Percentage of a number
x = a · p / 100%
18% of 25045
- Reduced by p %205
- Increased by p %295
Everyday sums
Percentages
Percentage of a number, percentage change, increases and discounts, percentage points.
A percent is one hundredth of the whole, so every calculation comes down to multiplying or dividing by 100. The difficulty starts where the base changes: a 20 % fall needs a 25 % rise to get back to the starting point, and two discounts one after the other do not add up to their plain sum. The calculators below work in both directions and show the multiplier behind the result.
Calculators
9What percentage one number is of another
p = a / b · 100 %a is18.00%
- Left to reach the whole82.00%
- Ratio a : b0.18 : 1
Percentage change
Δ = (b − a) / a · 100 %Increase+25.00%
- Absolute difference62.5
- Ratio b / a1.25
- To return to a−20.00%
Add a percentage to a number
x = a · (1 + p / 100)%
After the increase307.5
- Amount added57.5
- Multiplier1.23
Subtract a percentage from a number
x = a · (1 − p / 100)%
After the discount212.5
- Amount of the discount37.5
- Multiplier0.85
Value before the discount
a = b / (1 − p / 100)%
Before the discount248.75
- Amount of the discount49.75
Value before the increase
a = b / (1 + p / 100)%
Before the increase300
- Amount of the increase69
Percentage points
p.p. = p₂ − p₁%
%
Difference+1.70 pp
- Relative change+37.78%
Two changes in a row
combined = (1 + p₁)(1 + p₂) − 1%
%
Combined change−4.00%
- Combined multiplier0.96
- Naive sum (wrong)0.00%
- Difference from the plain sum-4.00%
Tables and cheat sheets
2Fraction, percentage and decimal
the most common values, worth remembering| Fraction | Percentage | Decimal | Of the number 240 |
|---|---|---|---|
| 1/1 | 100% | 1 | 240 |
| 1/2 | 50% | 0.5 | 120 |
| 1/3 | 33.33% | 0.333333 | 80 |
| 2/3 | 66.67% | 0.666667 | 160 |
| 1/4 | 25% | 0.25 | 60 |
| 3/4 | 75% | 0.75 | 180 |
| 1/5 | 20% | 0.2 | 48 |
| 2/5 | 40% | 0.4 | 96 |
| 1/6 | 16.67% | 0.166667 | 40 |
| 1/8 | 12.50% | 0.125 | 30 |
| 3/8 | 37.50% | 0.375 | 90 |
| 1/10 | 10% | 0.1 | 24 |
| 1/12 | 8.33% | 0.083333 | 20 |
| 1/16 | 6.25% | 0.0625 | 15 |
| 1/20 | 5% | 0.05 | 12 |
| 1/25 | 4% | 0.04 | 9.6 |
| 1/50 | 2% | 0.02 | 4.8 |
| 1/100 | 1% | 0.01 | 2.4 |
Percentage change multipliers
how much has to be added to get back to the starting point| Change | Multiplier | Back to the starting point |
|---|---|---|
| −50 % | 0.5 | +100.00% |
| −40 % | 0.6 | +66.67% |
| −30 % | 0.7 | +42.86% |
| −25 % | 0.75 | +33.33% |
| −20 % | 0.8 | +25.00% |
| −15 % | 0.85 | +17.65% |
| −10 % | 0.9 | +11.11% |
| −5 % | 0.95 | +5.26% |
| +5 % | 1.05 | −4.76% |
| +10 % | 1.1 | −9.09% |
| +15 % | 1.15 | −13.04% |
| +20 % | 1.2 | −16.67% |
| +23 % | 1.23 | −18.70% |
| +25 % | 1.25 | −20.00% |
| +30 % | 1.3 | −23.08% |
| +50 % | 1.5 | −33.33% |
| +100 % | 2 | −50.00% |
Worked examples with full solutions
3A hoodie marked down twice
A hoodie cost $249. It was first marked down by 30 %, then by a further 20 % off the new price in the sale. What does it cost now, and what is the total discount?
- First markdown: 30 % of $249
249 · 0.30 = $74.70 → 249 − 74.70 = $174.30 - The second markdown comes off $174.30
174.30 · 0.20 = $34.86 → 174.30 − 34.86 = $139.44 - Combined multiplier
0.70 · 0.80 = 0.56 → you pay 56 % of the original price - Combined markdown
100 % − 56 % = 44 %, that is 249 − 139.44 = $109.56
Answer: The final price is $139.44 and the combined discount is 44 %, not 50 %.
A fall and the climb back
A share price fell by 20 %. By what percentage does it now have to rise to get back to its original value?
- Take a starting value of 100
100 · (1 − 0,20) = 80 - Rise needed from 80 to 100
(100 − 80) / 80 = 0,25 - Convert to a percentage
0,25 · 100 % = 25 %
Answer: A 25 % rise is needed, because the percentage is taken from the new, lower base.
Percentage points vs percent
Unemployment rose from 4.5 % to 6.2 %. By how much did it go up?
- Difference in percentage points
6,2 − 4,5 = 1,7 p.p. - Relative change
(6,2 − 4,5) / 4,5 = 0,3778 - Converting to a percentage
0,3778 · 100 % ≈ 37,8 %
Answer: A rise of 1.7 percentage points, which is 37.8 % relative to the previous level.
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