Paper 32, worked through. Cambridge IGCSE Mathematics, May/June 2026: worked solutions

Eighty marks in an hour and a half, so just over a minute a mark. A calculator is allowed, and that changes what the marks are for: write down the calculation you type in, because a wrong answer with the right calculation beside it still earns method marks. Unless a question says otherwise, give answers that are not exact to three significant figures, and use the π button, not 3.14. You need a ruler and a protractor for two of the questions.

1. 1Writing a number in figures from its name in words.[1]
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   1. Four hundred thousand is 400 000.
   2. Fifty-three goes in the last two places. There are no thousands and no hundreds left over, so those places are filled with zeros.

   Answer400 053

   Say your answer back to yourself. 453 000 and 400 530 are the usual slips: the zeros hold the other digits in their places.

2. 2(a)Finding a multiple of 9 in a given range.[1]
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   1. The multiples of 9 are the 9 times table: 45, 54, 63 and so on.
   2. The only one between 50 and 60 is 54.

   Answer54

   Revise this: Types of number
3. 2(b)Listing every factor that two numbers share.[2]
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   1. A factor of a number divides into it exactly. Factors of 20: 1, 2, 4, 5, 10 and 20.
   2. Factors of 30: 1, 2, 3, 5, 6, 10, 15 and 30.
   3. The numbers on both lists are 1, 2, 5 and 10.

   Answer1, 2, 5 and 10

   "All" means every one, including 1. Find factors in pairs (1 × 20, 2 × 10, 4 × 5) so that none is missed.

   Revise this: Types of number
4. 3Working out how many items can be bought, and the change.[3]
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   1. Divide the money by the price: 35 ÷ 5.45 = 6.42…
   2. He cannot buy part of a vase, so round down: 6 vases.
   3. Cost of 6: 6 × 5.45 = $32.70.
   4. Change: 35 − 32.70 = $2.30.

   Answer6 vases and $2.30 change

   Always round down when counting how many you can afford, even if the decimal is above .5. Write money with two decimal places.

   Revise this: Time and money
5. 4(a)Finding a cube root with a calculator.[1]
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   1. The small 3 on the root sign means cube root: the number that, multiplied by itself three times, gives 1.728.
   2. Use the cube root key: the answer is 1.2.
   3. Check: 1.2 × 1.2 × 1.2 = 1.728.

   Answer1.2

   Without a cube root key, raise the number to the power 1 ÷ 3, using brackets around the power.

   Revise this: Squares, cubes and roots
6. 4(b)Working out a fraction with a calculation on the bottom.[1]
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   1. Work out the bottom first: 0.2² = 0.04, and 0.04 − 0.36 = −0.32.
   2. Then divide: 18 ÷ −0.32 = −56.25.

   Answer−56.25

   The fraction line acts like a pair of brackets around the whole of the bottom. Type 18 ÷ (0.2² − 0.36), or use the fraction key. Without the brackets the calculator divides by 0.04 only.

   Revise this: Fractions, decimals and the order of operations
7. 5(a)Finding the range of a whole group from facts about two smaller groups.[3]
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   1. Range = largest − smallest, so find the tallest and the shortest of the seven people.
   2. Tallest adult = shortest adult + the adults' range = 1.59 + 0.25 = 1.84 m.
   3. Shortest child = tallest child − the children's range = 1.43 − 0.38 = 1.05 m.
   4. Every adult is taller than every child, so these are the tallest and shortest of everyone. Range = 1.84 − 1.05 = 0.79 m.

   Answer0.79 m

   Adding the two ranges gives 0.63, which leaves out the gap between the tallest child and the shortest adult.

   Revise this: Data, averages and range
8. 5(b)Finding the median height of the seven people.[1]
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   1. The median is the middle value when they are in order. For 7 values it is the 4th.
   2. In order of height the three children come first, because the tallest child (1.43 m) is shorter than the shortest adult (1.59 m).
   3. So the 4th person is the shortest adult.

   Answer1.59 m

   "Write down" tells you no calculation is needed. You do not need to know anyone else's height.

   Revise this: Data, averages and range
9. 6Explaining why a number is not prime.[1]
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   1. A prime number has exactly two factors: 1 and itself.
   2. 39 = 3 × 13, so 3 and 13 are factors of it as well.

   Answer39 = 3 × 13, so it has factors other than 1 and 39.

   Name a factor. "It can be divided" is not enough: the reason has to mention 3 or 13.

   Revise this: Types of number
10. 7Finding the whole amount when a fraction of it is known.[2]
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    1. Three fifths of the wall takes 2.7 litres. One fifth takes 2.7 ÷ 3 = 0.9 litres.
    2. The whole wall is five fifths: 0.9 × 5 = 4.5 litres.

    Answer4.5 litres

    Find one part first, then build up to the whole. Multiplying 2.7 by 3/5 gives 1.62, which is less than the amount already used, so it cannot be right.

    Revise this: Fractions, decimals and the order of operations
11. 8Marking a point on a scale drawing from a distance and a bearing.[2]
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    1. Change the distance to the scale: 94 ÷ 20 = 4.7 cm.
    2. A bearing is measured clockwise from north. Put the centre of your protractor on T with its zero on the north line, and mark 112° round to the right.
    3. Rule a faint line from T through that mark. Measure 4.7 cm along it from T and mark V.

    AnswerV is 4.7 cm from T, at an angle of 112° clockwise from the north line.

    One mark is for the angle and one for the distance. 112° is more than 90°, so V has to be below the level of T, to the right. Measure on the printed paper: a screen is not life size.

    Revise this: Scale drawings and bearings
12. 9Changing money from one currency to another.[1]
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    1. Every dollar costs 0.95 euros. To find how many dollars 500 euros will buy, divide: 500 ÷ 0.95 = 526.315…
    2. Round to the nearest cent.

    Answer$526.32

    Multiply or divide? A dollar costs less than one euro, so he must get more than 500 dollars. Multiplying gives 475, which is fewer.

    Revise this: Rates and compound measures
13. 10Calculating simple interest.[2]
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    1. Simple interest pays the same amount every year: the percentage of the money first invested.
    2. One year: 4.5% of 8500 = 8500 × 4.5 ÷ 100 = $382.50.
    3. Three years: 382.50 × 3 = $1147.50.

    Answer$1147.50

    The question asks for the interest only. Adding it to the 8500 answers a different question and loses a mark.

    Revise this: Percentages
14. 11Finding the volume of a cuboid, to one decimal place.[2]
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    1. Volume of a cuboid = length × width × height.
    2. 10.8 × 3.7 × 4.6 = 183.816.
    3. To one decimal place: the digit after the 8 is 1, so it stays as 183.8.

    Answer183.8 cm³

    Write the full calculator answer first and round it afterwards. An unrounded 183.816 earns one of the two marks.

    Revise this: Volume and surface area
15. 12Factorising an expression with two terms.[1]
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    1. Factorising means writing it with brackets. Look for something that goes into both terms: x goes into x² and into 7x.
    2. Take x outside the brackets and write what is left inside: x² ÷ x = x, and 7x ÷ x = 7.

    Answerx(x + 7)

    Check by multiplying back out: x × x + x × 7 = x² + 7x.

    Revise this: Expanding and factorising
16. 13(a)Finding the power that makes a number equal to 1.[1]
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    1. Any number (other than zero) to the power 0 is 1.
    2. So 8 to the power 0 is 1.

    Answery = 0

    The power 1 leaves the number unchanged: 8 to the power 1 is 8, not 1.

    Revise this: Indices
17. 13(b)Multiplying two powers of the same letter.[1]
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    1. When you multiply powers of the same letter, add the powers.
    2. 4 + 5 = 9.

    Answerw = 9

    Add the powers, do not multiply them. Four x's times five more x's is nine x's multiplied together.

    Revise this: Indices in algebra
18. 14(a)Writing and simplifying an expression for a total.[2]
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    1. Add everything that is now in the jar: 60 + C + (2C + 7).
    2. Collect the letters: C + 2C = 3C. Collect the numbers: 60 + 7 = 67.

    Answer3C + 67

    An expression has no equals sign. Leaving it as 60 + C + 2C + 7 earns one of the two marks: "simplest form" means collect the terms.

    Revise this: Expanding and factorising
19. 14(b)(i)Writing an expression for an equal share.[1]
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    1. Sharing equally between two people means dividing the total by 2.
    2. Divide the whole of the expression from part (a) by 2.

    Answer(3C + 67) ÷ 2

    Use brackets, or a fraction line under the whole expression. 3C + 67 ÷ 2 would halve only the 67.

    Revise this: Expanding and factorising
20. 14(b)(ii)Forming an equation from the share and solving it.[4]
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    1. Each share is 86, so (3C + 67) ÷ 2 = 86.
    2. Multiply both sides by 2: 3C + 67 = 172.
    3. Subtract 67 from both sides: 3C = 105.
    4. Divide both sides by 3: C = 35.

    AnswerC = 35

    Check: 60 + 35 + (2 × 35 + 7) = 172, and 172 ÷ 2 = 86. Each line of working here earns a mark, so set them all out.

    Revise this: Solving linear equations
21. 15(a)Finding the first three terms from an nth term.[1]
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    1. Put n = 1, 2 and 3 into 3n − 1.
    2. 3 × 1 − 1 = 2. 3 × 2 − 1 = 5. 3 × 3 − 1 = 8.

    Answer2, 5, 8

    3n means 3 × n, so n = 2 gives 6, not 32.

    Revise this: Sequences
22. 15(b)Deciding which numbers in a list belong to the sequence.[2]
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    1. Every term is 1 less than a multiple of 3. So a number is in the sequence if adding 1 to it gives a multiple of 3.
    2. 326 + 1 = 327 = 3 × 109. 329 + 1 = 330 = 3 × 110. 332 + 1 = 333 = 3 × 111.
    3. The terms go up in threes, which fits: 326, 329, 332.

    Answer326, 329 and 332

    Or solve 3n − 1 = 326: 3n = 327, n = 109, a whole number, so it is a term. Once you have one term, keep adding 3.

    Revise this: Sequences
23. 16(a)Working out an expected number of wins.[1]
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    1. Expected number = probability × number of trials.
    2. 0.4 × 20 = 8.

    Answer8

    It is the average you would get over many sets of 20 matches, not a promise about this one.

    Revise this: Probability
24. 16(b)Completing a table of probabilities from a relationship between two of them.[3]
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    1. The three results cover everything that can happen, so their probabilities add up to 1. Lose and draw together: 1 − 0.4 = 0.6.
    2. Losing is three times as likely as drawing, so split 0.6 into 3 + 1 = 4 equal parts: 0.6 ÷ 4 = 0.15.
    3. Draw is one part: 0.15. Lose is three parts: 3 × 0.15 = 0.45.

    AnswerLose 0.45, draw 0.15

    Check: 0.4 + 0.45 + 0.15 = 1, and 0.45 is three times 0.15. Make sure the larger one goes under "lose".

    Revise this: Probability
25. 17(a)Finding the perimeter of an L-shaped figure with two lengths missing.[2]
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    1. The perimeter is the distance all the way round, so every side is needed. Two are not labelled.
    2. The short upright side: the right-hand side is 9 and the left-hand side is 6, so it is 9 − 6 = 3 cm.
    3. The bottom side: the top is 12 and the other horizontal side is 7, so it is 12 − 7 = 5 cm.
    4. Add all six sides: 12 + 9 + 5 + 3 + 7 + 6 = 42.

    Answer42 cm

    Mark the two missing lengths on the diagram first. A quick check: the perimeter of an L-shape like this equals that of the rectangle around it, 2 × (12 + 9) = 42.

    Revise this: Area and perimeter
26. 17(b)Finding the length of a sloping line across the corner of the shape.[2]
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    1. XY is the longest side of a right-angled triangle. Its other two sides are the missing lengths from part (a): 3 cm up and 5 cm across.
    2. Pythagoras' theorem: XY² = 3² + 5² = 9 + 25 = 34.
    3. XY = √34 = 5.830…

    Answer5.83 cm

    Square, add, then square root. Give three significant figures unless told otherwise.

    Revise this: Pythagoras' theorem
27. 17(c)Finding the side of a square with the same area as the compound shape.[3]
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    1. Split the shape into two rectangles: the top one is 12 by 6 and the small one below it is 5 by 3.
    2. Area = 12 × 6 + 5 × 3 = 72 + 15 = 87 cm².
    3. A square of side s has area s², so s² = 87 and s = √87 = 9.327…

    Answer9.33 cm

    Or take the big 12 by 9 rectangle and remove the missing 7 by 3 corner: 108 − 21 = 87. The area alone earns two of the three marks. Do not divide by 4: that is for perimeters.

    Revise this: Area and perimeter
28. 18(a)Completing a table of values for a quadratic.[3]
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    1. Put each x value into −x² + 3x + 5. The −x² means "square x, then make it negative".
    2. x = −1: −1 − 3 + 5 = 1. x = 0: 0 + 0 + 5 = 5.
    3. x = 1: −1 + 3 + 5 = 7. x = 2: −4 + 6 + 5 = 7. x = 4: −16 + 12 + 5 = 1.

    Answer1, 5, 7, 7, 1 (for x = −1, 0, 1, 2 and 4)

    On a calculator type −(x²), with the bracket, or the negative gets squared too. The values should be symmetrical: −5, 1, 5, 7, 7, 5, 1, −5. If yours are not, one is wrong.

    Revise this: Graphs of curves
29. 18(b)Plotting the points and drawing the curve.[4]
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    1. Plot the eight points from the table: (−2, −5), (−1, 1), (0, 5), (1, 7), (2, 7), (3, 5), (4, 1) and (5, −5).
    2. Join them with one smooth curve, drawn freehand. It is an upside-down U shape.
    3. The top of the curve is between x = 1 and x = 2. It must rise a little above 7 there: do not join (1, 7) to (2, 7) with a flat line.

    AnswerA smooth, symmetrical curve through all eight points, with a rounded top just above y = 7 between x = 1 and x = 2.

    Three marks are for the points and one for the curve. Never use a ruler for a curve. Turn the paper so that your hand is inside the curve as you draw.

    Revise this: Graphs of curves
30. 18(c)Reading the highest point from the graph.[1]
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    1. The curve is symmetrical, and the two points at height 7 are at x = 1 and x = 2. The top is halfway between them: x = 1.5.
    2. Read the height of your curve there: just above 7. The exact value is −(1.5)² + 3 × 1.5 + 5 = 7.25.

    Answer(1.5, 7.25). Any height above 7 and up to 7.5, read from your curve, is accepted.

    The highest point is not one of the plotted points. It sits between the two equal ones.

    Revise this: Graphs of curves
31. 18(d)Solving the equation from the graph.[2]
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    1. The left-hand side of the equation is the same as the graph's y. So the equation says y = 0, and that is the x-axis.
    2. Read the x values of the two points where your curve crosses the x-axis.
    3. They are close to x = −1.2 and x = 4.2.

    Answerx = −1.2 or x = 4.2 (read from your own curve)

    "Use your graph" means read it off, to the nearest small square. Each crossing is worth a mark. There are two because the curve crosses the axis twice.

    Revise this: Graphs of curves
32. 19Completing a pie chart from a percentage and a fraction.[4]
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    1. A full circle is 360°. Each group gets that share of it.
    2. Algebra: 35% of 360 = 0.35 × 360 = 126°.
    3. Geometry: 2/9 of 360 = 360 ÷ 9 × 2 = 80°.
    4. Statistics is the rest: 360 − 126 − 80 = 154°.
    5. Draw them with a protractor. Measure 126° from the line already drawn and rule a line from the centre. Measure 80° from that new line and rule another. What is left should measure 154°.

    AnswerThree sectors of 126° (algebra), 80° (geometry) and 154° (statistics), each labelled.

    Check that the three angles add up to 360° before drawing. Measure the last sector as a check: if it is not 154°, one of the other two is off. Angles must be within 2°.

    Revise this: Charts and scatter diagrams
33. 20Finding the percentage of a rectangle left after circles are cut from it.[4]
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    1. Area of the rectangle = 25 × 60 = 1500 cm².
    2. Area of one circle = π × 5² = 25π. Twelve circles: 12 × 25π = 300π = 942.47… cm².
    3. Metal left = 1500 − 942.47… = 557.52… cm².
    4. As a percentage of the rectangle: 557.52… ÷ 1500 × 100 = 37.16…

    Answer37.2%

    Keep the full calculator values until the end and round once. The percentage is of the original rectangle, so divide by 1500. Stopping at 62.8% gives the part cut out, not the part remaining.

    Revise this: Circles, arcs and sectors
34. 21Finding an arrival time from a distance and an average speed.[3]
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    1. Time = distance ÷ speed = 106 ÷ 40 = 2.65 hours.
    2. Change the decimal part into minutes: 0.65 × 60 = 39. So the journey takes 2 hours 39 minutes.
    3. Add it to the start: 14 30 + 2 hours = 16 30, and 39 minutes more is 17 09.

    Answer17 09

    2.65 hours is not 2 hours 65 minutes. There are 60 minutes in an hour, so multiply the decimal part by 60. Your calculator's time key will do it for you.

    Revise this: Rates and compound measures
35. 22Finding the lowest common multiple of two numbers.[2]
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    1. Write each number as a product of primes. 54 = 2 × 3 × 3 × 3. 90 = 2 × 3 × 3 × 5.
    2. The lowest common multiple needs every prime, the greatest number of times it appears in either: one 2, three 3s and one 5.
    3. 2 × 3 × 3 × 3 × 5 = 270.

    Answer270

    Or list multiples until one appears in both lists: 54, 108, 162, 216, 270 and 90, 180, 270. Do not mix it up with the highest common factor, which is 18 here.

    Revise this: Types of number
36. 23Giving the limits for a mass rounded to the nearest 50 grams.[2]
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    1. The mass could be up to half of 50 either side of 650. Half of 50 is 25.
    2. Smallest value: 650 − 25 = 625. Largest: 650 + 25 = 675.

    Answer625 ≤ m < 675

    Halve the unit you rounded to, then go that far each way. The upper limit is written 675, with the "less than" sign showing that 675 itself would round up to 700.

    Revise this: Rounding, estimating and bounds
37. 24Finding the surface area of a sphere, with units.[3]
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    1. Surface area of a sphere = 4πr², from the formula list.
    2. 4 × π × 6² = 144π = 452.38…
    3. Area is measured in square units.

    Answer452 cm²

    The unit has a mark of its own. cm² is for area and cm³ is for volume. Square the 6 before multiplying: 4 × π × 6 × 6.

    Revise this: Volume and surface area
38. 25Picking out the irrational numbers from a list.[1]
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    1. An irrational number cannot be written as a fraction. As a decimal it goes on for ever with no repeating pattern.
    2. Test each one. 331776 to the power 0.25 comes to exactly 24. 7/13 is already a fraction. 6432.89 to the power 17 is a decimal that ends, multiplied by itself, so it also ends.
    3. √0.4 = 0.6324… which never ends or repeats.

    Answer√0.4 only

    A root is irrational unless it comes out exactly. Do not confuse √0.4 with √0.04, which is exactly 0.2. Try each one on the calculator before deciding.

    Revise this: Types of number
39. 26Finding a side of a large right-angled triangle by going through a smaller one inside it.[4]
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    1. PR is in triangle PQR, where the only length missing is QR. Find that first, from the small triangle QRS.
    2. Triangle QRS has a right angle at R. QS = 23 cm is its hypotenuse, and QR is next to the 44° angle. Adjacent and hypotenuse means cosine: QR = 23 × cos 44° = 16.54… cm.
    3. Now the big triangle PQR. Its angle at Q is 31 + 44 = 75°. PR is opposite that angle and QR is next to it. Opposite and adjacent means tangent.
    4. PR = QR × tan 75° = 16.54… × tan 75° = 61.74…

    Answer61.7 cm

    Keep the unrounded QR in your calculator for the second step. Label each triangle's sides O, A and H before choosing sin, cos or tan. Make sure the calculator is in degrees.

    Revise this: Trigonometry in right-angled triangles

These explanations are Papermunch’s own, written to teach the method. The answers have been checked against the exam board’s mark scheme, which has the final say: open it above. The question paper and mark scheme belong toCambridge University Press & Assessment: each question shown here is drawn from the paper itself as you read, and is not kept on this site.