Percentages, interest and reverse percentages

Convert the percentage change into a single number to multiply by:

The last row is the one to keep separate. “Find 8% of $250” is $0.08\times 250=\mathrm{\$}20$; “increase $250 by 8%” is $1.08\times 250=\mathrm{\$}270$. Reading which is being asked is worth more marks than any technique on this page.

The denominator is always the original value — the one before the change — and getting that wrong turns a 25% rise into a 20% one. Profit and loss questions are the same formula with cost price as the original:

“Compound” means each year’s change applies to the new amount, not the original. $2 000 at 3% for 4 years is $2000\times {1.03}^4=\mathrm{\$}2251.02$. The interest earned is that minus the original — $\mathrm{\$}251.02$ — and questions ask for one or the other, so check which.

Depreciation and decay use the same formula with a multiplier below 1: a car worth $18 000 losing 15% a year is worth $18000\times {0.85}^5=\mathrm{\$}7986.70$ after five years. Population growth, bacterial growth and radioactive decay are all this formula with different words.

When the value you are given is the one after the change, you divide by the multiplier:

A coat costs $60 in a 20% off sale. The multiplier was $0.8$, so the original price was $60÷0.8=\mathrm{\$}75$. Taking 20% off $60 gives $48, which is wrong — and the check is immediate: 20% off $75 is $60. ✓

The signal words are “after”, “in a sale”, “including tax”, “had increased by” — anything saying the figure you have already contains the change. Extended papers also compound it: divide by the multiplier raised to $n$ to get back to the starting value $n$ years ago.

1. 1.Reverse percentage done by taking the percentage off

   If the figure is already after the change, divide by the multiplier. Then check by applying the change forwards — it takes five seconds and catches the error every time.

2. 2.Percentage change divided by the new value

   The denominator is the original. Change ÷ original × 100.

3. 3.Compound interest worked out year by year and rounded each time

   Use the power: $P\times {1.04}^3$ in one step. Rounding at each stage accumulates error and can lose the accuracy mark.

4. 4.Giving the total when the question wanted the interest

   “How much interest is earned” means final minus initial. “What is it worth” means the final value.

5. 5.Percentage changes added together

   A 10% rise then a 10% fall is not back to the start: $1.1\times 0.9=0.99$, a 1% loss overall. Multiply multipliers, never add percentages.