Percent, Ratio and Rates (2024)

| Percent - a special type of fraction | Percent Models | Ratios |
| Relationships:decimal fractions, common fractions, percent and ratio | Rates | Quick quiz |

Percent - a special type of fraction

The word percent come from the expression 'per cent' and literally means 'a part of one hundred'. A percent is a part, or fraction, out of 100. For example:

We can see that to write a percent as a fraction we express the percent as a fraction with a denominator of 100. We may then be able to simplify the fraction further.

For example, 75% = 75/100 = 3/4

To express a fraction as a percent we must first convert the fraction into hundredths (in simple cases we can do this by using equivalent fractions) and then replace '/100' by the percent '%' sign.

For example, 4/5 = 80/100 = 80%

We can see that we express a percent as a decimal by dividing by 100.

For example,

To express a decimal as a percent we multiply the decimal number by 100.

For example,

Some percents expressed as fractions and decimals

		= 0.125 (decimal)
		= 0.236 (decimal)
		= 0.333 (decimal, rounded to 3 decimal places)
		= 0.5 = 0.50 (decimal)
		= 0.667 (decimal, rounded to 3 decimal places)
		= 1.1 (decimal)
		= 1.5 (decimal)
		=2.0 (decimal)

Example 1: 30 out of 50 apples in a box are too bruised to sell. What percent of apples cannot be sold?

Example 2: Ryan spent 25 minutes in the bank, 11 minutes of which was spent waiting in a queue. What percent of time did he spend waiting in the queue?

Example 3: What percent is 7 cm of 20 cm?

Percent models

Dual-scale number line model

We can use the dual-scale number line, also called the proportional number line, to illustrate example 1 from above.

The dual-scale number line model is discussed further in the other pages of the Percent, Ratio and Rates topic.

Elastic tape measure model

The tape measure model is a good linear model of percent. Teachers can easily make these models using a ruler, such as a 1 metre ruler, and elastic. The elastic needs to be marked with a percent scale. It can then be stretched to the desired length.

For example, what is 60% of 50?

To find the answer we line up the zeros of the ruler and the elastic. We then stretch the elastic so that 100% lines up with the whole amount, which in this case is 50. We then look for 60% on the elastic and read the corresponding amount on the ruler. We can see below that 60% of 50 is 30.

The intention is NOT to use this model accurately. It is a good way of showing that percent always involves a proportional comparison of something to 100.

1 metre ruler

Elastic

By manipulating the tape measure, this model can be used for the 3 types of percent problems, discussed in Percent Examples. Examples of which are,

What is 20% of 50?
What percent is 10 of 50?
30% of what number is 15?

(Note: for a lesson, a teacher will need elastics tape measures of various lengths, because the elastic can only be stretched - it cannot be shrunk).

(This elastic tape measure model was developed by J. H. Weibe)

Ratios

A ratio is another way of comparing quantities. Each quantity must be measured in the same units. An advantage of ratios is that we can compare several things at once.

Although ratios must have each quantity measured in the same units, the units are not fixed. 1 litre of blue paint to 4 litres of the white paint represents the same ratio as 1 tin of blue paint to 4 tins of white paint, or 1 bucket of blue paint to 4 buckets of white paint. This fact makes ratios very versatile to use in everyday situations.

The order in which a ratio is written is very important. If we say the ratio of the number of girls to the number of boys is 5:2 this is very different to saying the ratio of the number of girls to the number of boys is 2:5.

A ratio can be written in different ways;

- in words - the ratio of the number of girls to the number of boys is 5 to 2, and this is the way we say it

- using a colon - number of girls : number of boys = 5 : 2

Example 4: Let's say I want to make the paint colour 'sky blue' and I know that the way to do this is to mix 1 part blue with 4 parts white. This means there is a ratio of blue to white of 1:4. In this case 1 litre of blue to 4 litres of white, making 5 litres of sky blue paint.

To make double the quantity of paint I can mix the blue to white as a ratio of 2:8. This will make the same colour. The ratios of 1:4 and 2:8 are equivalent, and worked out in the same way as equivalent fractions. We multiply each part of the original ratio by the same number and we can find equivalent ratios.

Sharing quantities in a given ratio

Example 5: We have a small inheritance of $15000 to be shared among 3 people in the ratio of 2 : 2 : 1, how much does each person receive?

Relationships - decimal fractions, common fractions, percent and ratio

We can use examples to illustrate the relationships between decimal fractions, common fractions, percent and ratio.

Example 6: An inheritance of $15000 is to be distributed among 3 people in the ratio of 2 : 2 : 1. ($15000 will be divided into 5 portions)

Example 7: Ratios and fractional parts.
A litre of mixed cordial requires 250 mls of cordial and 750 mls of water. How can we represent this as a ratio and a fraction?

Example 8: Ratios and fractional parts.
A group of 100 people is made up of 60 males and 40 females. How can we represent this as a ratio and a fraction?

In the early stages of introducing ratio at a primary level we generally discuss ratios in terms of a part to part or, quantity to quantity, comparison. At this stage part to whole relationships are often better represented by fractions or percents with which students already have some experience. When part to whole ratios are introduced care must be taken to ensure students clearly understand what is being represented.

Rates

We use rates when we are measuring one quantity or amount in relation to another quantity or amount. We use them to compare how quantities change, usually over a period of time. A significant difference between rates and ratios is that when we are forming rates, each quantity is measured in different units to form new composite units.

For example, let's say you are travelling at a rate of 60 kilometres per hour (km/h). Here we are measuring kilometres in relation to hours and the rate unit becomes 'kilometres per hour', often written as km/h.

Other examples of rates are, an athlete running at 10 metres per second (m/s), and a factory using water at a rate of 450 litres per hour (l/h).

Some rates we commonly use are:

Example 9: A hose is running water at the constant rate of 100 litres an hour.
1. How many litres will run in 2 hours?
2. How many hours will it take to run 350 litres?

Quick quiz

Extra questions:

Shade 20% of this diagram

(a) What percentage of the following shape is shaded?

(b) What percentage is not shaded?

3. What is the ratio of pink to white in the following figure?

4. A biro is 23 cm long and a ruler is 30 cm long.

(a) What is the ratio of the biro to the ruler?

(b) What is the ratio of the ruler to the biro?

5. Complete the following table.

6. A dog is three times as long as a bird. The dog is 90 cm long.

(a) How long is the bird?

(b) Write a ratio to show the relationship of the dog to the bird.

7. I have a 300g bag of sugar and a half a kilo of butter.

(a) What is the ratio of the sugar to the butter?

(b) What is the ratio of the butter to the sugar?

8. What two quantities are being measured when the units are km/hr?

To view the quiz answers, click here.