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

Short notes for the last weeks before the exam on OCR GCSE (9-1) Mathematics sections 3.03a and 3.03b, from the OCR specification Version 2.1 (August 2026). The study guide has the full explanations. Then test yourself with the practice questions, tick off topics on the checklist, and see the course hub. The free diagnostics show which skills to revisit first.

Fractions and multiples of π are for both tiers. Everything on surds is Higher tier only.

Tier map

Section	Skill	Tier
3.03a	Exact calculations with fractions, no calculator	Both tiers
3.03a	Exact calculations with multiples of π, no calculator	Both tiers
3.03a	Exact calculations with surds, no calculator	Higher tier only
3.03b	Simplify surd expressions, rationalise denominators	Higher tier only

Fractions

Operation	Method
Add or subtract	Common denominator, then combine numerators
Multiply	Cancel across, then numerator × numerator, denominator × denominator
Divide	Keep, flip, multiply
Mixed numbers	Convert to improper fractions first
a/b of N	(a × N) ÷ b
Find the whole	Part ÷ fraction

Method in steps

1. Write mixed numbers as improper fractions.
2. For + or −, find a common denominator. For ×, cancel first. For ÷, flip the second fraction.
3. Simplify the result.
4. Convert back to a mixed number only if the question asks for one.

Reminder: 2 3/8 − 7/12. Convert: 19/8 − 7/12 = 57/24 − 14/24 = 43/24 = 1 19/24.

Reminder: 4/9 of a quantity is 28, so the whole is 28 × 9/4 = 63.

Keep fractions rather than rounded decimals, because 1/3 is exact and 0.33 is not.

Multiples of π

Quantity	Formula
Circumference	πd or 2πr
Area of circle	πr²
Sector area	(θ/360) × πr²
Arc length	(θ/360) × 2πr
Cylinder volume	πr²h
Cylinder curved surface	2πrh

Method in steps

1. Substitute into the formula and keep π as a symbol.
2. Collect like terms: 9π + 4π = 13π.
3. Write the unit and “in terms of π” if asked.
4. Never replace π with a decimal until, and unless, a decimal is requested.

Reminder: a semicircle of diameter 8 has perimeter 4π + 8 and area (1/2) × π × 16 = 8π.

Reminder: the area of a circle is 100π, so r² = 100, r = 10 and the circumference is 20π.

Ratios of circle areas: radii 3 and 5 give 9π : 25π = 9 : 25. The π cancels.

Surds (Higher tier only)

A surd is an irrational root such as √2 or √11. A root that is a whole number, such as √49, is not a surd.

Rules

Rule	Example
√a × √b = √(ab)	√6 × √15 = √90 = 3√10
√a ÷ √b = √(a/b)	√98 ÷ √2 = √49 = 7
√a × √a = a	(2√3)² = 12
Roots do not split over addition	√(9 + 16) = 5, whereas √9 + √16 = 7

Simplify a surd

1. List square numbers: 4, 9, 16, 25, 36, 49, 64, 81, 100.
2. Find the largest that divides the number.
3. Take its root outside: √75 = √(25 × 3) = 5√3.

Add and subtract: simplify each surd first, then collect matching surds.

Expand: multiply every term by every term. Surd × same surd gives a whole number.

Difference of two squares: (a + √b)(a − √b) = a² − b. For example (√5 + 2)(√5 − 2) = 5 − 4 = 1.

Rationalising the denominator (Higher tier only)

Bottom of fraction	Multiplier (top and bottom)	Outcome
√k	√k	rational denominator k
a + √k	a − √k	denominator a² − k
a − √k	a + √k	denominator a² − k
√k − √m	√k + √m	denominator k − m

Method in steps

1. Simplify any surd in the denominator first.
2. Choose the multiplier: the surd itself, or the conjugate.
3. Multiply the numerator carefully, expanding every bracket.
4. Simplify the whole fraction, dividing through if possible.

Reminder: 10/√5 = 10√5/5 = 2√5.

Reminder: 3/(√7 + 2) = 3(√7 − 2)/(7 − 4) = 3(√7 − 2)/3 = √7 − 2.

More worked reminders

Fractions in sequence. Work out 3/4 × 5/6 and then 3/5 × 10/9.

3/4 × 5/6 = 15/24 = 5/8
3/5 × 10/9 = 30/45 = 2/3   (or cancel first: 3/9 = 1/3, then 1/3 × 2 = 2/3)

Mixed operations: 7/10 − 1/4 + 1/2 = 14/20 − 5/20 + 10/20 = 19/20.

A compound solid in terms of π. A cylinder has radius 2 and height 7. A hemisphere of radius 2 sits on top. If the volume of a hemisphere is (2/3)πr³, then:

Cylinder: π × 2² × 7 = 28π
Hemisphere: 2/3 × π × 8 = 16π/3
Total: 28π + 16π/3 = 84π/3 + 16π/3 = 100π/3

Answer: 100π/3 cubic units. A fraction multiple of π is still exact.

Squaring and expanding surds (Higher tier only).

(√7 + 3)² = 7 + 3√7 + 3√7 + 9 = 16 + 6√7
(5 - √2)(1 + √2) = 5 + 5√2 - √2 - 2 = 3 + 4√2

Rationalising with simplification (Higher tier only).

6/(√10 - 2) × (√10 + 2)/(√10 + 2)
Denominator: 10 - 4 = 6
Numerator: 6(√10 + 2)
6(√10 + 2)/6 = 2 + √10

Exact lengths (Higher tier only). A rectangle measures 3√2 by 2√6. Its area is 3√2 × 2√6 = 6√12 = 6 × 2√3 = 12√3. Its diagonal squared is 18 + 24 = 42, so the diagonal is √42. Leave √42 as it is, because 42 has no square factor.

Exact answers in the exam

• Read the instruction. “In terms of π”, “exact” and “without a calculator” all point to symbols, not decimals.
• Show each stage on separate lines. A short chain of lines that can be followed is better than one crowded line.
• Check a surd answer against a rough decimal in your head, for example √50 is a little over 7, so 5√2 is about 7.07. This is a sense check only, and the exact form is what you write down.
• If an answer asks for the form a + b√c, finish by writing it in exactly that form.
• When asked to “show that”, write every line of the argument, including the expansion and the conjugate, so that nothing is left to be assumed.

Must-know distinctions

• Exact (5/3, 12π, 4√5) is not the same as approximate (1.67, 37.7, 8.94). Follow the question wording.
• √9 + √16 = 7 but √25 = 5. Roots do not split over addition.
• Simplifying a surd is not the same as rationalising a denominator.
• Rational numbers can be written as a fraction of integers. Surds cannot.
• A fraction of an amount multiplies. Finding the whole divides.

Quick self-test

Do these without a calculator, then check the answers.

1. Work out 5/8 + 7/12.
2. Work out 2 2/5 ÷ 3/10.
3. A circle has diameter 12 cm. Give its circumference and its area in terms of π.
4. A circle has area 100π. Give its circumference in terms of π.
5. (Higher tier only) Simplify √75.
6. (Higher tier only) Simplify √6 × √15.
7. (Higher tier only) Expand and simplify (√5 + 2)(√5 − 2).
8. (Higher tier only) Rationalise 10/√5.
9. (Higher tier only) Rationalise 3/(√7 + 2).
10. (Higher tier only) Work out (2√3)².
11. (Higher tier only) Simplify 2√12 + √27.
12. A sector has angle 90° and radius 8 cm. Give its area and its arc length in terms of π.

Answers

1. 15/24 + 14/24 = 29/24 (1 5/24).
2. 12/5 × 10/3 = 8.
3. Circumference 12π cm. Area π × 6² = 36π cm². 12π cm and 36π cm².
4. r = 10, so circumference = 2π × 10 = 20π.
5. 5√3.
6. √90 = 3√10.
7. 5 − 4 = 1.
8. 10√5/5 = 2√5.
9. 3(√7 − 2)/3 = √7 − 2.
10. 4 × 3 = 12.
11. 2 × 2√3 + 3√3 = 4√3 + 3√3 = 7√3.
12. Area = 1/4 × π × 64 = 16π. Arc = 1/4 × 2π × 8 = 4π. 16π cm² and 4π cm.

Where marks are usually lost

• Adding fractions by adding the denominators.
• Failing to cancel before multiplying, which produces big numbers and errors.
• Dividing by the wrong fraction. Only the second fraction is flipped.
• Writing 3.14 or a calculator value for π when the question asks for an answer in terms of π.
• Using the diameter in a formula that needs the radius.
• Leaving √48 or √72 without checking for the largest square factor.
• Treating √(a + b) as √a + √b.
• Using only the surd to rationalise a two-term denominator.
• Sign errors in the conjugate or in the middle terms of an expansion.
• Not showing the working in an exact question. A bare decimal does not show an exact method.

Official syllabus

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