A percentage is a fraction out of 100. It lets you compare numbers on one common scale.
This lesson gives each standard formula and the reason it works. It then solves three examples and lists the traps that cost marks.
- Meaning: out of 100.
- Part: x% of N is x × N / 100.
- Change: change ÷ old value × 100.
- Successive: net % = x + y + xy/100.
- Trap: the base changes at every step.
What a Percentage Means
What does “per cent” mean?
Per cent means out of 100. So 40% is 40 out of 100, or 40/100.
The base of 100 is common to all. That is why you can compare marks, prices and rates on one scale.
How do fractions and percentages convert?
To turn a fraction into a percentage, multiply it by 100. So 3/5 × 100 = 60%.
To go back, write the percentage over 100 and cut it down. So 68% = 68/100 = 17/25.
| Fraction | Percentage |
|---|---|
| 1/2 | 50% |
| 1/4 | 25% |
| 1/5 | 20% |
| 1/8 | 12.5% |
| 3/8 | 37.5% |
| 7/8 | 87.5% |
Check 1
The fraction 3/5 equals 60%.
Core Formulas and Why They Work
How do you find a percentage of a number?
x% of N = (x/100) × N.
Why it works: percent means per 100. So you take x parts out of every 100 parts of N.
How do you find a percentage change?
% change = (new − old) ÷ old × 100.
Why it works: you compare the change with where you started. The old value is always the base.
How do you find a new value after a rise or fall?
After a rise of x%: new = old × (100 + x)/100. After a fall of x%: new = old × (100 − x)/100.
Why it works: the old value is 100%. A rise adds x to it. A fall takes x away.
How do successive changes combine?
Net % = x + y + xy/100. Write a fall as a negative number.
Why it works: the second change acts on the new value, not the old one. The xy/100 term is that extra effect.
How do price and consumption trade off?
If price rises R%, cut use by R/(100 + R) × 100.If price falls R%, raise use by R/(100 − R) × 100.
Both rules keep total spending the same.
Why it works: spending is price times quantity. When the price grows from 100 to 100 + R, the quantity must shrink in the same ratio.
How does yearly growth work?
After n years at R% a year: P × (1 + R/100)^n. For a fall, use (1 − R/100)^n.
Why it works: each year multiplies the last figure by the same number. Growth in one year becomes the base for the next.
What is the multiplier method?
A rise of x% multiplies by 1 + x/100. A fall of x% multiplies by 1 − x/100.
Why it works: 100% plus x% is (100 + x)% of the old value. As a decimal that is 1 + x/100.
For two changes, multiply the two multipliers. For the rise of 30% and fall of 15%, 1.30 × 0.85 = 1.105.
Check 2
A 20% rise followed by a 20% fall brings back the original value.
Three Solved Examples
Easy: what is 22% of 4210?
Find 1% first. 1% of 4210 is 42.1.
Now multiply by 22. 22 × 42.1 = 926.2.
So 22% of 4210 is 926.2.
Medium: a value rises 30%, then falls 15%. What is the net change?
Start at 100. After the rise it is 130.
A 15% fall takes off 19.5, which is 15% of 130. So 130 − 19.5 = 110.5.
The net change is a 10.5% rise. The formula agrees: 30 − 15 + (30 × −15)/100 = 10.5.
Exam level: a town of 25000 grows 10% a year. What is its size after 2 years?
Year 1: 25000 × 1.10 = 27500.
Year 2: 27500 × 1.10 = 30250.
So the population is 30250. Simple growth would give only 30000. The extra 250 is growth on growth.
Step by step: the price of sugar rises 25%. By how much must a family cut its use to keep the spending the same?
Step 1: let the old price be 100. The new price is 125.
Step 2: the spending stays fixed. So use × price must stay equal. Old use is 1 unit, so the new use is 100/125 = 0.8 units.
Step 3: the cut is 1 − 0.8 = 0.2, which is 20%.
Check with the formula: R/(100 + R) × 100 = 25/125 × 100 = 20%.
Check 3
A value growing 10% a year doubles in 10 years.
Common Traps and Shortcuts
Which traps cost marks?
Adding percentages with different bases. A 30% rise and a 15% fall is not a 15% rise.
A rise of x% then a fall of x% never gets back to the start. It leaves a loss of x²/100 percent.
Mixing up percent and percentage points. A move from 40% to 50% is 10 points. In relative terms it is a 25% rise.
A 100% rise in price needs a 50% cut in use to hold spending. The formula gives 100/200 × 100.
Which shortcuts save time?
Find 10% by moving the decimal point one place. Build other values from it. Half of 10% is 5%.
x% of y equals y% of x. So 4% of 75 is the same as 75% of 4, which is 3.
Use multipliers. A 25% rise is × 1.25. A 25% fall is × 0.75.
How do exam questions usually use percentages?
Most questions fall into four patterns. Find a part, find a change, chain two changes, or trade price against use.
Spot the pattern first. Then choose the tool: 1% method, multiplier, net formula or price-use formula.
Always ask what the base is. The base is the value the percentage is taken from. Most wrong answers come from a wrong base.
When two options are close, test with 100. It takes ten seconds and avoids sign errors.
Check 4
x% of y always equals y% of x.
MCQ on Percentage
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