Percentages and Ratios, Explained
After reading you will be able to answer the five everyday percentage questions by hand, check them against the calculator, and avoid the reversal errors that trip most people up.
A percentage is just a fraction with a fixed denominator of 100. When you say "15%", you mean 15/100, or 0.15. That single idea powers every mode on this page. A ratio is closely related: it compares two quantities directly, like 2 : 3, without forcing the denominator to 100.
Here is a quick hook. A shirt costs 80 and is marked "15% off". The discount is 0.15 \times 80 = 12, so you pay 68. That is the "What is p% of x?" mode, and it is the most common percentage question there is.
The five questions this page answers
Most percentage confusion comes from not naming the question precisely. There are really only a few shapes, and each has a mode:
- p% of x
- You know the percentage and the base, you want the part. Example: 15% of 80 is 12.
- a is what % of b
- You know the part and the base, you want the percentage. Example: 25 out of 200 is 12.5%.
- a is p% of what
- You know the part and the percentage, you want the base. Example: 30 is 15% of 200.
- % change from a to b
- You want the relative growth or drop between two values. Example: from 120 to 150 is +25%.
- ratio and proportion
- Simplify
a : b, or solve a/b = c/x for the missing term.
Deciding which mode you need is the real skill. The arithmetic is easy once the question is stated correctly.
The formulas and why they work
Every mode rests on the definition p\% = p/100. Convert once, and the rest is multiplication or division.
Here p is the percentage and x is the base (the whole). For 15% of 80: 0.15 \times 80 = 12.
To find what percent a is of b, divide the part by the base and scale to 100:
For a = 25 and b = 200: 25/200 = 0.125, times 100 gives 12.5%.
To recover the base when you know the part and the percentage, invert the first formula:
If 30 is 15% of some number, then x = 100 \times 30 / 15 = 200.
Percentage change compares the difference to the starting value, not to the ending value:
The starting value a is the reference. Choosing the wrong reference is the single most common mistake, and the next section shows exactly how much it costs.
Worked example: percentage change from 120 to 150
Reproduce the demo data
Set the mode to % change from a to b with a = 120 and b = 150, then press Calculate.
- Find the difference:
150 - 120 = 30. - Divide by the starting value:
30 / 120 = 0.25. - Scale to a percentage:
0.25 × 100 = 25.
The result is +25%. The plus sign matters: a positive change is growth, a negative change is a drop.
The reference is always the earlier value. Going from 120 up to 150 is a 25% increase, but going from 150 back down to 120 is -30/150 = -20\%. A 25% rise followed by a 20% fall returns you to the start. Percentage changes are not symmetric.
Why a rise and a fall do not cancel
People expect "up 25%, down 25%" to return to the start. It does not. Each percentage is applied to a different base. Up 25% from 120 gives 150. Down 25% from 150 gives 150 \times 0.75 = 112.5, which is below where you began.
To undo a 25% increase you must decrease by 30/150 = 20\%, not 25%. This asymmetry shows up in prices, populations, and investment returns.
Try it: how the reference changes the answer
Ratios and proportions
A ratio a : b compares two quantities. Simplifying a ratio means dividing both terms by their greatest common divisor. For 50 : 200, the divisor is 50, so the ratio reduces to 1 : 4. That reads as "one part for every four parts".
A proportion states that two ratios are equal, and you solve for the missing term by cross multiplication:
Suppose a map scale is 2 cm to 5 km and a road measures 3 cm. Set 2/5 = 3/x. Then x = (5 \times 3)/2 = 7.5 km. Keep the two ratios in the same order (map to ground on both sides) or the answer inverts.
| As a ratio | As a fraction | As a percentage |
|---|---|---|
| 1 : 4 | 1/4 | 25% |
| 1 : 2 | 1/2 | 50% |
| 3 : 4 | 3/4 | 75% |
| 1 : 5 | 1/5 | 20% |
A ratio a : b becomes a percentage of the whole only when you compare a part to the total. The part 1 in 1 : 4 is 20% of the total (1 out of 5 parts), not 25%. The 25% figure compares the first term to the second term, not to the total. State clearly which comparison you mean.
Common mistakes
These four errors account for most wrong answers:
- Percentage points versus percent. A rate rising from 4% to 6% is a 2 percentage point rise, but a 50% relative increase (2/4 = 0.5). Say which one you mean.
- Wrong reference in a change. Always divide by the starting value. From 120 to 150 is 25%, not 20%.
- Adding percentages of different bases. A 10% tax on 200 is 20, and a 10% tip on 220 is 22, so the combined add-on is 42, not 40. Order and base matter.
- Reversing a proportion. Keep both fractions in the same orientation before cross multiplying.
Related tools
When your percentages involve physical quantities like speeds, distances, or currencies with unit tracking, the Unit-Aware Scientific Calculator keeps the units attached through each step so you do not mix, say, minutes with hours.
Frequently asked questions
How do I calculate a percentage discount?
Multiply the price by the percentage as a decimal to get the discount, then subtract. A 15% discount on 80 is 0.15 × 80 = 12, so you pay 80 - 12 = 68. Equivalently, pay 0.85 × 80 = 68.
What is the difference between percent and percentage points?
Percentage points measure the raw gap between two percentages. Percent measures the relative change. Moving from 4% to 6% is 2 percentage points and a 50% relative increase. Reports that blur the two can be misleading.
Why is 25% up then 25% down not back to the start?
Each percentage applies to a different base. Up 25% from 120 gives 150. Down 25% from 150 subtracts 37.5, landing at 112.5. To reverse a 25% rise you must fall by 20%.
How do I turn a ratio into a percentage?
Decide what you are comparing. To express the first term as a percent of the second, use a/b × 100: 1 : 4 gives 25%. To express it as a percent of the total, use a/(a+b) × 100: 1 : 4 gives 20%.
How do I solve a proportion?
Cross multiply. For a/b = c/x, the missing term is x = b \cdot c / a. With a = 2, b = 5, c = 3, you get x = 15/2 = 7.5.
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