Money, Growth & Surds

Percentages: core skills

Every percentage question is one of three moves — and each is a single multiplication.

A percentage of a quantity

Write the percentage as a decimal multiplier and multiply: 17% of 240 is 0.17 × 240 = 40.8. Percentages above 100% behave identically: 150% of 60 is 1.5 × 60 = 90.

One quantity as a percentage of another

Divide the part by the whole, then multiply by 100. The whole is whatever the question compares to — the original amount, the cost price — and identifying it is where the mark is won.

Percentage increase and decrease

An increase of r% multiplies by (1 + r ÷ 100); a decrease by (1 − r ÷ 100). A 17.5% discount is one multiplication by 0.825. To find the percentage change itself, use (change ÷ original) × 100 — always dividing by the original.

Worked examples: a $840 jacket reduced by 17.5% has sale price 0.825 × 840 = $693.00. A lamp bought for $32 and sold for $44 has percentage profit (12 ÷ 32) × 100 = 37.5% — the divisor is the cost price.

Interest and reverse percentages

Interest is a percentage increase applied over time. Whether it is added or multiplied each year is the whole difference.

Simple interest

The same interest is added every period, always on the original principal: I = (P × R × T) ÷ 100, with P the principal, R the % per year and T the years. Worked example: $2400 at 4.5% for 5 years gives I = (2400 × 4.5 × 5) ÷ 100 = $540 interest earned, and a total value of 2400 + 540 = $2940.

Compound interest

Each year’s interest is calculated on the new total, so the multiplier applies n times: final value = P × (1 + R ÷ 100)ⁿ, and interest earned = final value − P. The same $2400 at 4.5% compounded for 5 years gives $2990.84 — $50.84 more than the simple scheme.

Reverse percentages

If a value is the result of a change, divide by the multiplier to recover the original. Worked example: after a 12% pay rise an engineer earns $33 600, so 33 600 = original × 1.12 and the original salary is 33 600 ÷ 1.12 = $30 000. Check: 30 000 × 1.12 = 33 600.

Using a calculator

The calculator is examined as a skill in its own right. Marks are lost not because the mathematics is wrong, but because a value was entered carelessly, rounded too early, or read off the display without being interpreted.

Enter the calculation, not the rounded steps

Work to full precision throughout and round once, at the end, to the accuracy the question demands. Over several stages, carry the previous answer forward with the calculator’s stored result rather than retyping a rounded version. Unless a question says otherwise, give the final answer to three significant figures, or exactly if it is exact.

Entering time and mixed units

Time is not decimal. Two hours 30 minutes is 2.5 hours, because 30 minutes is half an hour — but two hours 15 minutes is 2.25 hours, not 2.15. Divide the minutes by 60, or use the calculator’s degrees–minutes–seconds key.

Interpreting the display

The display shows digits; the question decides what they mean. A money answer of 4.8 is written $4.80. A time answer of 3.25 hours is 3 hours 15 minutes — the digits after the point are a fraction of an hour, not a count of minutes.

Worked example: a car travels 148 km in 2 h 15 min. Time = 15 ÷ 60 = 0.25, so 2.25 hours — not 2.15. Average speed = 148 ÷ 2.25 = 65.777… = 65.8 km/h. Rounding the time to 2.3 hours first gives 64.3 km/h — a wrong answer from correct arithmetic.

Time

Time is the one quantity here that is not decimal. Every calculation carries at 60 and at 24.

The 24-hour clock

Before noon, keep the hour and add a leading zero: 3.15 a.m. is 03 15. From noon, add 12: 3.15 p.m. is 15 15. Midnight is 00 00, and a 24-hour time is always four digits. The units: 60 s = 1 min, 60 min = 1 h, 24 h = 1 day, 1 year = 365 days.

Adding and subtracting times

Add the hours and the minutes separately, then carry any total of 60 minutes or more into the hours. Subtracting reverses this: borrow 60 minutes from the hour, not 100. Worked example: a train leaves at 18 55 and takes 3 h 27 min. Minutes: 55 + 27 = 82 = 1 h 22 min, so carry 1 hour; hours: 18 + 3 + 1 = 22. Arrival time = 22 22.

Timetables and time zones

Each column of a timetable is one service — read down a column, never across two. A time-zone difference is applied after the journey time, and only once. Worked example: a flight leaves Singapore at 23 40 local and takes 13 h 20 min; in Singapore time it arrives at 13 00 next day. London is 7 h behind, so 13 00 − 7 h = 06 00 London time.

Money

Money questions are rarely hard arithmetic. They are tested on convention: how an amount is written, how far it is rounded, and which way a currency conversion runs.

Writing money

An amount of money is written to the smallest unit of its currency — two decimal places for dollars, euros and pounds. A calculator display of 4.8 is $4.80; a display of 326.0869… is $326.09. Rounding to the nearest cent is the default, and it applies to the final answer only.

Converting between currencies

An exchange rate is given as "1 unit of A = k units of B". To convert A into B, multiply by k. To convert B back into A, divide by k. Writing the rate out in words before starting is the fastest way to avoid inverting it. Convert once, and round once — at the end.

Worked example: with $1 = €0.92, converting $250 into euros multiplies: 250 × 0.92 = €230.00. Converting €300 into dollars runs against the rate, so divides: 300 ÷ 0.92 = $326.09. A euro is worth less than $1, so a euro amount always converts to a larger number of dollars — a one-second check.

Exponential growth and decay

This section is Extended only. Compound interest is exponential growth with the finance stripped out: a principal becomes an initial amount a, years become periods t, and the plus becomes a plus-or-minus. Knowledge of e is not required.

ExtendedGrowth and decay side by side

The formula is y = a × (1 ± r ÷ 100)ᵗ, with + for growth and − for decay. A fish population rising by 8% each year multiplies by 1.08 every year; a car losing 15% of its value each year multiplies by 0.85 every year. A factor above 1 curves upwards without limit; a factor below 1 falls towards zero but never reaches it.

Worked example: a car bought for $18 000 depreciates by 15% each year. The multiplier is 1 − 15 ÷ 100 = 0.85, so after 4 years y = 18 000 × 0.85⁴ = 18 000 × 0.522006… = $9396 (nearest dollar). It loses 15% of a smaller value each year — not 15% of $18 000 four times.

Surds: simplifying and rationalising

This section is Extended only. A surd is an exact number that happens to be irrational. A decimal throws away precision the question wants kept.

ExtendedThe two rules

Roots multiply and divide term by term: √(ab) = √a × √b (a ≥ 0, b ≥ 0), and √(a ÷ b) = √a ÷ √b (a ≥ 0, b > 0). There is no rule for adding roots — √9 + √16 = 7, not √25. Roots add only as like terms: 3√2 + 5√2 = 8√2.

ExtendedSimplifying a surd

Split the number under the root into its largest square factor, then take that root outside. Worked example: √200 − √32. Since 200 = 100 × 2, √200 = 10√2; since 32 = 16 × 2, √32 = 4√2. Now like terms: 10√2 − 4√2 = 6√2. Expanding brackets: (3√2 + 1)(2 − √2) = 6√2 − 6 + 2 − √2 = 5√2 − 4, because √2 × √2 = 2.

ExtendedRationalising the denominator

A single-term denominator: multiply numerator and denominator by the surd itself. Since √b × √b = b, the denominator becomes rational in one step — 10 ÷ √6 = 10√6 ÷ 6 = 5√6 ÷ 3.

A two-term denominator needs the conjugate. Multiplying (√a + √b) by (√a − √b) is a difference of two squares: the cross terms cancel, leaving a − b. Worked example: 6 ÷ (√5 + √2). The conjugate is √5 − √2; the denominator becomes (√5 + √2)(√5 − √2) = 5 − 2 = 3, and the numerator is 6(√5 − √2), so the fraction is 2(√5 − √2) = 2√5 − 2√2. Multiplying by one root alone leaves a surd on the bottom.