Unit 1: Ratio, Rate, & Proportion

# Topic B: Rates

When a ratio is used to compare two different kinds of measure (e.g. apples and oranges, or meters and hours), it is called a rate. The denominator must be 1.

Example A

A car can drive 725 km on 55 L of gas. What is the rate in km per L? The ratio of this is $\dfrac{725\text{ km}}{55\text{ L}}$. Find the rate by making the denominator 1.

Divide $\dfrac{725}{55} \div \left(\dfrac{55}{55}\right) = \dfrac{725\div55}{55\div55}=\dfrac{13.18}{1}=13.18$

The rate is 13.18 km/L.

Example B

Sue bought 10 lb of oranges for \$4.99. What is the rate in cents per pound? The ratio is $\dfrac{4.99}{10\text{ lb}}=\dfrac{499\text{ cents}}{10 \text{ lb}}$. Find the rate by making the denominator 1.

Divide $\dfrac{499}{10} \div \left(\dfrac{10}{10}\right) =\dfrac{499\div10}{10\div10}=\dfrac{49.9}{1}=49.9$

The rate is 49.9 ¢/lb.

When talking about rate, use the word ‘per’.

In example A, say: “The fuel economy of the car is 13.18 kilometres per litre”.
In example B, say: “The oranges cost 49.9 cents per pound”.

Example C

It takes 60 ounces of grass seed to plant 30 m2 of lawn. What is the rate in ounces per square metre (m2)? The ratio is $\dfrac{60\text{ oz}}{30\text{ m}^2}$. Find the rate by making the denominator 1.

Divide $\dfrac{60}{30} \div \left(\dfrac{30}{30}\right) = \dfrac{60\div30}{30\div30}=\dfrac{2}{1}=2$

The rate is 2 oz/m2, or 2 ounces per square metre.

Exercise 1

Write the following ratios as rates, comparing distance to time.

1. 120 km, 3 hours
2. 27 km, 9 hours
3. 203 km, 29 seconds
4. 444 km, 48 seconds

1. 40 km/hour
2. 3 km/hour
3. 7 km/second
4. 9.25 km/second

Exercise 2

Write the following ratios as rates.

1. A leaky faucet can lose 52 litres of water in a week. What is the rate of litres lost per day? (round to two decimal places)
2. A ratio of distance travelled to time is called speed. What is the rate (speed) in kilometres per hour (km/h)?
1. 45 km, 3 hours
2. 129 km, 1.5 hours
3. 65 km, 13 hours

3. Vancouver Island has a population of 734,860, and a land mass of 32,134 square kilometres. What is the rate of number of people per square kilometre? (This is called population density.) Round your answer to the nearest whole number.
4. At rest, the heart beat of a mouse is 30,000 beats per 60 minutes. What is the rate of beats per minute?

1. 7.43 L/day
1. 15 km/hour
2. 86 km/hour
3. 5 km/hour
2. 23 people/km2
3. 500 beats/minute

# Topic B: Self-Test

Mark       /7                  Aim        6/7

1. Write the definition.
(1 mark)

1. Rate

2. Write the following ratios as rates. Round people to the nearest person.
(6 marks)

1. 12 cups water, 3 cups sugar
2. 72 metres, 24 seconds
3. 1,365,000 people, 4,000 km2
5. 12 cups of flour, 12 tsp. of baking powder
6. 8 litres of gas, 2 litres of oil

## Answers to Topic B Self-Test

1. A rate is used when a ratio compares two different kinds of measure, and when the denominator is 1.
1. 4 cups of water/cups of sugar
2. 3 m/second
3. 341 people km2
4. 20 cars/bike
5. 1 cup flour/tsp baking powder
6. 4 litres gas/litre oil 