Percentages in IGCSE Maths
Finding percentages, percentage increase and decrease, reverse percentages, compound interest and repeated percentage change.
0580 · C1.12 / E1.12Percentage basics
"Per cent" means "out of 100". So 35% means 35 out of 100, or \(\frac{35}{100}\), or 0.35.
To convert between percentages, decimals and fractions:
| Percentage | Decimal | Fraction |
|---|---|---|
| 5% | 0.05 | \(\frac{1}{20}\) |
| 15% | 0.15 | \(\frac{3}{20}\) |
| \(33.\overline{3}\)% | \(0.\overline{3}\) | \(\frac{1}{3}\) |
| 40% | 0.4 | \(\frac{2}{5}\) |
| 75% | 0.75 | \(\frac{3}{4}\) |
Percentage → decimal: divide by 100. Decimal → percentage: multiply by 100.
Finding a percentage of an amount
To find \(x\)% of an amount, multiply the amount by \(\frac{x}{100}\).
\[x\% \text{ of } A = \frac{x}{100} \times A\]Find 15% of $240.
A student scores 42 out of 60 on a test. What is this as a percentage?
Percentage change
You can also write this as:
\[\text{Percentage change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\]If the result is positive, it's a percentage increase. If it's negative, it's a percentage decrease.
A house was bought for $180,000 and sold for $207,000. Find the percentage profit.
Change = $207,000 − $180,000 = $27,000
\[\frac{27{,}000}{180{,}000} \times 100 = 15\%\]A car's value drops from $12,000 to $9,600. Find the percentage decrease.
Change = $12,000 − $9,600 = $2,400
\[\frac{2{,}400}{12{,}000} \times 100 = 20\%\]The denominator in the percentage change formula is always the original value. It is the value before the change happened. This is the most common error students make.
Percentage increase and decrease
To increase an amount by \(x\)%:
Work out \(x\)% of the amount, then add it to the original.
Multiply the amount by \((1 + \frac{x}{100})\). See the next section for details.
To decrease by \(x\)%, subtract instead of adding, or multiply by \((1 - \frac{x}{100})\).
A shirt costs $45. It is increased by 20%. Find the new price.
20% of $45 = \(0.20 \times 45 = \$9\)
New price = \(45 + 9 = \$54\)
\(45 \times 1.20 = \$54\)
A laptop costs $800. It is reduced by 15% in a sale. Find the sale price.
15% of $800 = \(0.15 \times 800 = \$120\)
Sale price = \(800 - 120 = \$680\)
\(800 \times 0.85 = \$680\)
The multiplier method
Every percentage change can be done in a single multiplication using a multiplier:
| Percentage change | Multiplier | Calculation |
|---|---|---|
| Increase by 20% | 1.20 | 100% + 20% = 120% = 1.20 |
| Increase by 5% | 1.05 | 100% + 5% = 105% = 1.05 |
| Decrease by 15% | 0.85 | 100% − 15% = 85% = 0.85 |
| Decrease by 3% | 0.97 | 100% − 3% = 97% = 0.97 |
| Increase by 12.5% | 1.125 | 100% + 12.5% = 112.5% = 1.125 |
Increase → multiplier greater than 1. Decrease → multiplier less than 1.
The multiplier method is essential for reverse percentages, compound interest, and repeated percentage change. Learn to convert any percentage change to a multiplier. The rest of this page depends on it.
Reverse percentages
For any percentage increase or decrease, the formula is:
\[\text{original} \times \text{multiplier} = \text{new value}\]Normally you know the original and want the new value, so you multiply. In a reverse percentage question, you know the new value and want the original, so you rearrange:
\[\text{original} = \frac{\text{new value}}{\text{multiplier}}\]It's just the same formula rearranged. You cannot simply apply the percentage in the other direction. That gives the wrong answer.
After a 20% increase, a price is $96. Find the original price.
20% increase → multiplier = 1.20
In a sale, a jacket is reduced by 30%. The sale price is $56. Find the original price.
30% decrease → multiplier = 0.70
If a price increases by 20% from $80 to $96, then decreasing $96 by 20% gives \(96 \times 0.80 = \$76.80\). Not $80. The 20% decrease is 20% of a different (larger) number, so it removes a different amount. You must divide by the multiplier to undo it correctly.
Compound interest
Compound interest means the interest is added to the balance each year, and the next year's interest is calculated on the new (larger) balance.
\[A = P \times r^n\]where \(A\) = final amount, \(P\) = principal (starting amount), \(r\) = multiplier per year, \(n\) = number of years (see indices for working with powers).
$5,000 is invested at 3% compound interest per year. Find the value after 4 years.
\(P = 5000\), \(r = 1.03\) (3% increase), \(n = 4\)
$2,000 is invested for 3 years at 5% per year. Find the difference between compound and simple interest.
Interest per year = \(0.05 \times 2000 = \$100\)
Total after 3 years = \(2000 + 3 \times 100 = \$2,300\)
\(2315.25 - 2300 = \$15.25\)
With simple interest, the interest is calculated on the original amount every year; it doesn't grow. The formula is:
\[A = P + P \times r \times n = P(1 + rn)\]where \(P\) is the principal, \(r\) is the rate as a decimal, and \(n\) is the number of years. For example, $2,000 at 5% simple interest for 3 years: interest = \(2000 \times 0.05 \times 3 = \$300\), total = $2,300.
Compound interest always gives a higher total than simple interest (for more than 1 year at the same rate), because each year's interest earns interest in the following years.
Repeated percentage change
The compound interest formula works for any repeated percentage change, not just money. Depreciation, population growth, radioactive decay, and inflation all use the same structure:
\[\text{Final} = \text{Start} \times r^n\]For growth, \(r > 1\). For decay/depreciation, \(r < 1\).
A car is worth $18,000. It depreciates by 12% each year. Find its value after 3 years.
12% decrease → multiplier = 0.88
\[\begin{aligned}\text{Value} &= 18{,}000 \times 0.88^3 \\[6pt] &= 18{,}000 \times 0.681472 \\[6pt] &= \$12{,}266.50\end{aligned}\]A population of 50,000 grows by 2% per year. After how many years will it exceed 60,000?
We need \(50{,}000 \times 1.02^n > 60{,}000\), so \(1.02^n > 1.2\).
Try values of \(n\):
\(1.02^9 = 1.1951...\) (not enough)
\(1.02^{10} = 1.2190...\) (exceeds 1.2)
The population exceeds 60,000 after 10 years.
Exam-style questions
A shop buys a jacket for $40 and sells it for $52. Calculate the percentage profit.
Show solution
Profit = $52 − $40 = $12
\[\frac{12}{40} \times 100 = 30\%\]After a 15% discount, a television costs $510. Calculate the original price.
Show solution
15% decrease → multiplier = 0.85
\[\text{Original} = \frac{510}{0.85} = \$600\]$8,000 is invested at 2.5% per year compound interest. Calculate the total amount after 5 years.
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A machine is bought for $25,000. It depreciates at 8% per year. Find its value after 6 years.
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8% decrease → multiplier = 0.92
\[\begin{aligned}\text{Value} &= 25{,}000 \times 0.92^6 \\[6pt] &= 25{,}000 \times 0.606355... \\[6pt] &= \$15{,}158.88\end{aligned}\]After 3 years of 4% compound interest per year, an investment is worth $5,624.32. Find the amount originally invested.
Show solution
Multiplier for 3 years at 4% = \(1.04^3 = 1.124864\)
\[\text{Original} = \frac{5624.32}{1.124864} = \$5,000\]Common mistakes
Percentage change = change ÷ original × 100. The denominator is always the original (the value before the change), not the new value. If a price goes from $80 to $100, the percentage increase is \(\frac{20}{80} \times 100 = 25\%\), not \(\frac{20}{100} \times 100 = 20\%\).
To undo a 20% increase, you cannot just decrease by 20%. You must divide by the multiplier (1.20). This is the single most common percentage error at IGCSE. See section 6 for the full explanation.
Compound interest grows the balance each year, so the interest amount increases. If a question says "compound interest", you must use \(P \times r^n\), not \(P + P \times r \times n\). Read the question carefully.
A 12% decrease has a multiplier of 0.88, not 0.12 and not 1.12. Think: you are keeping 88% of the value (100% − 12% = 88% = 0.88). If your multiplier is less than 0 or greater than 1 for a decrease, something has gone wrong.
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