Hard GCSE Fraction arithmetic Questions

Challenging, exam-style GCSE Fraction arithmetic questions with worked solutions. Stretch yourself on the hardest mixed numbers, multiplying fractions, cancelling, dividing fractions problems.

mixed numbersmultiplying fractionscancellingdividing fractionssubtracting fractionsadding fractions
GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
Work out 313−112×1193\frac{1}{3} - 1\frac{1}{2} \times 1\frac{1}{9}. Give your answer as a mixed number.
Show worked solution

Worked solution

  1. Apply the order of operations

    313−112×1193\frac{1}{3} - 1\frac{1}{2} \times 1\frac{1}{9}

    Multiplication comes before subtraction, so multiply first.

  2. Convert the first factor

    112=321\frac{1}{2} = \frac{3}{2}

    11 times 22 plus 11 is 33, so it is three halves.

  3. Convert the second factor

    119=1091\frac{1}{9} = \frac{10}{9}

    11 times 99 plus 11 is 1010, so it is ten ninths.

  4. Multiply

    32×109=3018\frac{3}{2} \times \frac{10}{9} = \frac{30}{18}

    Multiply the tops (3×10=30)(3\times 10=30) and the bottoms (2×9=18)(2\times 9=18).

  5. Simplify the product

    3018=53\frac{30}{18} = \frac{5}{3}

    Divide top and bottom by 66.

  6. Convert the first number

    313=1033\frac{1}{3} = \frac{10}{3}

    33 times 33 plus 11 is 1010, so it is ten thirds.

  7. Set up the subtraction

    103−53\frac{10}{3} - \frac{5}{3}

    Both fractions are already in thirds.

  8. Subtract the thirds

    103−53=53\frac{10}{3} - \frac{5}{3} = \frac{5}{3}

    Subtract the tops: 10−5=510 - 5 = 5.

  9. Convert to a mixed number

    53=123\frac{5}{3} = 1\frac{2}{3}

    Three thirds make one whole, leaving two thirds.

  10. Sense check with a decimal

    313−112×119≈1.6673\frac{1}{3} - 1\frac{1}{2} \times 1\frac{1}{9} \approx 1.667

    As a decimal this is about 1.6671.667, which is the right size — a quick reassurance that the answer is reasonable.

  11. Watch for the common error

    (313−112)×119≠123\left(3\frac{1}{3} - 1\frac{1}{2}\right) \times 1\frac{1}{9} \ne 1\frac{2}{3}

    Subtracting before multiplying breaks the order of operations.

  12. Notice the answer is exact

    53\frac{5}{3}

    Every step used exact fractions, so the answer is exact with no rounding.

  13. Sense check the size

    313−123≈1.673\frac{1}{3} - 1\frac{2}{3} \approx 1.67

    Taking about 1.671.67 from 3.333.33 leaves about 1.671.67, which matches one and two thirds.

  14. Reflect on the method

    multiply, then subtract

    Doing the multiplication first was essential to reach the correct answer.

  15. State the answer

    1231\frac{2}{3}

    The answer is 1231\frac{2}{3}.

Answer
1231\frac{2}{3}
Question 2
6 markschallenging
34\frac{3}{4} of a number, with 12\frac{1}{2} subtracted, gives 11. Work out the number.
Show worked solution

Worked solution

  1. Write the equation

    34n−12=1\frac{3}{4}n - \frac{1}{2} = 1

    Let n be the number. Three quarters of it, minus one half, equals 11.

  2. Undo the subtraction

    34n=1+12\frac{3}{4}n = 1 + \frac{1}{2}

    Add one half to both sides to isolate the term with n.

  3. Add the right-hand side

    1+12=321 + \frac{1}{2} = \frac{3}{2}

    One and a half is three halves.

  4. Rewrite

    34n=32\frac{3}{4}n = \frac{3}{2}

    So three quarters of n is three halves.

  5. Undo the multiplication

    n=32÷34n = \frac{3}{2} \div \frac{3}{4}

    Divide both sides by three quarters.

  6. Keep, flip, multiply

    n=32×43n = \frac{3}{2} \times \frac{4}{3}

    Flip three quarters to four thirds and multiply.

  7. Multiply straight across

    n=3×42×3=126n = \frac{3 \times 4}{2 \times 3} = \frac{12}{6}

    The top is 1212 and the bottom is 66.

  8. Simplify

    126=2\frac{12}{6} = 2

    Twelve sixths is exactly 22.

  9. Check the answer

    34×2−12=32−12=1\frac{3}{4} \times 2 - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1

    Three quarters of 22 is three halves; subtracting one half gives 11, as required.

  10. Sense check with a decimal

    32÷34=2\frac{3}{2} \div \frac{3}{4} = 2

    The answer works out to the whole number 22, which is a sensible size for this calculation.

  11. Reflect on the order

    undo subtraction, then division

    Reverse the operations in the opposite order to how they were applied.

  12. Notice the answer is exact

    22

    Every step used exact fractions, so the answer is exact with no rounding.

  13. Sense check the size

    34×2=112\frac{3}{4} \times 2 = 1\frac{1}{2}

    Three quarters of the answer is one and a half, comfortably above the one half being subtracted.

  14. State the number clearly

    22

    The unknown number is the whole number 22.

  15. State the answer

    22

    The number is 22.

Answer
22
Question 3
6 markschallenging
Work out 12+14+18+116\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}. Give your answer as a fraction.
Show worked solution

Worked solution

  1. Plan the calculation

    12+14+18+116\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}

    Put every fraction over the same denominator, then add.

  2. Find the common denominator

    LCD=16\text{LCD} = 16

    The denominators 22, 44, 88 and 1616 all divide into 1616.

  3. Convert one half

    12=816\frac{1}{2} = \frac{8}{16}

    Multiply top and bottom by 88.

  4. Convert one quarter

    14=416\frac{1}{4} = \frac{4}{16}

    Multiply top and bottom by 44.

  5. Convert one eighth

    18=216\frac{1}{8} = \frac{2}{16}

    Multiply top and bottom by 22.

  6. Keep the last fraction

    116=116\frac{1}{16} = \frac{1}{16}

    This one is already in sixteenths.

  7. Add the first two

    816+416=1216\frac{8}{16} + \frac{4}{16} = \frac{12}{16}

    Add the tops: 8+4=128 + 4 = 12.

  8. Add the third

    1216+216=1416\frac{12}{16} + \frac{2}{16} = \frac{14}{16}

    Add the next top: 12+2=1412 + 2 = 14.

  9. Add the fourth

    1416+116=1516\frac{14}{16} + \frac{1}{16} = \frac{15}{16}

    Add the last top: 14+1=1514 + 1 = 15.

  10. Sense check with a decimal

    12+14+18+116≈0.938\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} \approx 0.938

    As a decimal this is about 0.9380.938, which is the right size — a quick reassurance that the answer is reasonable.

  11. Confirm it is in simplest form

    gcd⁡(15,16)=1\gcd(15, 16) = 1

    The top and bottom share no common factor, so 1516\frac{15}{16} cannot be simplified any further.

  12. Spot the pattern

    1−116=15161 - \frac{1}{16} = \frac{15}{16}

    Each term is half the one before; the total is always one sixteenth short of a whole.

  13. Notice the answer is exact

    1516\frac{15}{16}

    Every step used exact fractions, so the answer is exact with no rounding.

  14. Sense check the size

    1516≈0.94\frac{15}{16} \approx 0.94

    The sum is just below 11, which fits the halving pattern approaching a whole.

  15. State the answer

    1516\frac{15}{16}

    The answer is 1516\frac{15}{16}.

Answer
1516\frac{15}{16}
Question 4
5 markschallenging
A plank 3123\frac{1}{2} m long is cut into pieces each 78\frac{7}{8} m long. How many pieces are there?
Show worked solution

Worked solution

  1. Set up the calculation

    312÷783\frac{1}{2} \div \frac{7}{8}

    To find how many pieces, divide the total length by the length of one piece.

  2. Convert the mixed number

    312=723\frac{1}{2} = \frac{7}{2}

    33 times 22 plus 11 is 77, so the plank is seven halves of a metre.

  3. Rewrite the division

    72÷78\frac{7}{2} \div \frac{7}{8}

    Now divide seven halves by seven eighths.

  4. Keep, flip, multiply

    72×87\frac{7}{2} \times \frac{8}{7}

    Flip seven eighths to eight sevenths and multiply.

  5. Multiply the tops and the bottoms

    7×82×7\frac{7 \times 8}{2 \times 7}

    Multiply the numerators together and the denominators together.

  6. Work out the product

    7×82×7=5614\frac{7 \times 8}{2 \times 7} = \frac{56}{14}

    The top is 5656 and the bottom is 1414.

  7. Simplify

    5614=4\frac{56}{14} = 4

    Fifty-six fourteenths is exactly 44.

  8. Check by multiplying back

    4×78=288=3124 \times \frac{7}{8} = \frac{28}{8} = 3\frac{1}{2}

    Four pieces each seven eighths long total three and a half metres — the whole plank.

  9. Sense check with a decimal

    312÷78=43\frac{1}{2} \div \frac{7}{8} = 4

    The answer works out to the whole number 44, which is a sensible size for this calculation.

  10. Interpret the division

    how many 78 in 312\text{how many } \frac{7}{8} \text{ in } 3\frac{1}{2}

    The division asks how many seven-eighth pieces fit into the plank: exactly 44.

  11. Common mistake to avoid

    divide, do not multiply

    The pieces come from dividing the total length by the piece length, not multiplying.

  12. Notice the answer is exact

    44

    Every step used exact fractions, so the answer is exact with no rounding.

  13. Sense check the size

    312÷1=3123\frac{1}{2} \div 1 = 3\frac{1}{2}

    Each piece is a bit under 11 m, so slightly more than 33 or 44 pieces is expected — 44 fits.

  14. Interpret in context

    4 pieces4 \text{ pieces}

    So there are 44 pieces.

  15. State the answer

    44

    There are 44 pieces.

Answer
44
Question 5
6 markschallenging
Work out 212×135−1142\frac{1}{2} \times 1\frac{3}{5} - 1\frac{1}{4}. Give your answer as a mixed number.
Show worked solution

Worked solution

  1. Apply the order of operations

    212×135−1142\frac{1}{2} \times 1\frac{3}{5} - 1\frac{1}{4}

    Multiplication comes before subtraction, so multiply first.

  2. Convert the first mixed number

    212=522\frac{1}{2} = \frac{5}{2}

    22 times 22 plus 11 is 55, so it is five halves.

  3. Convert the second mixed number

    135=851\frac{3}{5} = \frac{8}{5}

    11 times 55 plus 33 is 88, so it is eight fifths.

  4. Multiply

    52×85=4010\frac{5}{2} \times \frac{8}{5} = \frac{40}{10}

    Multiply the tops (5×8=40)(5\times 8=40) and the bottoms (2×5=10)(2\times 5=10).

  5. Simplify the product

    4010=4\frac{40}{10} = 4

    Forty tenths is exactly 44.

  6. Convert the last mixed number

    114=541\frac{1}{4} = \frac{5}{4}

    11 times 44 plus 11 is 55, so it is five quarters.

  7. Set up the subtraction

    4−544 - \frac{5}{4}

    Now subtract five quarters from 44.

  8. Write 44 as quarters

    4=1644 = \frac{16}{4}

    Four wholes is sixteen quarters.

  9. Subtract the quarters

    164−54=114\frac{16}{4} - \frac{5}{4} = \frac{11}{4}

    Subtract the tops: 16−5=1116 - 5 = 11.

  10. Convert to a mixed number

    114=234\frac{11}{4} = 2\frac{3}{4}

    Eight quarters make 22 wholes, leaving three quarters.

  11. Sense check with a decimal

    212×135−114≈2.752\frac{1}{2} \times 1\frac{3}{5} - 1\frac{1}{4} \approx 2.75

    As a decimal this is about 2.752.75, which is the right size — a quick reassurance that the answer is reasonable.

  12. Watch for the common error

    212×(135−114)≠2342\frac{1}{2} \times \left(1\frac{3}{5} - 1\frac{1}{4}\right) \ne 2\frac{3}{4}

    Subtracting before multiplying breaks the order of operations.

  13. Notice the answer is exact

    114\frac{11}{4}

    Every step used exact fractions, so the answer is exact with no rounding.

  14. Reflect on the method

    multiply, then subtract

    The product became a whole number, which made the final subtraction easy.

  15. State the answer

    2342\frac{3}{4}

    The answer is 2342\frac{3}{4}.

Answer
2342\frac{3}{4}

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