{6}\). Then \(2-1=1\) and \(\frac{8}{6}-\frac{5}{6}=\frac{3}{6}\). Answer \(1\frac{1}{2}\).
   2. Common denominator 10: \(4\frac{2}{10}-1\frac{7}{10}\). Write \(4\frac{2}{10}\) as \(3\frac{12}{10}\). Then \(2\) and \(\frac{5}{10}\). Answer \(2\frac{1}{2}\).
   3. Write 5 as \(4\frac{8}{8}\). Then \(4-2=2\) and \(\frac{8}{8}-\frac{3}{8}=\frac{5}{8}\). Answer \(2\frac{5}{8}\).
   4. Write \(6\frac{1}{4}\) as \(5\frac{5}{4}\) and \(2\frac{1}{2}\) as \(2\frac{2}{4}\). Then \(3\) and \(\frac{3}{4}\). Answer \(3\frac{3}{4}\).

5. Answer to Question 5

   1. Multiply the numerators and multiply the denominators: \(\frac{1\times1}{2\times3}=\) \(\frac{1}{6}\). Half of a third is a sixth.
   2. Multiply the numerators and the denominators: \(\frac{2\times3}{3\times4}=\frac{6}{12}\), which simplifies to \(\frac{1}{2}\). You can also cancel the 3s first to get \(\frac{2}{4}=\frac{1}{2}\).
   3. Cancel 4 with 8 (giving 1 and 2) and 3 with 9 (giving 1 and 3): \(\frac{1}{2}\times\frac{1}{3}=\) \(\frac{1}{6}\).
   4. Find one quarter, then three quarters: \(20\div4=5\) and \(5\times3=\) 15.
   5. One sixth of 48 is \(48\div6=8\), so five sixths is \(8\times5=\) 40.

6. Answer to Question 6

   1. Dividing by 3 shares each half into 3 equal parts, so the pieces become sixths: \(\frac{1}{2\times3}=\) \(\frac{1}{6}\).
   2. Multiply the denominator by 2: \(\frac{3}{4\times2}=\) \(\frac{3}{8}\). Half of three quarters is three eighths.
   3. The numerator 4 divides exactly by 4, so the quickest way is \(4\div4=1\): the answer is \(\frac{1}{5}\) (the same as \(\frac{4}{20}\) simplified).
   4. 2 does not divide by 5, so multiply the denominator: \(\frac{2}{3\times5}=\) \(\frac{2}{15}\).
   5. 6 divides by 3, so divide the numerator: \(6\div3=2\), giving \(\frac{2}{7}\). Check: \(\frac{2}{7}\times3=\frac{6}{7}\).

7. Answer to Question 7

   1. Sam: \(12\div4=3\) slices. Mia: \(12\div3=4\) slices. Dad: \(12\div6=2\) slices.
   2. \(\frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\) \(\frac{7}{12}\).
   3. \(\frac{7}{12}+\frac{2}{12}=\frac{9}{12}\) eaten. Left: \(1-\frac{9}{12}=\frac{3}{12}=\) \(\frac{1}{4}\).
   4. \(\frac{1}{4}\div2=\) \(\frac{1}{8}\) of the whole pizza each. (3 slices shared by 2 means 1.5 slices each, and \(\frac{1.5}{12}=\frac{1}{8}\).)

8. Answer to Question 8

   1. \(\frac{3}{4}\div6=\frac{3}{24}=\frac{1}{8}\) of a litre. One litre is 1 000 ml, so \(1\,000\div8=\) 125 ml in each glass.
   2. \(\frac{2}{3}\div4=\frac{2}{12}\), which simplifies to \(\frac{1}{6}\) of a litre (about 167 ml).
   3. Use the common denominator 12: \(\frac{9}{12}+\frac{8}{12}=\frac{17}{12}\). As a mixed number this is \(1\frac{5}{12}\) litres.
   4. \(2=1\frac{12}{12}\), so \(1\frac{12}{12}-1\frac{5}{12}=\) \(\frac{7}{12}\) of a litre more.

9. Answer to Question 9

   1. \(\frac{3}{5}\) of 40: \(40\div5=8\) and \(8\times3=\) 24 children.
   2. \(\frac{1}{4}\) of 24 = 6 children.
   3. \(\frac{3}{5}\times\frac{1}{4}=\) \(\frac{3}{20}\). Check: \(\frac{3}{20}\) of 40 = 6.
   4. \(\frac{1}{8}\) of 40 = 5 children. Fraction by other means: \(1-\frac{3}{5}-\frac{1}{8}=\frac{40}{40}-\frac{24}{40}-\frac{5}{40}=\) \(\frac{11}{40}\).

10. Answer to Question 10

    1. Write the mixed number as an improper fraction: \(2\frac{1}{2}=\frac{5}{2}\). Then \(\frac{5}{2}\times\frac{3}{4}=\frac{15}{8}=\) \(1\frac{7}{8}\) kg.
    2. \(2=1\frac{8}{8}\), so \(1\frac{8}{8}-1\frac{7}{8}=\) \(\frac{1}{8}\) kg, which is 1 000 ÷ 8 = 125 g.
    3. \(\frac{1}{3}\times\frac{15}{8}=\frac{15}{24}\). Divide top and bottom by 3: \(\frac{5}{8}\) kg (625 g).

11. Answer to Question 11

    1. \(\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\), so lawn \(=1-\frac{5}{6}=\) \(\frac{1}{6}\).
    2. Vegetables: 60 m². Flowers: 40 m². Lawn: 20 m².
    3. \(\frac{3}{4}\) of 60 = 45 m².
    4. \(\frac{1}{4}\times\frac{1}{6}=\) \(\frac{1}{24}\) of the garden, which is 5 m².

12. Answer to Question 12

    1. Change to eighths: \(2\frac{6}{8}+1\frac{5}{8}\). Wholes: 3. Eighths: \(\frac{11}{8}=1\frac{3}{8}\). Total \(4\frac{3}{8}\) km.
    2. \(3\frac{1}{2}=3\frac{4}{8}\). Wholes: 4 + 3 = 7; eighths: \(\frac{3}{8}+\frac{4}{8}=\frac{7}{8}\). Total \(7\frac{7}{8}\) km.
    3. Count on from \(7\frac{7}{8}\): \(\frac{1}{8}\) more makes 8, and 2 more make 10. He needs \(2\frac{1}{8}\) km more.
    4. \(\frac{1}{5}\) of 10 km is 2 km, so \(\frac{2}{5}\) is 2 × 2 = 4 km.

13. Answer to Question 13

    1. \(\frac{1}{5}\) of £45 is \(45\div5=9\), so \(\frac{2}{5}\) is \(9\times2=\) £18.
    2. \(\frac{1}{3}\) of £45 is \(45\div3=\) £15.
    3. Common denominator 15: \(\frac{6}{15}+\frac{5}{15}=\) \(\frac{11}{15}\) of her money.
    4. \(45-18-15=\) £12 left. As a fraction: \(1-\frac{11}{15}=\) \(\frac{4}{15}\). Check: \(45\div15=3\) and \(3\times4=12\).

14. Answer to Question 14

    1. Three bows of \(\frac{3}{4}\) m: \(3\times\frac{3}{4}=\frac{9}{4}=\) \(2\frac{1}{4}\) m.
    2. \(3\frac{1}{2}=3\frac{2}{4}\), so \(3\frac{2}{4}-2\frac{1}{4}=\) \(1\frac{1}{4}\) m.
    3. \(1\frac{1}{4}=\frac{5}{4}\), and \(\frac{5}{4}\div5=\) \(\frac{1}{4}\) m, which is 25 cm.
    4. No. A bow needs \(\frac{3}{4}\) m but each piece is only \(\frac{1}{4}\) m, three times too short. She would need three pieces joined together.

15. Answer to Question 15

    1. Sprint: \(120\div4=30\). Relay: \(120\div3=40\). Long jump: \(120\div6=20\).
    2. \(\frac{1}{4}+\frac{1}{3}+\frac{1}{6}=\frac{3}{12}+\frac{4}{12}+\frac{2}{12}=\frac{9}{12}\). The hurdles are \(1-\frac{9}{12}=\frac{3}{12}=\) \(\frac{1}{4}\).
    3. \(\frac{1}{4}\) of 120 = 30 children. Check: 120 − 30 − 40 − 20 = 30.
    4. \(\frac{3}{5}\) of 40: \(40\div5=8\), \(8\times3=\) 24 children.
    5. Sprint girls: \(\frac{1}{2}\) of 30 = 15. Long jump girls: \(\frac{2}{5}\) of 20 = 8. Total 23 girls.
    6. Sprint and relay together are \(\frac{1}{4}+\frac{1}{3}=\frac{7}{12}\), and \(\frac{7}{12}\) is more than \(\frac{6}{12}=\frac{1}{2}\). She is right. (30 + 40 = 70 children, which is more than 60.)

16. Answer to Question 16

    1. Saturday: \(240\div8=30\) and \(30\times3=90\) pages. Sunday: \(240\div4=60\) pages.
    2. \(\frac{3}{8}+\frac{1}{4}=\frac{3}{8}+\frac{2}{8}=\) \(\frac{5}{8}\).
    3. \(1-\frac{5}{8}=\) \(\frac{3}{8}\), which is \(240-150=\) 90 pages.
    4. \(\frac{1}{3}\) of 90 = 30 pages. As a fraction of the whole book: \(\frac{1}{3}\times\frac{3}{8}=\) \(\frac{1}{8}\).
    5. \(90-30=\) 60 pages. As a fraction: \(\frac{3}{8}-\frac{1}{8}=\frac{2}{8}=\) \(\frac{1}{4}\) of the book.

17. Answer to Question 17

    1. \(\frac{3}{4}\times5\frac{1}{2}=\frac{3}{4}\times\frac{11}{2}=\frac{33}{8}=\) \(4\frac{1}{8}\) litres.
    2. \(5\frac{1}{2}-4\frac{1}{8}=5\frac{4}{8}-4\frac{1}{8}=\) \(1\frac{3}{8}\) litres.
    3. \(3\frac{3}{4}-1\frac{1}{2}=3\frac{3}{4}-1\frac{2}{4}=\) \(2\frac{1}{4}\) litres.
    4. \(1\frac{3}{8}+2\frac{1}{4}=1\frac{3}{8}+2\frac{2}{8}=3\frac{5}{8}\), so \(3\frac{5}{8}\) litres.
    5. There are \(2\frac{1}{4}=\frac{9}{4}\) litres of milk. Each bottle is \(\frac{1}{4}\) litre, so \(9\) bottles. She fills 9 bottles.

18. Answer to Question 18

    1. No. Ali has added the numerators and the denominators. The common denominator is 12: \(\frac{4}{12}+\frac{3}{12}=\) \(\frac{7}{12}\). (A sum of two fractions cannot be smaller than \(\frac{1}{3}\), and \(\frac{2}{7}\) is smaller.)
    2. Yes. \(3\times2=6\), \(4\times5=20\), and \(\frac{6}{20}=\frac{3}{10}\) when divided by 2. Bea is right.
    3. No. Chen multiplied by 4 instead of dividing. Dividing by 4 makes the fraction smaller: \(\frac{2}{3\times4}=\frac{2}{12}=\) \(\frac{1}{6}\).
    4. Dee subtracted the smaller fraction from the bigger fraction instead of the other way round. \(3\frac{1}{4}=2\frac{5}{4}\) and \(1\frac{1}{2}=1\frac{2}{4}\), so the answer is \(1\frac{3}{4}\). The correct answer is \(1\frac{3}{4}\).

19. Answer to Question 19

    1. \(240\div8=30\) and \(30\times5=\) 150 litres.
    2. \(\frac{2}{5}\) of 150: \(150\div5=30\), \(30\times2=\) 60 litres.
    3. \(150-60=\) 90 litres.
    4. \(\frac{1}{6}\) of 240 = 40 litres.
    5. \(90+40=130\) litres. As a fraction: \(\frac{130}{240}=\) \(\frac{13}{24}\). Half of the tank is 120 litres, so it is more than half full.

20. Answer to Question 20

    1. \(1\frac{1}{12}=\frac{13}{12}\). \(\frac{13}{12}-\frac{3}{12}=\frac{10}{12}=\) \(\frac{5}{6}\).
    2. \(\frac{5}{6}\times\frac{3}{5}=\frac{15}{30}=\) \(\frac{1}{2}\).
    3. \(\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\), so the third fraction is \(1-\frac{5}{6}=\) \(\frac{1}{6}\). It is a unit fraction because its numerator is 1.