Percentage conversions
Percentage means "out of $100$100". When calculating with percentages, conversions between fractions, decimals and percentages are often necessary.
Converting percentages to fractions
To convert from a percentage to a fraction:
- Write the percentage amount as the numerator.
- Write $100$100 as the denominator.
- Then simplify the fraction.
Worked examples
example 1
Convert $20%$20% into a fraction.
Think: We write $20$20 in the numerator, $100$100 in the denominator and then simplify.
Do:
| $20%$20% | $=$= | $\frac{20}{100}$20100 |
| $=$= | $\frac{1}{5}$15 |
example 2
Convert $282%$282% into a fraction.
Think: We write $282$282 in the numerator, $100$100 in the denominator and then simplify.
Do:
| $282%$282% | $=$= | $\frac{282}{100}$282100 |
| $=$= | $\frac{141}{50}$14150 |
Reflect: We can also express the fraction as a mixed fraction (or mixed number).
| $\frac{141}{50}$14150 | $=$= | $2\frac{41}{50}$24150 |
example 3
Convert $26.5%$26.5% into a fraction.
Think: We write $26.5$26.5 in the numerator, $100$100 in the denominator and then simplify.
Do:
| $26.5%$26.5% | $=$= | $\frac{26.5}{100}$26.5100 |
|
| $=$= | $\frac{265}{1000}$2651000 |
Multiply the top and bottom by $10$10 |
|
| $=$= | $\frac{53}{200}$53200 |
Simplify |
Converting fractions to percentages
To convert from a fraction to a percentage:
- Multiply the fraction by $100%$100% (which means multiplying by $100$100 and attaching the percent symbol $%$% at the end).
- Then simplify the percentage.
Worked examples
example 4
Convert $\frac{23}{100}$23100 into a percentage.
Think: Multiply the fraction by $100%$100%. Notice that this will cancel with $100$100 in the denominator.
Do:
| $\frac{23}{100}$23100 | $=$= | $\frac{23}{100}\times100%$23100×100% |
| $=$= | $23%$23% |
example 5
Convert $\frac{3}{20}$320 into a percentage.
Think: Multiply the fraction by $100%$100%. Then simplify.
Do:
| $\frac{3}{20}$320 | $=$= | $\frac{3}{20}\times100%$320×100% |
| $=$= | $3\times5%$3×5% | |
| $=$= | $15%$15% |
example 6
Convert $\frac{250}{400}$250400 into a percentage.
Think: Multiply the fraction by $100%$100%. Then simplify.
Do:
| $\frac{250}{400}$250400 | $=$= | $\frac{250}{400}\times100%$250400×100% |
| $=$= | $\frac{250}{4}%$2504% | |
| $=$= | $62.5%$62.5% |
Reflect: Notice that we can also divide the top and bottom by $4$4 so that the denominator is $100$100. Then the percentage is just the numerator.
example 7
Convert $\frac{3}{16}$316 into a percentage.
Think: Multiply the fraction by $100%$100%. Then simplify.
Do:
| $\frac{3}{16}$316 | $=$= | $\frac{3}{16}\times100%$316×100% |
| $=$= | $\frac{3}{4}\times25%$34×25% | |
| $=$= | $\frac{75}{4}%$754% |
Reflect: We can also express the value of the percentage as a decimal, which gives $18.75%$18.75%.
example 8
Convert $4\frac{3}{4}$434 into a percentage.
Think: Let's express this as an improper fraction first, and then multiply by $100%$100%.
Do:
| $4\frac{3}{4}$434 | $=$= | $\frac{19}{4}$194 |
| $=$= | $\frac{19}{4}\times100%$194×100% | |
| $=$= | $19\times25%$19×25% | |
| $=$= | $475%$475% |
Converting between percentages and decimals
To convert a percentage to a decimal, divide by $100$100 (move the decimal two places over to the left) and drop the $%$% symbol. For instance, $72%=0.72$72%=0.72 and $112%=1.12$112%=1.12.
To convert a decimal to a percentage, multiply by $100$100 and attach the $%$% symbol. For instance, $0.85=85%$0.85=85%, $0.2=20%$0.2=20% and $2.5=250%$2.5=250%.
Finding percentages of a quantity
Finding a percentage of a quantity is the same as finding a fraction of a quantity. For example, to find $\frac{2}{3}$23 of $60$60 just multiply the two numbers together $\frac{2}{3}\times60=40$23×60=40.
The same is done with percentages as a percentage can be converted into a fraction. For example, to find $71%$71% of $526$526, so let's multiply them together.
| $71%$71% of $526$526 | $=$= | $71%\times526$71%×526 |
Remember, "of means" means multiply |
| $=$= | $\frac{71}{100}\times526$71100×526 |
Converting the percentage into a fraction |
|
| $=$= | $373.46$373.46 |
Simplify the multiplication |
Alternatively, using the calculator you could multiply by the percentage converted to a decimal–in this case $0.71$0.71.
Practice questions
Question 1
Evaluate $28%$28% of $5000$5000.
Question 2
Evaluate $51.3%$51.3% of $240$240
Express your answer as a decimal.
Question 3
Lisa scored $70%$70% on her Maths exam, which was marked out of $140$140. What was Lisa's actual mark out of $140$140?
Expressing the percentage of an amount
Exploration
For example say there is a rainwater tank with $24$24 kL of water in it but it can hold $50$50 kL, what percentage of the tank is full?
Start with the fraction of water in the tank. Remember that in a fraction the numerator represents how much there is and the denominator represents the total amount. So the fraction is $\frac{24}{50}$2450.
Then convert it into a percentage:
| $\frac{24}{50}$2450 | $=$= | $\frac{24}{50}\times\frac{2}{2}$2450×22 |
| $=$= | $\frac{48}{100}$48100 | |
| $=$= | $48%$48% |
Therefore $48%$48% of the tank is full of water.
Different units
Exploration
When calculating percentages, ensure that both numbers are in the same units. For example, what percentage is $65$65 cm of $3$3 m?
First convert them both to the same unit. $65$65 cm is already in centimetres, and $3$3 m is $3\times100=300$3×100=300 cm.
Then find the percentage as usual:
| $\frac{65}{300}$65300 | $=$= | $\frac{65}{300}\times100%$65300×100% |
| $=$= | $\frac{65}{3}%$653% | |
| $=$= | $21\frac{2}{3}%$2123% |
Practice questions
Question 4
When Bart looked at the bill from the mechanic, the total cost of repairs was $\$800$$800. $\$640$$640 of this was for labour and the rest was for replacement of parts.
What percentage of the cost of repairs was for labour?
What percentage of the cost of repairs was for replacement of parts?
Question 5
Ben is going to purchase some sports gear on layby. This involves paying some money as a deposit, and paying the remainder later. The price of the gear is $\$65$$65.
If he needs to pay $25%$25% deposit, how much is this? As we are dealing with dollars and cents, give your answer to two decimal places.
What is the remaining balance on the layby purchase? Give your answer to two decimal places.
Question 6
What percentage is $134$134 L of $536$536 L?
Question 7
What percentage is $385$385 metres of $4$4 km?
Question 8
What percentage is $24$24 minutes of $2$2 hours?
Don't forget to include the percentage symbol where required.