Percentages

Percentages can be converted to fractions and decimals, and vice versa. Here are the conversion rules:

• To convert a percentage to a fraction: Remove the % sign and divide by 100
• To convert a fraction to a percentage: Multiply by 100 and add the % sign
• To convert a percentage to a decimal: Remove the % sign and divide by 100
• To convert a decimal to a percentage: Multiply by 100 and add the % sign

Examples:

• 25% = $\frac{25}{100}=\frac{1}{4}=0.25$
• $\frac{3}{5}=0.6=60\%$
• 75% = $\frac{75}{100}=\frac{3}{4}=0.75$

A percentage can also be thought of as a ratio comparing a part to the whole, where the whole is 100.

For example:

• 25% means 25 parts out of 100 parts
• 50% means 50 parts out of 100 parts, or 1 part out of 2 parts
• 100% means all parts, or the whole

This is why percentages are useful for comparing parts of different wholes. For example, if 25% of students in one class passed a test and 30% of students in another class passed the same test, we can directly compare these percentages even if the classes have different numbers of students.

To find a percentage of a number, multiply the number by the percentage (expressed as a decimal).

Formula: $\mathrm{part}=\mathrm{whole}\times \mathrm{percentage}$

Examples:

• 25% of 80 = $80\times 0.25=20$
• 60% of 150 = $150\times 0.6=90$

To find what percentage one number is of another, divide the part by the whole and multiply by 100.

Formula: $\mathrm{percentage}=\frac{\mathrm{part}}{\mathrm{whole}}\times 100$

Examples:

• What percentage is 20 of 80? $\frac{20}{80}\times 100=25\%$
• What percentage is 90 of 150? $\frac{90}{150}\times 100=60\%$

To find the whole when given a percentage and the part, divide the part by the percentage (expressed as a decimal).

Formula: $\mathrm{whole}=\frac{\mathrm{part}}{\mathrm{percentage}}$

Examples:

• 20 is 25% of what number? $\frac{20}{0.25}=80$
• 90 is 60% of what number? $\frac{90}{0.6}=150$

To increase a number by a percentage, multiply the number by (1 + the percentage as a decimal).

Formula: $\mathrm{new\; value}=\mathrm{original\; value}\times (1+\mathrm{percentage})$

Examples:

• Increase 80 by 25%: $80\times (1+0.25)=80\times 1.25=100$
• Increase 150 by 60%: $150\times (1+0.6)=150\times 1.6=240$

To decrease a number by a percentage, multiply the number by (1 - the percentage as a decimal).

Formula: $\mathrm{new\; value}=\mathrm{original\; value}\times (1-\mathrm{percentage})$

Examples:

• Decrease 80 by 25%: $80\times (1-0.25)=80\times 0.75=60$
• Decrease 150 by 60%: $150\times (1-0.6)=150\times 0.4=60$

To find the original value after a percentage increase or decrease, divide the new value by (1 ± the percentage as a decimal).

For an increase: $\mathrm{original\; value}=\frac{\mathrm{new\; value}}{1+\mathrm{percentage}}$

For a decrease: $\mathrm{original\; value}=\frac{\mathrm{new\; value}}{1-\mathrm{percentage}}$

Examples:

• If a price increased by 25% to $100, what was the original price? $\frac{100}{1+0.25}=\frac{100}{1.25}=80$
• If a price decreased by 60% to $60, what was the original price? $\frac{60}{1-0.6}=\frac{60}{0.4}=150$

To find the percentage difference between two numbers, divide the difference by the original number and multiply by 100.

Formula: $\mathrm{percentage\; difference}=\frac{\mathrm{new\; value}-\mathrm{original\; value}}{\mathrm{original\; value}}\times 100$

Examples:

• If a price increased from $80 to $100, what is the percentage increase? $\frac{100-80}{80}\times 100=\frac{20}{80}\times 100=25\%$
• If a price decreased from $150 to $60, what is the percentage decrease? $\frac{60-150}{150}\times 100=\frac{-}{90}\times 100=-60\%$

Note: A negative percentage difference indicates a decrease, while a positive percentage difference indicates an increase.

For percentage problems, you can set up a proportion where one ratio is the percentage (as a fraction) and the other ratio is the part to the whole.

Formula: $\frac{\mathrm{percentage}}{100}=\frac{\mathrm{part}}{\mathrm{whole}}$

Examples:

• What is 25% of 80?
• Set up the proportion: $\frac{25}{100}=\frac{x}{80}$
• Cross-multiply: $25\times 80=100\times x$
• Solve for x: $x=\frac{25\times 80}{100}=20$

• 20 is what percentage of 80?
• Set up the proportion: $\frac{x}{100}=\frac{20}{80}$
• Cross-multiply: $x\times 80=100\times 20$
• Solve for x: $x=\frac{100\times 20}{80}=25$
• Therefore, 20 is 25% of 80.

You can also use proportions to find the whole when given a percentage and the part.

Example:

• 20 is 25% of what number?
• Set up the proportion: $\frac{25}{100}=\frac{20}{x}$
• Cross-multiply: $25\times x=100\times 20$
• Solve for x: $x=\frac{100\times 20}{25}=80$
• Therefore, 20 is 25% of 80.

If you know that a number is a certain percentage of another number, you can find the other number using a proportion.

Example:

• If 30 is 60% of a number, what is the number?
• Set up the proportion: $\frac{60}{100}=\frac{30}{x}$
• Cross-multiply: $60\times x=100\times 30$
• Solve for x: $x=\frac{100\times 30}{60}=50$
• Therefore, 30 is 60% of 50.

When a value changes by a percentage and then changes again by another percentage, the total change is not simply the sum of the two percentages.

Example:

• If a price increases by 20% and then increases again by 10%, what is the total percentage increase?
• First increase: $100\times 1.2=120$
• Second increase: $120\times 1.1=132$
• Total increase: $\frac{132-100}{100}\times 100=32\%$
• Therefore, the total increase is 32%, not 30%.

This is because the second percentage is applied to the new value after the first increase, not the original value.

When you need to find a percentage of a percentage, multiply the two percentages (expressed as decimals).

Example:

• What is 25% of 40%?
• $0.25\times 0.4=0.1=10\%$
• Therefore, 25% of 40% is 10%.