Percentages are the most frequently tested idea in the quantitative section, and the one most often failed by candidates who can actually do the arithmetic. The failure is almost always the same: the percentage is applied to the wrong base. A pay rise of 10 percent followed by a cut of 10 percent does not leave the pay unchanged — and a candidate who computes it as "unchanged" has made a reasoning error, not a calculation error.
Once you handle percentages as multipliers, most of these questions collapse into one or two multiplications, and the trap disappears.
Replace every percentage with a multiplier and the questions become multiplication problems.
| Change | Multiplier | Change | Multiplier |
|---|---|---|---|
| +10% | x 1.10 | -10% | x 0.90 |
| +25% | x 1.25 | -25% | x 0.75 |
| +50% | x 1.50 | -50% | x 0.50 |
Two consequences follow immediately. First, a sequence of changes is just a product of multipliers, and multiplication order does not matter — so +20 percent then -20 percent is 1.2 x 0.8 = 0.96, a net fall of 4 percent, not zero. Second, "find the original" is a division: if the final value is 660 after a 25 percent discount, the original is 660 divided by 0.75, which is 880.
The model is incomplete in one way worth naming: percentage change is always measured against the starting value. Identify that base before you touch the numbers, every single time.
Percentage of a quantity. Split into tens where possible. Eighteen percent of 250: 10 percent is 25 and 8 percent is 20, so 18 percent is 25 + 20 = 45.
Percentage change. Use the form:
A price moves from 800 to 920: the change is 120, and 120 divided by 800 is 0.15, so the increase is 15 percent. Note that dividing by the old value is what makes it a percentage increase rather than a percentage of the new price.
Successive changes. Multiply the multipliers. A 25 percent increase followed by a 20 percent discount is 1.25 x 0.80 = 1.00 — the price returns exactly to where it started. This is why shops can advertise a discount after a rise without losing anything, and it is a favourite question shape.
Reverse percentage. When the final value and the change are known and the original is wanted, divide by the multiplier rather than subtracting a percentage of the final value. After a 25 percent discount the price is 660, so the original is 660 / 0.75 = 880. Check it forward: 25 percent of 880 is 220, and 880 - 220 = 660.
Profit and loss percent is a percentage change with a vocabulary. A shop buys at 400 and sells at 500: profit is 100, and 100 / 400 = 25 percent. Profit is measured against cost, not against selling price.
Percentage points are not percentages. If a pass rate rises from 40 percent to 46 percent, that is an increase of 6 percentage points, but a relative increase of 6 / 40 = 15 percent. Read the question to see which one is being asked.
A shop's price for a tool is raised by 25 percent at the start of the season. At the end of the season the shop offers a 20 percent discount on the new price. A customer says the tool now costs exactly what it did before. Is the customer right?
Work with multipliers and no original price is needed. The rise gives a multiplier of 1.25; the discount gives a multiplier of 0.80. Their product is 1.25 x 0.80 = 1.00, so the final price equals the original price: the customer is right.
Now change one number and watch what happens. Keep the discount at 20 percent but reduce the rise to 10 percent: 1.10 x 0.80 = 0.88, a net fall of 12 percent — larger than either individual change, because both now pull the price down from the higher intermediate base. Make the discount equal to the rise instead, 25 percent each way, and the multiplier is 1.25 x 0.75 = 0.9375, a net fall of 6.25 percent. The relationship is asymmetric, which is exactly why the test keeps asking it: only one pair of numbers (a rise of n percent followed by a discount of n/(1+n)) returns the price exactly home.
A quantity is increased by 20 percent and then decreased by 20 percent. What is the net change?
After a 25 percent discount a jacket costs 660. What was its price before the discount?
A shop buys an item for 400 and sells it for 500. What is the profit percentage?
Without looking back: what is the net effect of a 20 percent rise followed by a 20 percent fall, and how would you find the original price from a final price after a 30 percent discount?
The answer is: a net fall of 4 percent, because 1.2 x 0.8 = 0.96. To reverse a 30 percent discount, divide the final price by 0.70.
Fill in the blanks from memory:
A 15 percent increase is the multiplier . A rising pass rate that moves from 40 percent to 46 percent has risen by 6 percentage points, which is a relative increase of percent. Eighteen percent of 250 is .
A salary is 60,000. It is raised by 10 percent, and later cut by 10 percent. What is the final salary, and what single percentage change would have produced the same result in one step?
Multiply the multipliers: 1.10 x 0.90 = 0.99. The final salary is 60,000 x 0.99 = 59,400. The equivalent single change is a fall of 1 percent.
Notice the two things this example is designed to expose. The absolute loss is 600, which is small enough that a candidate working in whole numbers might round it away and answer "unchanged" — the same mistake the multiplier method prevents. And the single-step equivalent is not 1 percent of the original change; the compounding is what produces exactly 0.99, and the size of the net effect depends on the size of the changes, not on their difference.
Percentages handle comparisons against a single base. The next technique handles comparisons between quantities — how one part relates to another, how a total splits, and how a rate scales when one input changes. That is ratio and proportion, and it is the machinery behind most word problems in this section.