Here is a problem type that shows up everywhere in JEE arithmetic, banking, discount math, and everyday shopping. You see two or more percentage changes applied one after another to the same quantity, and you have to find the net effect.

"A shopkeeper marks up a dress by 40\% and then offers a 25\% discount. What is the net percentage change over the original cost?"

"An investment grows by 20\% in year one, loses 10\% in year two, and grows another 15\% in year three. What is the total growth?"

"A city's population increased by 5\% in 2023 and decreased by 5\% in 2024. What is the net change?"

If you reach for the operator "+" the moment you see these, you will get a wrong answer. The correct reflex is multiplication of factors, and it takes all of five seconds to execute once you have it.

The recognition pattern

Whenever you see two or more percentage changes chained onto the same base quantity, recognise the pattern:

\text{successive percentage changes} \implies \text{multiply the factors}

A percentage change of p\% becomes a factor:

The net multiplier for the whole chain is the product of all the individual factors. Never the sum.

The three-step execution

Step 1. Read off each percentage change and write its factor.

For "marked up 40\% then discounted 25\%," the factors are 1.40 and 0.75.

Step 2. Multiply the factors together.

1.40 \times 0.75 = 1.05

Step 3. Read the net percentage change from the product.

A product of 1.05 means an overall increase of 5\%. (A product of 0.96 would mean a 4\% decrease, because 0.96 = 1 - 0.04.)

So the shopkeeper's markup-then-discount gives a net 5\% profit over cost — not zero (as the naive "+40 and -25 = +15" or "+40 - 25 = +15" suggests), and not 15\% either. The multiplicative composition of the two changes gives exactly 5\%.

Markup of forty percent then discount of twenty-five percent gives a five percent net gainA flowchart with three boxes connected by arrows. The leftmost box shows a factor of one point forty labelled plus forty percent. A multiplication sign leads to a middle box showing the factor zero point seventy-five labelled minus twenty-five percent. An equals sign leads to the rightmost box showing one point zero five, which is highlighted and labelled net plus five percent. Below the flow, a comparison shows that naively adding the percentages would give plus fifteen percent, which is wrong. 1.40 +40% markup × 0.75 −25% discount = 1.05 net +5% multiply the factors → 1.40 × 0.75 = 1.05 → gain of 5% never add: +40 + (−25) = +15% is the wrong answer reason: the 25% discount is on the marked-up price, a different base
A $40\%$ markup followed by a $25\%$ discount composes as $1.40 \times 0.75 = 1.05$, a net gain of $5\%$. Adding the percentages gives $+15\%$, which is wrong because the two operations act on different bases — the $25\%$ discount is $25\%$ of the already-marked-up price, not $25\%$ of the original.

Why "multiply factors" is the right reflex

Every percentage change is secretly a multiplication. "+40\%" is "multiply by 1.40." "-25\%" is "multiply by 0.75." Successive changes are successive multiplications, and repeated multiplications compose by multiplying the factors. That is arithmetic, not convention.

Adding the percentages fails because it assumes all percentages act on the same base. They don't — each step acts on whatever price survived the previous step. The multiplicative view quietly respects this: x \times 1.40 \times 0.75 computes the new price directly, no matter what x was, and the final-to-original ratio is always 1.05. The answer is base-free, and that is exactly what a "net percentage change" should be.

A three-step chain: practice the reflex

Try the investment example: +20\%, then -10\%, then +15\%.

Factors: 1.20, 0.90, 1.15.

Product: 1.20 \times 0.90 \times 1.15 = 1.08 \times 1.15 = 1.242.

So the investment grew by 24.2\% overall — not +25\% (which is what naive addition 20 - 10 + 15 gives) and not +45\% (which is what you would get if you forgot the loss). The multiplicative product captures the real compounding, including the fact that the 10\% loss in year two was 10\% of a larger amount than the starting value.

The "+5\% then -5\%" trap

The classic trap: "A price rose by 5\% and then fell by 5\%. What is the net change?" Your instinct says zero. The correct answer is -0.25\%.

Factors: 1.05 and 0.95. Product: 1.05 \times 0.95 = 0.9975. That is 0.25\% below 1, a tiny net loss.

The gap — 0.0025 — is exactly 0.05^2, the square of the percentage expressed as a decimal. This is always true for "+p\% then -p\%" chains: the net multiplier is (1+p)(1-p) = 1 - p^2, a net loss of p^2 (as a fraction). For p = 0.05, you lose 0.0025 = 0.25\%. For p = 0.20, you lose 0.04 = 4\%. For p = 0.50, you lose 0.25 = 25\%. Volatility is quadratically expensive.

Why this squared gap appears: (1+p)(1-p) = 1 - p^2 is a difference-of-squares identity. The two percentage changes almost cancel — they would exactly cancel if they were additive — but their multiplicative combination leaves a -p^2 residue. The residue is small for small p but noticeable for large p, and it always points "downward" (i.e., the net change is a loss).

Converting a product back to a percentage

Once you have the net multiplier, read off the net percentage change by subtracting one and shifting the decimal two places.

In exam conditions you can often leave the answer as the multiplier, with the percentage implicit. "A net multiplier of 0.75" and "a 25\% net discount" say the same thing.

What the rule does not say

Important caveat: the "multiply factors" rule applies when the changes are successive on the same quantity. If two percentages act on different bases — "the tax is 18\% of the subtotal, and the shipping is 10\% of the original price" — then you cannot just multiply the factors. You have to compute each contribution to the total separately.

So the triggering pattern, in full, is: successive percentage changes stacking on the same running quantity → multiply the factors.

Reflex checklist

When you spot a problem with two or more percentage changes:

  1. Is each change applied to the result of the previous one? If yes, this rule applies.
  2. Convert each change to a factor: (1 + p/100) for increases, (1 - p/100) for decreases.
  3. Multiply all the factors.
  4. Read off the net percentage from the product.

This reflex replaces the wrong instinct "+" with the right one "\times." Once you run it a few times, you will stop seeing successive percentages as an addition problem and start seeing them as what they are — a product.

Related: Percentages and Ratios · Price Up 20% Then Down 20% — Why You End Up Below Where You Started · Why You Can't Just Add Percentages When Discounts Stack · Plus 20% Then Minus 20%: Why You End Up Below Where You Started