OCR GCSE Mathematics: Exact calculations with fractions, pi and surds (J560)

This guide is for OCR GCSE (9-1) Mathematics, aligned to the OCR specification Version 2.1 (August 2026). It covers sections 3.03a and 3.03b, exact calculations. Fractions and multiples of π are assessed at both tiers. Surds are Higher tier only. These sections are worded “without a calculator”, so you should be able to do every skill here by hand. Each tier has papers with and without a calculator, and the specification adds that any content can be assessed on either type.

Check your progress with the printable checklist, browse the OCR GCSE Mathematics hub, then use the revision notes and the practice questions. For wider fraction work see fractions, decimals and percentages. Not sure where you stand? Try the free diagnostics.

What this unit covers

Section	What you must be able to do	Tier
3.03a	Use fractions in exact calculations without a calculator	Both tiers
3.03a	Use multiples of π in exact calculations without a calculator	Both tiers
3.03a	Use surds in exact calculations without a calculator	Higher tier only
3.03b	Simplify expressions with surds, including rationalising denominators	Higher tier only

“Exact” means no rounding. An answer such as 5/3, 20π or 4√5 is exact. The decimals 1.666…, 62.83 and 8.944 are approximations. When a question says “give your answer in terms of π”, “give an exact answer” or “do not use a calculator”, leave the answer in fraction, π or surd form.

Exact calculations with fractions (3.03a)

Fractions stay exact because you never convert to a decimal that might recur. Work with them like this.

Add and subtract. Use a common denominator. Change mixed numbers to improper fractions first if that is easier.

Multiply. Multiply the numerators and the denominators. Cancel common factors before you multiply to keep the numbers small.

Divide. Keep the first fraction, flip the second, then multiply.

Fraction of an amount. “a/b of N” means (a/b) × N. To find the whole from a part, divide the part by the fraction.

Worked example 1: adding mixed numbers

Work out 3 5/6 + 1 7/9. Give the answer as a mixed number.

3 5/6 = 23/6        1 7/9 = 16/9
Common denominator 18:
23/6 = 69/18        16/9 = 32/18
69/18 + 32/18 = 101/18
101 ÷ 18 = 5 remainder 11

Answer: 5 11/18.

Worked example 2: multiplying and dividing together

Work out 2 1/4 ÷ 1 1/8 × 5/6.

2 1/4 = 9/4        1 1/8 = 9/8
9/4 ÷ 9/8 = 9/4 × 8/9 = 2
2 × 5/6 = 10/6 = 5/3

Answer: 5/3 (or 1 2/3). Working left to right matters because the division and multiplication have equal rank.

Worked example 3: fractions of an amount

A monthly budget is £84. 2/7 goes on rent and 1/3 on food. What fraction is left, and how much money is that?

Spent: 2/7 + 1/3 = 6/21 + 7/21 = 13/21
Left: 1 - 13/21 = 8/21
Amount: 8/21 × 84 = 8 × 4 = 32

Answer: 8/21, which is £32.

Reverse case: if 3/5 of a length is 21 cm, the whole is 21 ÷ 3/5 = 21 × 5/3 = 35 cm.

Exact calculations with multiples of π (3.03a)

π is a number, so you can add, subtract and multiply multiples of it like any other term. 4π + 3π = 7π and 6 × 5π = 30π. You do not replace π with 3.14 until the final line, and often not at all.

Circles

Circumference = πd = 2πr. Area = πr². Keep π as a symbol throughout.

Worked example 4: area of a ring

A circular path has an outer radius of 9 m and an inner radius of 5 m. Find the area of the path in terms of π.

Outer circle: π × 9² = 81π
Inner circle: π × 5² = 25π
Path: 81π - 25π = 56π

Answer: 56π m².

Worked example 5: a cylinder

A tin is a cylinder with radius 6 cm and height 7 cm. Work out its volume and its curved surface area, both in terms of π.

Volume = πr²h = π × 36 × 7 = 252π
Curved surface area = 2πrh = 2 × π × 6 × 7 = 84π

Answer: 252π cm³ and 84π cm².

Worked example 6: a sector

A sector has angle 72° and radius 5 cm. Find its area, then find the ratio of its area to the whole circle.

Sector area = 72/360 × π × 5² = 1/5 × 25π = 5π
Whole circle = 25π
Ratio = 5π : 25π = 1 : 5

Answer: 5π cm². The π cancels in a ratio, which is why exact methods are neat.

Working backwards

If a circumference is 14π cm, then 2πr = 14π, so r = 7. The area is π × 7² = 49π cm². Divide both sides by π and you never need a decimal.

Exact calculations with surds (3.03a, Higher tier only)

A surd is a root whose value is not a fraction of two integers, such as √2, √7 or √10. √16 = 4 is not a surd because it is a whole number. Surds are irrational, so their decimals never end or repeat. Leaving them in root form keeps the answer exact.

Manipulating surds (3.03b, Higher tier only)

Rules for multiplying and dividing

Rule	Example
√a × √b = √(ab)	√6 × √15 = √90
√a ÷ √b = √(a/b)	√48 ÷ √3 = √16 = 4

Also √a × √a = a, so (√7)² = 7 and (3√2)² = 9 × 2 = 18.

These rules do not work for addition. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5.

Simplifying a surd

Find the largest square number that divides into the number under the root, then take its root outside.

√12 = √(4 × 3) = √4 × √3 = 2√3
√72 = √(36 × 2) = 6√2

If you spot only a smaller square factor, simplify in two stages: √72 = √(9 × 8) = 3√8 = 3 × 2√2 = 6√2.

Adding and subtracting

Simplify each surd first. You can then collect terms with the same surd, just like 3x + 2x.

Worked example 7

Simplify √45 + √20 − √5.

√45 = 3√5     √20 = 2√5
3√5 + 2√5 - √5 = 4√5

Answer: 4√5.

Multiplying and expanding

Expand brackets as normal and use √a × √a = a.

Worked example 8

Expand and simplify (2 + √3)(5 − √3).

2 × 5 = 10
2 × (-√3) = -2√3
√3 × 5 = 5√3
√3 × (-√3) = -3
Total: 10 - 3 + (-2√3 + 5√3) = 7 + 3√3

Answer: 7 + 3√3.

Squaring a bracket needs two full brackets: (3 + √2)² = 9 + 3√2 + 3√2 + 2 = 11 + 6√2.

Rationalising a denominator: one surd

To remove a surd from the bottom, multiply top and bottom by that surd. This multiplies the fraction by 1, so its value does not change.

1/√3 = (1 × √3)/(√3 × √3) = √3/3

Simplify the surd first if you can, because it keeps numbers smaller.

Worked example 9

Write 4/√12 with a rational denominator.

√12 = 2√3
4/(2√3) = 2/√3
2/√3 × √3/√3 = 2√3/3

Answer: 2√3/3.

Rationalising a denominator: two terms

For a denominator such as a + √b, multiply top and bottom by the conjugate a − √b (change the sign in the middle). The denominator becomes a difference of two squares: (a + √b)(a − √b) = a² − b, which is rational.

1/(√3 + 1) × (√3 - 1)/(√3 - 1) = (√3 - 1)/(3 - 1) = (√3 - 1)/2

Worked example 10

Rationalise 7/(√6 − 2).

Conjugate of √6 - 2 is √6 + 2
Denominator: (√6 - 2)(√6 + 2) = 6 - 4 = 2
Numerator: 7(√6 + 2) = 7√6 + 14

Answer: (7√6 + 14)/2.

Worked example 11

Write (1 + √5)/(3 − √5) in the form a + b√5.

Multiply by (3 + √5)/(3 + √5)
Numerator: (1 + √5)(3 + √5) = 3 + √5 + 3√5 + 5 = 8 + 4√5
Denominator: 9 - 5 = 4
(8 + 4√5)/4 = 2 + √5

Answer: a = 2, b = 1.

Worked example 12: exact lengths

A right-angled isosceles triangle has hypotenuse 10 cm. Find the exact length of each short side and the area.

x² + x² = 100, so x² = 50
x = √50 = 5√2
Area = 1/2 × 5√2 × 5√2 = 1/2 × 25 × 2 = 25

Answer: 5√2 cm and 25 cm². The area is a whole number even though the sides are surds.

Common errors

• Adding denominators: 1/3 + 1/4 is not 2/7.
• Writing 3.14 for π in an answer that asks for terms of π. Leave π as a symbol.
• Treating √(a + b) as √a + √b.
• Stopping at √72 because it “looks done”. Always check for a square factor.
• Rationalising 1/(√3 + 1) by multiplying by √3 only. A two-term denominator needs its conjugate.
• Sign slips in the conjugate. Write the middle sign explicitly and expand fully.
• Giving a final answer such as 6/√2 with a surd still in the denominator when the question says to rationalise.
• Rounding part-way through a calculation and losing the exact form.

Official syllabus

OCR GCSE (9-1) Mathematics (J560), Version 2.1 (August 2026), for first assessment in 2017, published by OCR. This guide covers sections 3.03a and 3.03b.