AQA GCSE Mathematics: Number — Revision Notes

Condensed for the final weeks. For the full explanation, use the Number study guide.

Tier: fractional indices (N7), surds (N8), recurring decimals to fractions (N10) and calculating with upper and lower bounds (N16) are Higher tier only, and are marked (Higher) below. Everything else is Foundation content, assessed at both tiers.

Index laws

a^m x a^n = a^(m+n)         a^m / a^n = a^(m-n)         (a^m)^n = a^(mn)
a^0 = 1                     a^-n = 1 / a^n              a^(1/n) = nth root of a
a^(m/n) = (nth root of a)^m

Integer indices, including negative ones, are Foundation. (Higher) For a fractional index, take the root first, then the power — the numbers stay small.

8^(2/3) = (cube root of 8)^2 = 2^2 = 4

A negative index means reciprocal, not a negative answer: 2⁻³ = 1/8, never −8. (Higher) For a negative fractional index, flip first, then apply: (4/9)^(−1/2) = (9/4)^(1/2) = 3/2.

Surds (Higher)

sqrt(a) x sqrt(b) = sqrt(ab)        sqrt(a) / sqrt(b) = sqrt(a/b)

Simplify by extracting the largest square factor: √50 = √25 × √2 = 5√2.

Rationalising the denominator:

single term:   3/sqrt(2)  ->  multiply top and bottom by sqrt(2)   =  3 sqrt(2) / 2

two terms:     1/(3 + sqrt(2))  ->  multiply by the CONJUGATE (3 - sqrt(2))
               = (3 - sqrt(2)) / (9 - 2) = (3 - sqrt(2)) / 7

The conjugate works because (a + √b)(a − √b) = a² − b, which contains no surd.

√a + √b ≠ √(a+b) — check with √9 + √16 = 7, not √25 = 5.

Standard form

A x 10^n        where  1 <= A < 10

Multiplying and dividing: handle the numbers and the powers separately, then re-adjust if A falls outside the range. 40 × 10⁵ must become 4 × 10⁶.

Percentages

increase by 15%:   x 1.15          decrease by 15%:   x 0.85
reverse percentage: DIVIDE by the multiplier
compound interest:  P x (multiplier)^n

Reverse percentage is the most commonly failed question. If a price is £69 after a 15% increase, the original is 69 ÷ 1.15 = £60 — not 69 × 0.85, which gives £58.65. Divide, never subtract.

A 15% rise followed by a 15% fall does not return you to the start: 1.15 × 0.85 = 0.9775, a 2.25% net loss.

Fractions

Multiply: multiply across. Divide: multiply by the reciprocal of the second fraction. Add and subtract: common denominator first. Always convert mixed numbers to improper fractions before multiplying or dividing.

Bounds

For a value rounded to the nearest unit u, add and subtract u/2.

length 24 cm to nearest cm:   23.5 <= L < 24.5

Writing this error interval is Foundation (N15). (Higher) Combining bounds — the rule that decides the marks:

Operation	For the maximum	For the minimum
Add	UB + UB	LB + LB
Subtract	UB − LB	LB − UB
Multiply	UB × UB	LB × LB
Divide	UB ÷ LB	LB ÷ UB

Subtraction and division cross over. That is the entire difficulty of the topic.

HCF and LCM

Use prime factorisation. HCF = product of the lowest power of each common prime. LCM = product of the highest power of every prime appearing.

Worked example. Find the HCF and LCM of 60 and 72.

60 = 2^2 x 3 x 5
72 = 2^3 x 3^2

HCF = lowest power of each shared prime = 2^2 x 3 = 12
LCM = highest power of every prime present = 2^3 x 3^2 x 5 = 360

Reverse percentage for a decrease

A jacket costs £68 after a 15% reduction. The sale price is 85% of the original, so the multiplier is 0.85: original = 68 ÷ 0.85 = £80. Check by working forwards: 80 × 0.85 = 68.

The trap here is finding 15% of £68 and adding it on — that gives £78.20, which is wrong, because the percentage applies to the original price, not the sale price. Estimating by rounding each value to 1 significant figure before a calculation is also useful for sanity-checking whether an answer is plausible.

Exam traps

• Reading a negative index as a negative answer.
• Subtracting a percentage instead of dividing in reverse-percentage problems.
• Assuming an increase then an equal decrease cancels out.
• (Higher) Using UB ÷ UB for a maximum quotient.
• (Higher) Writing √a + √b = √(a+b).
• (Higher) Leaving a surd in the denominator when an exact answer is required.

Self-test

1. (Higher) Evaluate 8^(2/3) and (4/9)^(−1/2).
2. (Higher) Rationalise 1/(3 + √2).
3. A price is £69 after a 15% increase. What was it before?
4. Give the error interval for a length of 24 cm measured to the nearest cm.
5. (Higher) How do you find the maximum value of a ÷ b from bounds?
6. A jacket costs £68 after a 15% reduction. Find the original price, and explain the common error to avoid.
7. Find the HCF and LCM of 60 and 72 using prime factorisation.

Answers: 1. 4; and 3/2. 2. Multiply top and bottom by (3 − √2) to get (3 − √2)/7. 3. 69 ÷ 1.15 = £60. 4. 23.5 ≤ L < 24.5. 5. Upper bound of a divided by the lower bound of b. 6. £80, since the multiplier for an 85% sale price is 0.85 and 68 ÷ 0.85 = £80; the common error is finding 15% of £68 and adding it on, which wrongly applies the percentage to the sale price rather than the original. 7. 60 = 2² × 3 × 5 and 72 = 2³ × 3², so HCF = 2² × 3 = 12 and LCM = 2³ × 3² × 5 = 360.