Exact calculation: fractions, π, surds
WJEC Higher Unit 1 (non-calculator) is built around exact answers. Decimal approximations lose marks here.
Working with fractions
- Add/subtract: common denominator first.
- Multiply: numerators × numerators, denominators × denominators, then simplify.
- Divide: multiply by the reciprocal.
- Simplify by dividing top and bottom by their HCF.
Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
Working in multiples of π
When circle calculations are required exactly, leave π in the answer.
- Circumference = 2πr or πd
- Area = πr²
- Sector area = (θ/360) × πr²
- Arc length = (θ/360) × 2πr
Example: a circle of radius 6 has area 36π cm² (NOT 113.10).
Working with surds
A surd is an irrational root that cannot be simplified to a rational number, e.g. √2, √3, √7.
Rules:
- √a × √b = √(ab)
- √a / √b = √(a/b)
- √(a²b) = a√b
- (√a)² = a
Simplifying surds
Find the largest perfect-square factor.
- √50 = √(25 × 2) = 5√2.
- √75 = √(25 × 3) = 5√3.
- √200 = √(100 × 2) = 10√2.
Rationalising the denominator
To remove a surd from the bottom of a fraction, multiply top and bottom by that surd:
3 / √5 = 3√5 / 5.
For "(1 + √a)" denominators, multiply by the conjugate "(1 − √a)".
Adding and subtracting surds
Only "like surds" can be added: 3√2 + 5√2 = 8√2. Different surds stay separate: √2 + √3 cannot be simplified.
WJEC exam tip
If the question says "give your answer in the form a + b√c where a, b, c are integers", the marker is looking for those exact values. Convert to a decimal and you lose every A1 mark in the question.
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