Percentages

New Zealand

Level 5

Lesson

In mathematics not only do we sometimes have to find percentages of things, but also need to express amounts of things in percentages. For example we know there is a rainwater tank has $24$24L of water in it but can hold $50$50L, how do we know what percentage is full?

The key here is look at a situation carefully, gather all the relevant information and extract the correct fraction that we can use to convert into a percentage. Let's go back to our rainwater tank example. What do you think the fraction would be if we wanted to find how full the tank was?

Remember that in a fraction the **numerator** represents how much there is and the **denominator **represents the total capacity. So here our fraction must be $\frac{24}{50}$2450!

Another way this question might be asked without the tank is: what percentage is $24$24 of $50$50? To work out this question we would ALSO need the fraction $\frac{24}{50}$2450.

Then to convert it into a percentage it is as easy as multiplying it by $100%$100%:

$\frac{24}{50}\times100%$2450×100% | $=$= | $\frac{12}{25}\times100%$1225×100% | simplify the fraction |

$=$= | $\frac{12\times100}{25}$12×10025 $%$% | write as one fraction | |

$=$= | $\frac{12\times4\times25}{25}$12×4×2525 $%$% | split up the $100$100, so I can cancel out common factors | |

$=$= | $12\times4%$12×4% | cancel the $25$25's | |

$=$= | $48%$48% | evaluate! |

So what percentage of the tank is empty?

Complementary parts are the parts that go together to make up a total quantity. For example:

$\text{amount with water}+\text{empty amount}=\text{full tank}$amount with water+empty amount=full tank

Remember that $100%$100% represents a total amount or a whole quantity. So if $48%$48% of our tank is full, we find the percentage that is empty by evaluating $100-42$100−42. This means $52%$52% of the tank is empty.

We get the same value when we convert the fraction of the tank that is empty to a percentage:

$\frac{26}{50}\times100=52%$2650×100=52%

Examples

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.

a) What percentage of the cost of repairs was for labour?

Think: We need to express the labour cost as a fraction of the total cost and multiply it by $100$100 to convert it into a percentage.

Do: $\frac{640}{800}\times100=80%$640800×100=80%

b) What percentage of the cost of repairs was for replacement of parts?

Think: The total repair cost was made up of labour and parts, so these values are complementary.

Do: $100-80=20%$100−80=20%

Jack is going to layby some sports gear. This involves paying some money as a deposit, and paying the remainder later. The price of the gear is $\$20$$20.

a) If he needs to pay $50%$50% deposit, how much is this? As we are dealing with dollars and cents, give your answer to two decimal places.

b) What is the remaining balance on the layby purchase? Give your answer to two decimal places.

What percentage is $134$134L of $536$536L?

In mathematics not only do we sometimes have to find percentages of things, but also need to express amounts of things in percentages.

For example we know there is a rainwater tank has $24$24L of water in it but can hold $50$50L, how do we know what percentage is full?

The key here is look at a situation carefully, gather all the relevant information and extract the correct fraction that we can use to convert into a percentage.

Let's go back to our rainwater tank example. What do you think the fraction would be if we wanted to find how full the tank was?Remember that in a fraction the **numerator** represents how much there is and the **denominator **represents the total capacity. So here our fraction must be $\frac{24}{50}$2450!

Another way this question might be asked without the tank is: what percentage is $24$24 of $50$50?

To work out this question we would ALSO need the fraction $\frac{24}{50}$2450.

Then to convert it into a percentage it is as easy as multiplying it by $100%$100%: $\frac{24}{50}\times100%=\frac{12}{25}\times100%$2450×100%=1225×100%. This becomes $\frac{1200%}{25}=48%$1200%25=48%, therefore the answer is $48%$48%.

When calculating percentages in real life you might come across different units of measurements.

For example, I might want to find out what percentage $65$65cm is of $3$3m.

Metres and centimetres are clearly different units, so how do we deal with them? Well it's actually a simple case of converting them both to the same unit.

Let's convert both to centimetres. $65$65cm is already in centimetres, and $3$3m is $3\times100$3×100m = $300$300m.

Then we just use our old multiplication technique to find the percentage:

$\frac{65}{300}\times100%$65300×100% | $=$= | $\frac{6500%}{300}$6500%300 |

$=$= | $\frac{65%}{3}$65%3 | |

$=$= | $21\frac{2}{3}$2123 $%$% |

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?

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.

What percentage is $134$134 L of $536$536 L?

What percentage is $385$385 metres of $4$4 km?

What percentage is $24$24 minutes of $2$2 hours?

Don't forget to include the percentage symbol where required.

Understand operations on fractions, decimals, percentages, and integers