Hard GCSE Exact calculation Questions

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

mixed numbersdividing fractionscancellingmultiplying fractionsarea contextinverse problem
GCSE Foundation34 questionsStep-by-step solutions
Question 1
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.
Show worked solution

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}
Question 2
6 markschallenging
The perimeter of a semicircle is 5π+105\pi + 10 cm. Work out its radius.
Show worked solution

Worked solution

  1. Recall the semicircle perimeter formula

    P=πr+2rP = \pi r + 2r

    A semicircle's perimeter is the curved half-circumference (πr)(\pi r) plus the straight diameter (2r)(2r).

  2. Set up the equation

    πr+2r=5π+10\pi r + 2r = 5\pi + 10

    Set the formula equal to the given exact perimeter.

  3. Compare the pi terms

    πr=5π\pi r = 5\pi

    The pi parts on each side must match, because pi cannot mix with plain numbers.

  4. Solve from the pi terms

    r=5r = 5

    Dividing by pi gives radius 55.

  5. Compare the plain-number terms

    2r=102r = 10

    The number parts must match too.

  6. Solve from the number terms

    r=5r = 5

    Halving 1010 also gives radius 55 — both comparisons agree.

  7. Why matching works

    π is irrational\pi \text{ is irrational}

    A multiple of pi can never equal a plain number, so the two kinds of term must balance separately.

  8. Formal alternative: factorise

    r(π+2)=5π+10r(\pi + 2) = 5\pi + 10

    Factor the left side, and factor the right side as 55 lots of (π+2)(\pi + 2).

  9. Factor the right side

    5π+10=5(π+2)5\pi + 10 = 5(\pi + 2)

    Both terms on the right share the factor 55... revealing the same structure as the left.

  10. Divide both sides

    r=5(π+2)π+2=5r = \frac{5(\pi+2)}{\pi+2} = 5

    Dividing by (π+2)(\pi + 2) confirms r=5r = 5 rigorously.

  11. Check

    π×5+2×5=5π+10\pi \times 5 + 2 \times 5 = 5\pi + 10

    Radius 55 reproduces the given perimeter exactly. ✓

  12. Watch for the common error

    P≠πrP \ne \pi r

    Forgetting the diameter term would wrongly give r=5r = 5 from an incomplete equation — here it happens to agree, but the method must include both terms.

  13. Attach the units

    r=5 cmr = 5 \text{ cm}

    The radius is in centimetres.

  14. Reflect

    match π terms and number terms\text{match } \pi \text{ terms and number terms}

    Exact answers in the form aπ\pi + b can be reverse-engineered by comparing the two kinds of term.

  15. State the answer

    r=5 cmr = 5 \text{ cm}

    The radius is exactly 55 cm.

Answer
r=5 cmr = 5 \text{ cm}
Question 3
6 markschallenging
Work out 212×135÷2232\frac{1}{2} \times 1\frac{3}{5} \div 2\frac{2}{3}. Give your answer as a mixed number.
Show worked solution

Worked solution

  1. Plan the chain

    212×135÷2232\frac{1}{2} \times 1\frac{3}{5} \div 2\frac{2}{3}

    Convert all three mixed numbers to improper fractions, then work left to right.

  2. Convert the first mixed number

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

    2×2+1=52 \times 2 + 1 = 5 halves.

  3. Convert the second mixed number

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

    1×5+3=81 \times 5 + 3 = 8 fifths.

  4. Convert the third mixed number

    223=832\frac{2}{3} = \frac{8}{3}

    2×3+2=82 \times 3 + 2 = 8 thirds.

  5. Start with the multiplication

    52×85\frac{5}{2} \times \frac{8}{5}

    Multiplication and division have equal priority, so work left to right.

  6. Cancel the 55s

    512×851\frac{\cancel{5}^1}{2} \times \frac{8}{\cancel{5}_1}

    The 55s cancel completely.

  7. Cancel the 88 with the 22

    121×841\frac{1}{\cancel{2}_1} \times \frac{\cancel{8}^4}{1}

    22 divides into 88 four times.

  8. Complete the multiplication

    52×85=4\frac{5}{2} \times \frac{8}{5} = 4

    The product collapses to exactly 44.

  9. Now the division

    4÷834 \div \frac{8}{3}

    Divide the running total by eight thirds.

  10. Keep, flip, multiply

    4×384 \times \frac{3}{8}

    Dividing by eight thirds means multiplying by three eighths.

  11. Cancel the 44 with the 88

    41×382\frac{\cancel{4}^1 \times 3}{\cancel{8}_2}

    44 divides into 44 and into 88.

  12. Complete the division

    32\frac{3}{2}

    The result is three halves.

  13. Convert to a mixed number

    32=112\frac{3}{2} = 1\frac{1}{2}

    Two halves make 11 whole, leaving one half.

  14. Sense check with decimals

    2.5×1.6÷2.67≈1.52.5 \times 1.6 \div 2.67 \approx 1.5

    The decimal estimate lands on about 1.51.5, matching exactly.

  15. State the exact answer

    1121\frac{1}{2}

    The exact value is one and a half.

Answer
1121\frac{1}{2}
Question 4
5 markschallenging
Work out (34)3\left(\frac{3}{4}\right)^3. Give your answer as a fraction.
Show worked solution

Worked solution

  1. Understand the power

    (34)3=34×34×34\left(\frac{3}{4}\right)^3 = \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4}

    Cubing means three copies of the fraction multiplied together.

  2. Recall the rule for powers of fractions

    (ab)3=a3b3\left(\frac{a}{b}\right)^3 = \frac{a^3}{b^3}

    Cube the top and the bottom separately.

  3. Cube the numerator

    33=273^3 = 27

    3×3×3=273 \times 3 \times 3 = 27.

  4. Cube the denominator

    43=644^3 = 64

    4×4×4=644 \times 4 \times 4 = 64.

  5. Write the result

    (34)3=2764\left(\frac{3}{4}\right)^3 = \frac{27}{64}

    The cube is twenty-seven sixty-fourths.

  6. Check for simplification

    gcd⁡(27,64)=1\gcd(27, 64) = 1

    2727 is a power of 33 and 6464 a power of 22 — no common factor.

  7. Step-by-step check: first product

    34×34=916\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}

    Two copies give nine sixteenths.

  8. Step-by-step check: second product

    916×34=2764\frac{9}{16} \times \frac{3}{4} = \frac{27}{64}

    The third copy gives twenty-seven sixty-fourths. ✓

  9. Sense check the size

    2764<34\frac{27}{64} < \frac{3}{4}

    Cubing a number between 00 and 11 makes it smaller — 27/6427/64 is indeed less than 3/43/4.

  10. Decimal comparison

    0.753=0.421875=27640.75^3 = 0.421875 = \frac{27}{64}

    The decimal cube matches the fraction exactly (this one terminates).

  11. Watch for the common error

    (34)3≠274\left(\frac{3}{4}\right)^3 \ne \frac{27}{4}

    Cubing only the top is the classic slip — the bottom must be cubed too.

  12. Another common error

    (34)3≠912\left(\frac{3}{4}\right)^3 \ne \frac{9}{12}

    Multiplying top and bottom by 33 is not cubing — that just makes an equivalent fraction.

  13. Real-world connection

    probability of 3 events at 34\text{probability of 3 events at } \frac{3}{4}

    Cubes of fractions appear in repeated-probability problems later in the course.

  14. Reflect

    (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

    Powers act on numerator and denominator alike, keeping everything exact.

  15. State the exact answer

    2764\frac{27}{64}

    The exact cube is twenty-seven sixty-fourths.

Answer
2764\frac{27}{64}
Question 5
5 markschallenging
A cylinder has volume 45π45\pi cm3^3 and radius 33 cm. Work out its height.
Show worked solution

Worked solution

  1. Recall the volume formula

    V=πr2hV = \pi r^2 h

    The volume of a cylinder is pi times radius squared times height.

  2. Substitute what is known

    π×32×h=45π\pi \times 3^2 \times h = 45\pi

    The volume and radius are given; the height is the unknown.

  3. Square the radius

    32=93^2 = 9

    33 squared is 99.

  4. Rewrite the equation

    9πh=45π9\pi h = 45\pi

    Nine pi times the height equals forty-five pi.

  5. Divide both sides by pi

    9h=459h = 45

    The pi cancels from both sides — exact form keeps this clean.

  6. Divide both sides by 99

    h=45÷9h = 45 \div 9

    Undo the multiplication by 99.

  7. Evaluate

    h=5h = 5

    4545 divided by 99 is exactly 55.

  8. Check

    π×9×5=45π\pi \times 9 \times 5 = 45\pi

    Radius 33 and height 55 give volume 45π45\pi, matching the question. ✓

  9. Attach the units

    h=5 cmh = 5 \text{ cm}

    The height is in centimetres.

  10. Why exact form helped

    45π9π=5\frac{45\pi}{9\pi} = 5

    Because both sides kept their pi, the division was a simple whole-number calculation with no rounding.

  11. Contrast with the decimal route

    141.37…28.27…=5.000…?\frac{141.37\ldots}{28.27\ldots} = 5.000\ldots?

    Working in rounded decimals risks a slightly-off answer; the exact route cannot drift.

  12. Watch for the common error

    h≠453h \ne \frac{45}{3}

    Dividing by the radius instead of the radius squared gives 1515 — remember the r2r^2 in the formula.

  13. Reflect on the structure

    volume÷cross-section=height\text{volume} \div \text{cross-section} = \text{height}

    The height is the volume divided by the circular cross-section area 9π9\pi.

  14. Confirm

    h=5h = 5

    All checks agree.

  15. State the answer

    h=5 cmh = 5 \text{ cm}

    The height is exactly 55 cm.

Answer
h=5 cmh = 5 \text{ cm}

Unlock 29 more Exact calculation questions

Create a free account to work through every GCSE Exact calculation question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Exact calculation practice

Practice QuestionsWorked Solutions

Related Number topics

Upper and lower boundsError intervalsEstimating powers and rootsEstimation and checkingFactors, multiples and primesFDP equivalence