2.15 Problem solving with percentages

Percentages are used for a variety of things, usually when we want to describe how much of something there is. For example, perhaps you only want 50\% of the juice in you cup or when the car dashboard says that the fuel tank is only 20\% full. However, 50\% of the water in a 100\text{ L} swimming pool is obviously very different to 50\% of the 2\text{ L} milk in your fridge. Let's take a look at how we can figure out how much there actually is when we hear about percentages.

We already know how to find a  fraction of a quantity  through multiplication. For example, we know to find \dfrac{2}{3} of 60 all we do is multiply the two numbers together, so \dfrac{2}{3} \times 60=40 is our answer. We can do the same with percentages as we know how to turn them into fractions with 100 as the denominator.

For example, we want to find what 25\% of 84 is, so let's multiply them together. 25\% \times 84 can be rewritten as \dfrac{25}{100}\times 84, and we can simplify the fraction and get \dfrac{1}{4}\times 84=21.

A lot of the time it's hard for us to accurately calculate percentages of amounts in real life, so we'll have to estimate. Because percentages are expressed as something out of a hundred, we can also express them in diagrams of 5,\,10,\,100 things or more.

Examples

Example 1

Consider the following:

a

Express 40\% as a decimal.

Worked Solution

Create a strategy

To write the percentage as a decimal divide by 100 .

Apply the idea

\displaystyle 40\%	\displaystyle =	\displaystyle \frac{40}{100}	Divide by 100
	\displaystyle =	\displaystyle 0.4	Convert to a decimal

Watch question walkthrough

b

Now find 40\% of 90 kilograms.

Worked Solution

Create a strategy

Convert percentage to a decimal and multiply by the mass.

Apply the idea

\displaystyle 40\% \ \text{of}\ 90

\displaystyle =

\displaystyle 40\% \times 90

Multiply the percentage and mass

\displaystyle \\	\displaystyle =	\displaystyle 0.4\times 90	Convert the percentage to a decimal
\displaystyle \\	\displaystyle =	\displaystyle 36\ \text{kg}	Evaluate

Watch question walkthrough

Example 2

Ellie bought a 454\text{ mL} drink that claimed to be orange juice. In the ingredients list it said that orange juice made up 17\% of the drink. To estimate the amount of orange juice in the drink, which of the following would give the closest answer?

A

10\% \times 454

B

20\% \times 454

C

10\% \times 400

Worked Solution

Create a strategy

Choose the option which uses both a percentage and amount closest to the real values.

Apply the idea

The percentage closest to 17\% is 20\% which is in Option B. Options A and B use the correct amount. So Option B will give the closest answer.

Watch question walkthrough

Example 3

In a census, people are asked their gender and age. The graph shows the results: the percentage of females and males in each age group.

a

To the nearest 1\%, what percentage of females are between 5 and 9 years of age?

A

7\%

B

2\%

C

11\%

Worked Solution

Create a strategy

Read across for the given age group.

Apply the idea

The females are represented by the red bars. Read across to the red bar next to the age group of 5-9 which goes up to 11\%. The correct option is C.

Watch question walkthrough

b

To the nearest 1\%, what percentage of males are between 30 and 34 years of age?

A

7\%

B

4\%

C

2\%

Worked Solution

Create a strategy

Read across for the given age group.

Apply the idea

The males are represented by the blue bars. Read across to the blue bar that corresponds to the age group of 30-34 which is\ 7\%. The correct option is A.

Watch question walkthrough

c

The percentage of females between the ages of 20 and 29 is about:

A

7\%

B

2\%

C

11\%

Worked Solution

Apply the idea

The percentages that corresponds to the age group (20-24) is 8\%, and the age group (24-29) is 7\%. So we must add these percentages.

\displaystyle \text{Total percentage}	\displaystyle =	\displaystyle 8\%+7\%	Add the two percentages
\displaystyle \\	\displaystyle =	\displaystyle 15\%	Evaluate the addition

Watch question walkthrough

d

The percentage of males below 20 years of age is about:

A

15\%

B

10\%

C

30\%

D

50\%

Worked Solution

Create a strategy

Examine the graph and evaluate the sum of all age groups below 20.

Apply the idea

Add the percentages that corresponds to the male age group (0-4),\ (5-9), \ (10-14) and \ (15-19). Their percentages are 13\%, 13\%,13\%, and 11\% respectively.

\displaystyle \text{Total percentage}	\displaystyle =	\displaystyle 13\% + 13\% + 13\% + 11\%	Add all percentages
\displaystyle \\	\displaystyle =	\displaystyle 50\%	Evaluate

Watch question walkthrough