Problems on Ages
- Let the present age be x; n years ago it was x − n and n years hence it will be x + n.
- The difference between two people's ages never changes. If the ratio of ages is 3 : 1 now and the father is 30 years older, then 2 parts = 30, so the ages are 45 and 15.
- The total age of a group rises by (number of people × years). The average age of a group of 5 rises by exactly 1 each year.
- Ratio questions: write the ages as 3k and 5k (say), apply the change to both, and solve the new ratio for k.
Worked example
Ages are in the ratio 4 : 5. After 6 years the ratio is 6 : 7. Then (4k + 6)/(5k + 6) = 6/7 gives 28k + 42 = 30k + 36, so k = 3 and the ages are 12 and 15.
Square Roots and Cube Roots
- A perfect square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8. A number ending in an odd count of zeros is not a perfect square.
- Square root by prime factors: pair the factors. 1764 = 2² × 3² × 7², so √1764 = 2 × 3 × 7 = 42. Cube root: group in threes. 2744 = 2³ × 7³, so ∛2744 = 14.
- Smallest number to multiply by to make a perfect square: the product of the unpaired prime factors. 180 = 2² × 3² × 5, so multiply by 5.
- Roots of decimals: count decimal places in pairs. √0.0009 = 0.03 and √0.09 = 0.3.
Surds
A surd is a root that cannot be simplified to a rational number, such as √2 or ∛5. Only like surds can be added: 3√2 + 5√2 = 8√2, but √2 + √3 stays as it is.
| Rule | Example |
|---|---|
| √a × √b = √(ab) | √2 × √8 = √16 = 4 |
| √(a²b) = a√b | √72 = √(36 × 2) = 6√2 |
| (√a + √b)(√a − √b) = a − b | (√5 + √3)(√5 − √3) = 2 |
| Rationalise 1/(√a + √b): multiply by (√a − √b) | 1/(√3 + √2) = √3 − √2 |
Comparing surds
Raise them to a common power. To compare ∛4 and √3, take the 6th power: ∛4 gives 16 and √3 gives 27, so √3 is larger.
Laws of Indices
| Law | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 25 |
| (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| a⁰ = 1, a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| a^(m/n) = (ⁿ√a)ᵐ | 8^(2/3) = 2² = 4; 27^(−1/3) = 1/3 |
Common Mistakes
Writing √(a + b) as √a + √b (√9 + √16 = 7, but √25 = 5); applying an age change to only one person; and treating a⁰ as 0 instead of 1.