Imagine two stores both raise their prices. Store A raises prices by $10. Store B raises prices by $10. Same change, right? Not necessarily — if Store A's original price was $20, that's a 50% increase. If Store B's original price was $200, that's only a 5% increase. The raw dollar amount tells you nothing useful on its own. This is exactly why percentage change exists: it expresses how big a change is relative to where you started, so you can compare changes fairly across completely different scales.
This comes up constantly: a salary negotiation ('I got a 5% raise'), a product discount ('20% off'), a health statistic ('cases increased by 30%'), an investment return ('up 12% this year'). In every case, the percentage tells you the story the raw number can't. The most common misconception here is treating percentage increases and decreases as symmetrical — that a 20% increase followed by a 20% decrease gets you back to where you started. It doesn't, and understanding why will be one of the most useful things you take from this lesson.
A percentage is a ratio with 100 as the fixed denominator. Saying '25%' is just a shorthand for '25 out of every 100 parts'. The reason we standardize on 100 is comparison: if one class has 10 absent students out of 40, and another has 15 absent out of 60, which class has the worse attendance problem? It's not obvious from the raw numbers. But 10/40 = 25% and 15/60 = 25%, so they're identical — the percentage makes that immediately visible.
This is the core purpose of percentages: they give every quantity the same denominator (100) so comparisons become instant. When you read that unemployment is 4.2%, you don't need to know the size of the workforce to understand what that means proportionally.
A quick translation guide: 25% = 25/100 = 0.25. To go from a percentage to a decimal, divide by 100 (move the decimal two places left). To go from a decimal to a percentage, multiply by 100 (move it two places right). You'll do this automatically in every percentage problem, so the conversion should feel like second nature.
Common mistake: confusing 'percentage' with 'percentage points'. If a tax rate goes from 20% to 25%, it increased by 5 percentage points — but that's a 25% increase in the rate itself (5/20 × 100 = 25%). These are completely different statements. In this lesson, we're working with percentage change — the relative change — not the raw difference in percentage points.
Think of a percentage as a universal translator. It converts any ratio, no matter how large or small the original quantities, into a common language everyone can immediately understand.
A percentage fixes the denominator at 100, making any two ratios directly comparable.
0.25
Percentage increase answers the question: 'How large is this growth compared to what we started with?' The formula has three logical steps that follow directly from that question.
First, find the actual increase: new value − original value. This is the raw amount that was added. Second, ask 'what fraction of the original is that increase?' — so divide the increase by the original value. Third, convert that fraction to a percentage by multiplying by 100.
Why do we divide by the original, not the new value? Because percentage increase describes the change relative to the starting point. The original value is the reference — it's what existed before anything changed. Compare: a $10 increase on a $50 price gives 10/50 × 100 = 20%. The same increase on a price gives 10/500 × 100 = 2%. Same absolute change, wildly different percentage changes. That's exactly the point — the original value puts the change in context.
Here's the most common error: dividing by the new value instead of the original. For the $50 → $60 example, that would give 10/60 × 100 ≈ 16.7% instead of the correct 20%. If you use the new value, you're comparing the change to the wrong baseline — the state of affairs after the change rather than before it. Always anchor to the original.
A quick self-check: if a value doubled, you should get a 100% increase. Test it: original 50, new 100. Increase = 50. 50/50 × 100 = 100%. ✓ If your formula gives anything other than 100% for a doubling, you've used the wrong denominator.
Mental model: percentage increase is like asking 'what fraction of the original value was added?' If the answer is one-fifth, that's a 20% increase. You're measuring the new addition against the old baseline.
Divide the increase by the original — you're measuring how large the change is relative to where you started.
20%
Percentage decrease works by exactly the same logic as percentage increase — you're still measuring a change relative to the original. The only difference is the direction: instead of the new value exceeding the original, the new value is less.
The formula: (original − new) / original × 100. We subtract new from original (rather than original from new) so we get a positive number — a decrease is reported as a positive percentage, not a negative one. Some textbooks write this as |new − original| / original × 100 (using absolute value), which also works.
Again, the denominator is always the original. A shirt drops from $80 to $60: decrease is $20, and 20/80 × 100 = 25%. A different shirt drops from $80 to $40: decrease is $40, and 40/80 × 100 = 50%. Same original price, different absolute drops — the percentage tells you immediately which is the bigger relative reduction.
The classic trap here: students sometimes divide by the new (lower) value. For the $80 → $60 example, that gives 20/60 × 100 ≈ 33.3% instead of the correct 25%. This error makes decreases look larger than they actually are. A useful intuition check: a 100% decrease should bring the value to zero (you lost everything). Test it: 80 − 0 = 80, and 80/80 × 100 = 100%. ✓ If you divided by the new value (0), you'd have division by zero — an obvious impossibility that exposes the error.
Now here's the asymmetry that trips up almost everyone: if a price increases by 25% and then decreases by 25%, you do NOT end up where you started. Starting at $80, a 25% increase gives $100. A 25% decrease on gives $75. You're short of the original $80. This happens because the decrease is calculated on the new (larger) base, not the original one. This is why percentage changes are not reversible — and it will matter enormously when you encounter compound percentage changes.
Think of percentage decrease as asking: 'What fraction of the original was removed?' Same question as increase, same formula, same reference point — just pointing in the opposite direction.
Divide the reduction by the original — percentage decrease measures what fraction of the starting value was removed.
25%
Percentage change appears in almost every domain where something moves over time — prices, populations, test scores, investment values, disease rates. The calculation is always the same; what changes is how you interpret the result and what you do with it.
For finding the new value after a percentage change (rather than finding the percentage itself), there's a faster method than calculating the change separately and adding or subtracting. For a 15% discount on a $200 jacket: instead of computing 15% of $200 = $30 and then subtracting $30, you can multiply directly by the 'multiplier'. A 15% decrease leaves 85% of the original, so: $200 × 0.85 = $170. One step, same answer. For an increase: a 10% increase means you end up with 110% of the original, so multiply by 1.10.
This multiplier approach is especially powerful for multi-step problems. If a price increases 10% and then decreases 20%, the multipliers are 1.10 and 0.80. Combined: 1.10 × 0.80 = 0.88, meaning the final price is 88% of the original — a net 12% decrease. Notice that 10% up and 20% down is not a net 10% down. The sequence and the changing base make the arithmetic non-intuitive, which is exactly why the multiplier method prevents errors.
A practical note on interpretation: when reading real-world percentage claims, always ask 'percentage of what?' A headline like 'sales up 50%' is almost meaningless without knowing the baseline. A startup going from 2 customers to 3 is a 50% increase; that's not the same story as a major retailer doing the same. The percentage is only as meaningful as the context around the original value.
Mental model: each percentage change is a scaling operation. Multiplying by 1.20 scales a value up by 20%. Multiplying by 0.75 scales it down by 25%. Chained percentage changes are chained multiplications — and multiplications are commutative (the order doesn't change the final result), but each step still uses the current value as its base, so the absolute dollar amounts at each stage will differ.
Each percentage change multiplies the current value — increases use multipliers above 1, decreases use multipliers below 1.
A laptop originally costs $700. The price increases to $840. What is the percentage increase?
The percentage increase
A pair of shoes costs $120 and is on sale for $90. What is the percentage decrease?
The percentage decrease
A store increases the price of a jacket by 10% and then applies a 20% discount. The original price is $200. What is the final price?
The final price after increase and discount
A percentage expresses a quantity as a fraction of 100, making it possible to compare proportions across different-sized totals — without percentages, comparing a change of $10 on a $50 item versus a $10 change on a $500 item would require a calculation every time.
Percentage increase = (new − original) / original × 100. The denominator is always the original value because you're measuring how much the starting point grew, not how the ending point compares to itself.
Percentage decrease = (original − new) / original × 100. Same denominator rule: divide by the original, not the new value. The percentage tells you what fraction of the starting value was removed.
Percentage increases and decreases are not symmetrical: a 25% increase followed by a 25% decrease does not return to the original value. The second change operates on a different base than the first, so the absolute amounts differ even when the percentages match.
For multi-step problems, convert each percentage to a multiplier (increase of x% → multiply by 1 + x/100; decrease of x% → multiply by 1 − x/100) and chain them. This prevents the common error of applying a second change to the wrong base.
Percentage increase and decrease is the gateway to almost every quantitative concept in personal finance and data analysis. Simple interest is a repeated percentage increase on a fixed principal. Compound interest is a repeated percentage increase on a growing principal — and understanding why those two give different results comes directly from what you've learned here about changing bases. Tax calculations, profit and loss, markup and markdown in retail, population growth models, and investment returns all use this same framework. In statistics, percentage change is how you communicate whether a trend is meaningful. Master this now and every financial or data topic that follows will feel like a natural extension.
Practice exclusively with problems where you must identify the original value before calculating — create your own examples by picking two numbers and asking 'what is the percentage change from X to Y?' then verify by applying your percentage back to X to confirm you get Y.
Deliberately practice the asymmetry trap: take a value, increase it by some percentage, then calculate what percentage decrease is needed to return to the original. The decrease percentage will always be smaller than the increase percentage, and working through several examples until this feels obvious is the goal.
For every multi-step problem, refuse to combine the percentages mentally — always write out the intermediate value after each step. Once you've done this ten times correctly, the habit of 'find the new base first' will be automatic.
Test your intuition with extreme cases: a 100% increase should double the value; a 100% decrease should bring it to zero; a 50% decrease followed by a 100% increase should return to the original. Run these checks on your formula until every result is immediate — these benchmarks let you detect errors instantly on real problems.
A phone originally costs $500 and is now $550. What is the percentage increase?
A jacket costs $120 and is discounted to $90. What is the percentage decrease?
A population of 800 grows to 880. What is the percentage increase?
A product's price drops from $250 to $200. Calculate the percentage decrease.
A TV costs v$400. Its price is increased by 15% and then decreased by 10%. What is the final price?
An investment of $1000 grows by 8% the first year and 12% the second year. What is the total value after two years?
How do you value this lesson?