Percentages and Percentage Change
Converting between fractions, decimals, and percentages, and correctly combining successive percentage increases and decreases.
What you'll learn
- Convert a value between fraction, decimal, and percentage form
- Compute the result of a single percentage increase or decrease applied to a value
- Combine two successive percentage changes into one equivalent net percentage change
Prerequisites
Explanation
A percentage is just a fraction with a fixed denominator of 100 — "20%" literally means 20/100, or 0.2 as a decimal. Converting between the three forms is mechanical: fraction → decimal by dividing, decimal → percentage by multiplying by 100, percentage → decimal by dividing by 100. The fraction 3/8, for instance, is 0.375 as a decimal, which is 37.5% — three-eighths of anything is just over a third of it.
Applying a percentage change to a value means multiplying by (1 + percent/100) for an increase, or (1 − percent/100) for a decrease. A price of ₹250 increased by 20% becomes 250 × 1.20 = ₹300. A price of ₹250 decreased by 15% becomes 250 × 0.85 = ₹212.50. This single multiplying-factor trick is worth internalizing because it's also exactly how successive percentage changes work.
Successive percentage changes do not simply add. This is the single most common percentage mistake in aptitude tests. If a value increases by 20% and then decreases by 10%, the net change is not +10%. Instead, you multiply the two factors: 1.20 × 0.90 = 1.08, meaning a net increase of 8%, not 10%. The general formula for combining two successive percentage changes p1 and p2 into one net percentage change is:
net% = p1 + p2 + (p1 × p2)/100
Check it against the example above: 20 + (−10) + (20 × −10)/100 = 20 − 10 − 2 = 8. Matches. The correction term (p1 × p2)/100 is exactly what people forget when they just add the two percentages.
Worked example. A shop increases the price of an item by 25%, then later decreases the new price by 20%. What's the net change? Using the formula: 25 + (−20) + (25 × −20)/100 = 25 − 20 − 5 = 0%. Despite the numbers looking like they should roughly cancel and give a small net change, they cancel exactly here — the 25% increase and 20% decrease exactly offset each other, because 1.25 × 0.80 = 1.00 precisely. That's not a coincidence for these particular numbers; it's a useful pattern to recognize: an increase of x% is exactly undone by a subsequent decrease of x/(100+x) × 100%, not by a decrease of x% (25/(125)×100 = 20, which is exactly the 20% used here).
The habit to build: whenever a problem says "increased by A% then decreased by B%" (or vice versa), reach for multiplying factors — (1 + A/100)(1 − B/100) — rather than adding or subtracting the raw percentages.
Example
Applying a percent increase or decrease multiplies the original value by (1 + percent/100).
function percentChange(value, percent) {
return Math.round(value * (1 + percent / 100) * 100) / 100;
}
// Example: percentChange(200, 20) -> 240Guided exercise
Guided exercise
Write applyPercentChange(value, percent) that returns value after a single percentage change is applied (a positive percent means increase, a negative percent means decrease), rounded to 2 decimal places.
Checks: Correctly applies a percentage increase · Correctly applies a percentage decrease · plus 1 hidden check
Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.
Stuck? Get a hint.
Independent exercise
Independent exercise
Write successivePercentChange(p1, p2) that returns the single equivalent net percentage change (rounded to 2 decimal places) that results from applying percentage change p1 followed by percentage change p2 to some value. Use the formula net% = p1 + p2 + (p1 * p2) / 100.
Checks: Correctly nets a mixed increase and decrease · Correctly nets two successive increases · plus 1 hidden check
Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.
Stuck? Get a hint.
Common mistakes
- Adding successive percentage changes directly (treating +20% then -10% as a net +10%) instead of using multiplying factors, which gives the correct +8%.
- Confusing a 'percentage point' change with a 'percent' change — moving from 40% to 44% is a 4 percentage-point change, which happens to also be a 10% relative increase, but these describe different things.
- Using the wrong base (original) value when computing a percentage increase or decrease, especially after the value has already changed once in a multi-step problem.
Knowledge check
Takeaway
Percentages are fractions of 100 in disguise, and successive percentage changes must be multiplied together as factors, never simply added.
Summary
A percentage converts freely between fraction, decimal, and percent form by scaling with 100. Applying a percentage change means multiplying by (1 + percent/100), and combining two successive changes requires multiplying their factors together (or the equivalent net% = p1 + p2 + p1*p2/100 formula), not adding the raw percentages.
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