GCSE Exact calculation Practice Questions

Free GCSE Exact calculation practice questions with full step-by-step worked solutions. Covers adding fractions, same denominator, subtracting fractions, simplifying. Practise exam-style problems and check your method.

adding fractionssame denominatorsubtracting fractionssimplifyingmultiplying fractionsmultiples of pi
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
Work out 15+25\frac{1}{5} + \frac{2}{5}. Give your answer as a fraction.
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Worked solution

  1. Check the denominators

    15+25\frac{1}{5} + \frac{2}{5}

    Both fractions are in fifths, so they can be added straight away.

  2. Add the numerators

    1+25=35\frac{1+2}{5} = \frac{3}{5}

    When the denominators match, add the tops and keep the bottom the same.

  3. State the exact answer

    35\frac{3}{5}

    Three fifths is already in its simplest form — an exact answer, not a rounded decimal.

Answer
35\frac{3}{5}
Question 2
1 markeasy
Work out 4×294 \times \frac{2}{9}.
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Worked solution

  1. Write 44 as a fraction

    4=414 = \frac{4}{1}

    A whole number times a fraction is easiest with both as fractions.

  2. Multiply tops and bottoms

    4×21×9=89\frac{4 \times 2}{1 \times 9} = \frac{8}{9}

    44 times 22 is 88 on top; 11 times 99 is 99 on the bottom.

  3. State the exact answer

    89\frac{8}{9}

    Eight ninths is already in simplest form.

Answer
89\frac{8}{9}
Question 3
2 marksintermediate
Work out 23÷4\frac{2}{3} \div 4. Give your answer as a fraction.
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Worked solution

  1. Write 44 as a fraction

    4=414 = \frac{4}{1}

    Write the whole number over 11 so the division rule applies.

  2. Keep, flip, multiply

    23÷41=23×14\frac{2}{3} \div \frac{4}{1} = \frac{2}{3} \times \frac{1}{4}

    Dividing by 44 is the same as multiplying by one quarter.

  3. Cancel the 22 with the 44

    213×142\frac{\cancel{2}^1}{3} \times \frac{1}{\cancel{4}_2}

    22 divides into both 22 and 44.

  4. Multiply what remains

    13×2=16\frac{1}{3 \times 2} = \frac{1}{6}

    The product is one sixth.

  5. Sense check

    23÷4<23\frac{2}{3} \div 4 < \frac{2}{3}

    Sharing two thirds into 44 parts must give something smaller — one sixth is. ✓

  6. State the exact answer

    16\frac{1}{6}

    The exact quotient is one sixth.

Answer
16\frac{1}{6}
Question 4
3 markshard
Work out 513×385\frac{1}{3} \times \frac{3}{8}.
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Worked solution

  1. Convert the mixed number

    513=1635\frac{1}{3} = \frac{16}{3}

    5×3+1=165 \times 3 + 1 = 16, so five and a third is sixteen thirds.

  2. Set up the multiplication

    163×38\frac{16}{3} \times \frac{3}{8}

    Multiply the improper fraction by three eighths.

  3. Cancel the 33s

    1631×318\frac{16}{\cancel{3}_1} \times \frac{\cancel{3}^1}{8}

    The 33s cancel completely.

  4. Cancel the 1616 with the 88

    1621×181\frac{\cancel{16}^2}{1} \times \frac{1}{\cancel{8}_1}

    88 divides into 1616 twice and into 88 once.

  5. Multiply what remains

    2×11×1=2\frac{2 \times 1}{1 \times 1} = 2

    Everything cancels to exactly 22.

  6. Check with decimals

    5.33…×0.375=25.33\ldots \times 0.375 = 2

    Roughly 5.335.33 times 0.3750.375 is 22 — consistent.

  7. Reflect on cancelling

    16×33×8=4824=2\frac{16 \times 3}{3 \times 8} = \frac{48}{24} = 2

    Without cancelling the same answer appears through bigger numbers.

  8. Why the answer is whole

    38×16=6, 63=2\frac{3}{8} \times 16 = 6,\ \frac{6}{3} = 2

    Three eighths of sixteen thirds collapses neatly because 88 divides 1616 and 33 divides 33.

  9. Sense check

    38≈0.4,513×0.4≈2.1\frac{3}{8} \approx 0.4,\quad 5\tfrac{1}{3} \times 0.4 \approx 2.1

    A rough estimate lands near 22. ✓

  10. State the exact answer

    22

    The exact product is 22.

Answer
22
Question 5
6 markschallenging
Work out 3π4+5π6−π3\frac{3\pi}{4} + \frac{5\pi}{6} - \frac{\pi}{3}. Give your answer as an exact multiple of π\pi.
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Worked solution

  1. Treat pi like a unit

    3π4+5π6−π3\frac{3\pi}{4} + \frac{5\pi}{6} - \frac{\pi}{3}

    Every term is a fraction of pi, so add and subtract the fractions and carry the pi along.

  2. Extract the fraction problem

    34+56−13\frac{3}{4} + \frac{5}{6} - \frac{1}{3}

    Work with the plain fractions first; the pi returns at the end.

  3. Find the lowest common denominator

    LCD of 4,6,3=12\text{LCD of } 4, 6, 3 = 12

    Twelfths work for all three fractions.

  4. Convert the first fraction

    34=912\frac{3}{4} = \frac{9}{12}

    Multiply top and bottom by 33.

  5. Convert the second fraction

    56=1012\frac{5}{6} = \frac{10}{12}

    Multiply top and bottom by 22.

  6. Convert the third fraction

    13=412\frac{1}{3} = \frac{4}{12}

    Multiply top and bottom by 44.

  7. Combine the twelfths

    9+10−412\frac{9 + 10 - 4}{12}

    Add and subtract the numerators in order.

  8. Evaluate the numerator

    9+10−4=159 + 10 - 4 = 15

    99 plus 1010 is 1919, minus 44 is 1515.

  9. Write the fraction

    1512\frac{15}{12}

    The fraction total is fifteen twelfths.

  10. Simplify

    1512=54\frac{15}{12} = \frac{5}{4}

    Divide top and bottom by 33.

  11. Reattach the pi

    54π=5π4\frac{5}{4}\pi = \frac{5\pi}{4}

    The exact answer is five quarters of pi.

  12. Check with decimals of pi

    0.75+0.833…−0.333…=1.250.75 + 0.833\ldots - 0.333\ldots = 1.25

    The multipliers of pi total 1.251.25, which is five quarters. ✓

  13. Watch for the common error

    3+5−14+6−3π=77π=π?\frac{3+5-1}{4+6-3}\pi = \frac{7}{7}\pi = \pi?

    Adding tops and bottoms separately gives π\pi — wrong, because that is not how fractions combine.

  14. Reflect

    aπ±bπ=(a±b)πa\pi \pm b\pi = (a \pm b)\pi

    Fractions of pi follow exactly the same rules as ordinary fractions.

  15. State the exact answer

    5π4\frac{5\pi}{4}

    The exact value is five pi over four.

Answer
5π4\frac{5\pi}{4}

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