Aptitude · Percentages · Model 2

Down 20% then up 20% leaves you down 4%

Two changes in a row never add, because the second acts on what the first left. The formula A + B + AB/100 is the whole model, and the product term is exactly what the naive sum forgets.

Set two changes and watch the product term appear →

Net change from successive changes of A% and B% is A + B + AB/100. The product term is why they never simply add.

01 The idea

The second change acts on a different number

An income falls 20% and then rises 20%. It feels like it should be back where it started, and it is not — it is down 4%. The fall took 20% of the original; the rise gave 20% of the smaller figure that was left. Different bases, so they cannot cancel.

Work it in multipliers. Down 20% is × 0.8, up 20% is × 1.2, and 0.8 × 1.2 = 0.96. So you keep 96% of what you had — a 4% loss. That is the whole computation, and it generalises to any number of changes because multipliers chain.

Expanding the product gives the standard formula: for changes of A% and B%, the net change is A + B + AB/100, with the signs carried. For −20 and +20 that is −20 + 20 − 400/100 = −4%. The first two terms are the naive sum; the third is the correction the naive sum omits.

One special case is worth knowing outright. Equal and opposite changes of x% always leave a loss of x²/100 per cent — 4% for 20%, 9% for 30%, 1% for 10%. It is always a loss, whichever order you apply them in, and never zero.

Turn each change into a multiplier and multiply. The net percentage is the product minus one, and it is never the sum of the two percentages.

Successive changeTwo or more percentage changes applied one after another, each acting on the result of the previous one.

The product termAB/100 in the net-change formula. It is the change-on-the-change, and it is exactly what adding the percentages leaves out.

Equal and oppositeA rise and fall of the same x% leaves a loss of x²/100 per cent. Always a loss, and independent of the order.

02 Worked example

Income down 20%, then up 20%

The case everyone gets wrong first. A person’s income decreases by 20% and then increases by 20%. What is the net change?

Take the income as 100The answer is a percentage, so the actual figure does not matter and 100 is easiest.income = 100

Apply the 20% fallTwenty per cent of 100 is 20, so the income drops to 80.100 × 0.8 = 80

Apply the 20% rise to 80, not to 100This is the whole point. Twenty per cent of 80 is 16, not 20.80 × 1.2 = 96

Compare with where you startedNinety-six against an original hundred.96 − 100 = −4 ⇒ a 4% decrease

Check against the formulaA = −20 and B = +20, carried with their signs.−20 + 20 + (−20 × 20)/100 = −4% ✓

The lost 4% is exactly the 20% that was not restored: the rise gave back 16 where the fall had taken 20. Note also that reversing the order gives the same answer — up 20% to 120, then down 20% to 96. Multipliers commute, so the order of successive changes never affects the final value.

03 The method

The formula, and the consumption question built on it

One formula covers every successive-change question. The second box is the model’s other half, and it is the same idea read backwards.

Net change = A + B + AB/100, signs carried. Equivalently, multiply the multipliers: (1 + A/100)(1 + B/100) − 1.

The price-and-consumption question. If a price falls by R%, consumption may rise by R/(100 − R) × 100 per cent with no change in expenditure — because the two multipliers must multiply to 1. A 20% price fall allows a 25% consumption rise; a 50% fall allows 100%. And if income and expenditure both change by the same x%, savings change by exactly x% too, which is worth spotting on sight.

Changes	Naive sum	True net
−20% then +20%	`0%`	`−4%`
−30% then +30%	`0%`	`−9%`
+15% then −20%	`−5%`	`−8%`
+20% then −30%	`−10%`	`−16%`
+25% then −20%	`+5%`	`0%`
Price −20%	—	consumption may rise 25%
Price −50%	—	consumption may rise 100%

05 Cheat sheet

Model 2 on one page

The formula, the two special cases, and the consumption rule.

Case	Route	Worked
Two changes	`A + B + AB/100`	−20, +20 → −4%
As multipliers	`(1+A/100)(1+B/100) − 1`	0.8 × 1.2 = 0.96
Equal and opposite x%	`loss of x²/100 %`	20% → 4% loss
Price falls R%, consumption	`R/(100−R) × 100`	20% → 25%
Income and expenditure both +x%	`savings also +x%`	no calculation needed
Order of the changes	`irrelevant`	multipliers commute
Adding the percentages	`always wrong`	misses AB/100

Equal and opposite always losesA rise and fall of the same x% leaves a loss of x²/100 per cent — never zero, and never a gain, whichever order you use.

The order never mattersThe final value is the starting value times both multipliers, and multiplication commutes. Questions that stress the order are checking whether you know that.

Same-percentage income and expenditureIf both change by x%, savings change by x% too. It falls straight out of the algebra and saves the whole calculation.

06 Where & why

Where Model 2 shows up

One of the most heavily set percentage models, because the intuitive answer is wrong and is always in the options.

Bank PO · SSC CGL

Income, expenditure and savings

“Income up 20%, expenditure up 15% — find the change in savings.” Work in 100s and it is four short lines.

TCS NQT · Infosys

Price and consumption

“Price falls 10% — by how much may consumption rise?” The answer is 11.11%, and 10% is the trap option.

Discounts and markups

The same arithmetic, different chapter

Successive discounts use A + B + AB/100 with both signs negative. A markup then a discount is the same formula with mixed signs.

Population, inflation, depreciation

Any quantity changing repeatedly

Percentage changes over successive years multiply. This is also why compound interest is a power rather than a product.

Whenever two percentage changes appear in one sentence, write both multipliers before doing anything. Adding the percentages is wrong every single time, and the wrong answer is always offered.

07 Interview questions

What gets asked

Ten, and the first two are the misconception the model exists to break.

An income falls 20% and then rises 20%. What is the net change?

A 4% decrease. The fall takes 100 to 80, and the 20% rise then gives 20% of 80 — which is 16, not 20 — taking it to 96. The two changes act on different bases, so they cannot cancel.

Give the general formula.

Net change = A + B + AB/100, with signs carried. For −20 and +20 that is −20 + 20 − 400/100 = −4%. Equivalently multiply the multipliers: 0.8 × 1.2 = 0.96.

What happens with equal and opposite changes of x%?

A loss of x²/100 per cent, always. Twenty per cent gives a 4% loss, thirty gives 9%, ten gives 1%. It is never zero and never a gain, and the order makes no difference.

Does the order of the two changes matter?

No. The final value is the start times both multipliers, and multiplication commutes. Up 25% then down 20% gives exactly the same result as down 20% then up 25%. Questions that emphasise the order are testing whether you know it is irrelevant.

Petrol falls 10% in price. By how much may consumption rise with no change in expenditure?

11.11%. Expenditure is price times consumption, so the two multipliers must multiply to 1. A 10% fall is × 9/10, so consumption must be × 10/9, a rise of 1/9. The formula is R/(100 − R) × 100.

And if the price falls 50%?

Consumption may double — a 100% rise. Half the price means twice the quantity for the same money, which is obvious once stated but catches people who reach for 50%. The formula gives 50/50 × 100 = 100%.

An income rises 20% and expenditure rises 15%, with 75% originally spent. What happens to savings?

They rise 35%. Take income as 100, so expenditure is 75 and savings 25. New income is 120, new expenditure is 75 × 1.15 = 86.25, so savings become 33.75 — an increase of 8.75 on 25, which is 35%.

Is there a shortcut when income and expenditure change by the same percentage?

Yes: savings change by that same percentage too. If both rise 20%, savings rise 20%, with no calculation needed. It follows because savings are income minus expenditure and both terms are being scaled by the same factor.

What two changes leave a quantity exactly unchanged?

Hard

A share price rises 25%, then falls 20%, then rises 40%. (a) Find the single equivalent percentage change. (b) Find the percentage fall now needed to bring it exactly back to its original price. (c) Show that no sequence of a rise of x% and a fall of x% can ever return a price to its starting value, and give the general expression for the shortfall. (d) State what would have to be true of the two percentages for the price to return exactly.

Follow-up

Parts (c) and (d) are the model made general. Equal percentages can never cancel, and the exact condition for a genuine reversal is that the multipliers are reciprocals — so a rise of x% needs a fall of 100x/(100+x) per cent, not x%. That single expression covers the whole reverse-pair family.

Show the hint

Work in multipliers throughout. For (d), ask what the product of the two multipliers must equal.