AQA GCSE Mathematics: Number — Revision Notes

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

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

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. And for a negative fractional index, flip first, then apply: (4/9)^(−1/2) = (9/4)^(1/2) = 3/2.

Surds

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

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.
• Using UB ÷ UB for a maximum quotient.
• Writing √a + √b = √(a+b).
• Leaving a surd in the denominator when an exact answer is required.

Self-test

1. Evaluate 8^(2/3) and (4/9)^(−1/2).
2. Rationalise 1/(3 + √2).
3. A price is £69 after a 15% increase. What was it before?
4. Give the bounds for a length of 24 cm measured to the nearest cm.
5. 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.