These four calculations show up constantly in everyday life — sale prices, recipe scaling, splitting a bill, reading a news headline about a rate change — and each one has a specific, common mistake that catches people out. Here's the correct math for each, and the trap to watch for.
Percentages: the "reverse doesn't undo it" trap
This part is simple — 25% of 80 is 20. The trap comes with sequential percentage changes:
| Step | Calculation | Result |
|---|---|---|
| Start | — | 80 |
| Increase 25% | 80 × 1.25 | 100 |
| Then decrease 25% | 100 × 0.75 | 75 |
A 25% increase followed by a 25% decrease does not return you to 80 — it lands at 75. This happens because the second percentage is calculated on the new, larger number (100), not the original one (80). The same trap applies to any pair of equal-and-opposite percentage changes: the decrease is always calculated on a bigger base than the increase was, so it removes more in absolute terms than the increase added.
Percentage points vs. percent change
These get conflated constantly, especially in news reporting. If an interest rate goes from 4% to 5%, that's a 1 percentage point increase — but it's a 25% change relative to the original rate (1 ÷ 4 = 25%). Both descriptions are accurate; they're just answering different questions, and mixing them up in a sentence can misstate a change by a large margin.
Discounts: percentage off vs. final price
Worked example: a $80 item at 25% off:
| Step | Calculation | Result |
|---|---|---|
| Savings | 80 × 0.25 | $20 |
| Sale price | 80 − 20 | $60 |
The common mistake here is stacking two discounts by simply adding the percentages — "25% off, plus an extra 10% off" is not 35% off the original price. The second discount applies to the already-reduced price, so the combined effect is smaller than adding the two percentages suggests (a 25% discount followed by a 10% discount is a 32.5% total reduction, not 35%).
Ratios: always simplify to compare fairly
where GCD is the greatest common divisor of both numbers. Worked example: a class has 18 boys and 24 girls:
| Step | Calculation | Result |
|---|---|---|
| GCD of 18 and 24 | — | 6 |
| Simplified ratio | 18÷6 : 24÷6 | 3 : 4 |
An unsimplified ratio isn't wrong, but it makes comparison harder — "3:4" is immediately easier to reason about than "18:24," and simplified ratios are what most people mean when they ask for "the ratio" in a clean form.
Fractions: common denominators, not just adding across
The single most common fraction mistake is adding numerators and denominators straight across (1/4 + 1/6 ≠ 2/10). The correct approach finds a common denominator first:
| Step | Calculation | Result |
|---|---|---|
| Common denominator | 4 × 6 | 24 |
| Convert and add | 6/24 + 4/24 | 10/24 |
| Simplified | 10÷2 / 24÷2 | 5/12 |
Always simplify the final answer using the same GCD approach as ratios — an unsimplified fraction like 10/24 is mathematically correct but not the standard way to present a result.
Common mistakes, summarized
- Assuming a percentage increase and an equal percentage decrease cancel out. They don't — the decrease is always calculated on a larger base.
- Confusing percentage points with percent change. "Up 1 percentage point" and "up 25%" can both describe the same move from 4% to 5%.
- Adding stacked discounts instead of applying them sequentially. 25% off + 10% off is not 35% off.
- Adding fractions straight across without a common denominator. 1/4 + 1/6 is 5/12, not 2/10.
- Leaving ratios or fraction results unsimplified. Always reduce by the greatest common divisor for the standard, comparable form.
Try the calculators
Each of these has a dedicated calculator: the percentage calculator (including percentage change), the discount calculator, the ratio calculator, and the fraction calculator for add, subtract, multiply, and divide.
Frequently asked questions
Does a 25% increase followed by a 25% decrease get you back to the original number?
No. Increasing 80 by 25% gives 100, but decreasing 100 by 25% gives 75, not 80. This happens because the second percentage is calculated on the new, larger number rather than the original one.
What is the difference between percentage points and percent change?
A percentage point difference is a simple subtraction of two percentages (e.g., going from 40% to 50% is a 10 percentage point increase). A percent change describes that same move relative to the starting value (a 25% increase, since 10 is 25% of 40). The two numbers are rarely equal and are easy to mix up.
How do you calculate a discount?
Multiply the original price by the discount percentage (as a decimal) to find the savings, then subtract that from the original price to get the sale price. For example, 25% off $80 is $20 in savings, for a sale price of $60.
How do you simplify a ratio?
Divide both numbers in the ratio by their greatest common divisor (GCD). For example, 18:24 has a GCD of 6, simplifying to 3:4.