Percentages are multipliers
Most percentage errors come from adding percentages that should be multiplied. Write each change as a multiplier and the rest is arithmetic.
Rise 20%, then fall 20%:
1.20 × 0.80 = 0.96 -> overall 4% fall, not 0%
Shortcut for two changes a% and b%:
a + b + (a × b) / 100
20 + (-20) + (20 × -20)/100 = -4 -> 4% fall, same answer
Reverse percentages use the same idea. If a price after a 25% rise is 150, the original is 150 ÷ 1.25 = 120. Taking 25% off 150 gives 112.5, which is wrong because the 25% was a share of the old price, not the new one. Expect 112.5 among the options, because it is the instinctive answer.
The direction of comparison also matters. If A is 50 and B is 40, A is 25% more than B (10 ÷ 40), but B is 20% less than A (10 ÷ 50). The difference is the same; the base changes.
Percentage points and percentage change
When the quantity is itself a percentage, two different answers are both called "the increase". An interest rate moving from 4% to 5% has risen by 1 percentage point, and by 25% in relative terms, since 1 ÷ 4 = 0.25. Read the question for which one it asks. If the options include both 1 and 25, the setter is testing exactly this distinction.
Ratios: scale to a common term
A ratio is a set of parts. Divide the total by the number of parts and multiply out. If A:B = 3:5 and the total is 64, one part is 64 ÷ 8 = 8, so A = 24 and B = 40.
To combine two ratios that share a term, make the shared term equal. With A:B = 2:3 and B:C = 4:5, B is 3 in one and 4 in the other. Scale the first by 4 and the second by 3, giving A:B = 8:12 and B:C = 12:15, so A:B:C = 8:12:15.
Data tables: estimate first, then check
Data interpretation questions in numerical reasoning tests are mostly about reading the right cell and choosing the right comparison. The arithmetic is usually simple. Take this table of regional revenue:
| Region | Last year | This year | Change | % change |
|---|
| North | 240 | 276 | +36 | 15.0% |
| South | 180 | 216 | +36 | 20.0% |
| East | 410 | 459 | +49 | 12.0% |
| West | 95 | 118 | +23 | 24.2% |
"Which region grew most?" has two answers. East had the largest absolute increase, 49. West had the largest percentage increase, about 24.2%, while having the smallest absolute increase. The question wording decides, and the wrong reading is usually offered as an option.
Estimation narrows the field before you calculate. West rose by 23 on a base of about 100, so roughly 23%. East rose by about 50 on about 400, roughly 12%. That is enough to rank them without a calculator.
For a single figure, break the percentage into easy pieces. To find 18.7% of 4,920:
10% of 4,920 = 492
5% of 4,920 = 246
3.7% of 4,920 = 3.7 × 49.2 = 182.04
18.7% total = 492 + 246 + 182.04 = 920.04
Estimate check: 20% of 4,900 = 980, a little high, as expected.
Accurate arithmetic on the wrong comparison
The trap in numerical reasoning is not slow arithmetic. It is calculating correctly on the wrong base: percentage points when the question meant relative change, the new value when it meant the old, absolute growth when it asked about percentage growth. These errors give a clean, confident number that matches an option, so nothing feels wrong.
The fix is a five-second habit before calculating. Say what the base is ("of last year's figure") and what unit the answer is in ("percent change, not points"). Then estimate, then calculate. If the exact answer and the estimate disagree by more than you expect, one of them used the wrong base.
Pacing against the clock
Many numerical reasoning tests are timed tightly enough that finishing every question is hard. Check the invitation for whether a calculator is allowed, since that changes how much estimation you need. If one question needs four table lookups and another needs one, take the short one first when the test lets you move between questions. Practise with a timer, and track errors by type rather than by count.
Name the base and the unit before you calculate; most wrong options in numerical reasoning tests are right arithmetic on the wrong comparison.
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