The conversions worth keeping
Most percentage questions on a placement or aptitude paper are not hard arithmetic. They are slow arithmetic. If you already know a short list of fraction conversions, you can replace a long division with one multiplication.
Keep these in memory and use them without recalculating:
- 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%
- 1/6 = 16.67%, 1/8 = 12.5%, 1/10 = 10%, 1/20 = 5%
- 1/25 = 4%, 3/4 = 75%, 2/5 = 40%, 3/8 = 37.5%
So 12.5% of 240 is the same as one-eighth of 240, which is 30. You do not need to write 12.5/100 and multiply.
Swap the two numbers
A second shortcut is easy to forget under a timer: x% of y equals y% of x. The product is the same either way.
Take 25% of 84. That is 84% of 25. Ten percent of 25 is 2.5, so 80% is 20, and 4% is 1. The answer is 21. The same idea helps when one of the numbers is a round 100, 50, or 25 and the other is awkward.
Successive change is not ordinary addition
If a value rises by a% and then by b%, the net change is not a + b. Use:
Net percentage change = a + b + (a × b) / 100
The second change acts on the new value, so the cross term is required. Put a fall in as a negative number.
A price rises 20% and then falls 20%. Net change = 20 + (−20) + (20 × −20) / 100 = −4. The price is 4% below the start, not back where it began. On 500, the path is 600 and then 480.
The same formula covers two rises. A quantity grows 10% and then 20%. Net change = 10 + 20 + (10 × 20) / 100 = 32%, not 30%.
A question that uses both ideas
A number is increased by 25% and the result is 180. What was the original number?
125% of the original is 180, so the original is 180 × 100 / 125. Since 125% is 5/4, dividing by 5/4 means multiplying by 4/5: 180 × 4 / 5 = 144.
Check: 25% of 144 is 36, and 144 + 36 = 180.
The mistake that costs the mark
When a question says “increased by 10% and then decreased by 10%,” do not cancel the two changes. Compute the net with the formula, or apply each step to the running total. Also watch the base. “20% more than last year” uses last year as the base. “20% of the new price” uses the new price. Read which number the percentage hangs on before you calculate.
Practice the fraction list until it is automatic, then use the swap and the successive-change formula on mixed questions. Those three tools cover a large share of the percentage section.