Percent, ratio & rates home

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Rates examples

Example 1:
(a) A freight truck travels 110 kilometres in 1 hour, write this as a rate.
(b) A smaller freight truck travels 180 kilometres in 2 hours, write this as a rate.

Working Out Thinking

rate = kilometres ÷ hours
= 110 km ÷ 1 hour
= 110 km/h


rate = kilometres ÷ hours
= 180 km ÷ 2 hours
= 90 km ÷ 1 hour
= 90 km/h

(a) To write '110 kilometres in 1 hour' as a rate I need to convert these quantities into kilometres per hour. Knowing what my units will be, km/h, actually helps me to work out the answer.

(b) To write '220 kilometres in 2 hours' as a rate of km/h I need to convert these quantities into kilometres per hour. My units will be, km/h. In this case I know how many kilometres are travelled in 2 hours, so I need to divide the quantities by 2 to find out how many kilometres are travelled in 1 hour, that is, km travelled per hour.

Example 2: A company wants to find how many customers come in their shop per week. If on average 9 people were found to come into the shop each half hour and the shop is open 45 hours per week, what is the expected rate of customers per week?

Working Out Thinking

Rate = customers ÷ time

= 9 customers in 1/2 hour
= 18 customers in 1 hour

The rate of customers per week,

= 18 customers x 45 hours
= 810 customers per week

Note that 9 customers in half an hour is 9 ÷ (1/2) in one hour.

We want to find out the number of customers per week which will be the units in our answer. We could first find out the number of customers per hour and then we can easily convert this rate into customers per week (because we know the shop is open for 45 hours per week). We know that there are 9 customers per half hour, therefore the customer rate per hour is 9x2 = 18. We know that the shop is open 45 hours per week, so the customer per week rate is 18 customers x 45 hours. The units for our answer are 'customers per week'.


Converting Units

Working with rates is usually fairly straightforward, since the units give us clues about what our answer will be. Usually we want to convert quantities into standard rates such as km/h (as we have done above) and m/s and knowing this can help us work out our calculations.

Example 3: Movie - finding a rate

Ahmad has been typing out an essay on his computer continuously for the last 20 minutes. Ahmad then ran a 'word count' and discovered that we had typed 1320 words so far. He is under pressure to finish typing the rest of his essay (about 3600 words) in about an hour and wants to work out his typing rate of words per minute.

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Movie - using the calculated rate in a problem

If Ahmad needs to type about 3600 more words and has just over an hour before the deadline, will he make it?

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Example 4: Movie - finding a rate

The local GP sees on average 35 patients in a day. If she works 7 hours a day, what is her rate of patients per hour?

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Movie - using the calculated rate in a problem

How many minutes does she spend with each patient?

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Example 5: Isabel shoots on average 12 goals per match. If there are 15 matches in a season, what is Isabel's goal rate per season?

If Isabel shoots an average of 12 goals per match then her rate of goals per match is 12. As there are 15 matches in a season her rate of goals per season is 12 x 15.



Another common application of rates is comparing rates. To do so we may have to convert one rate into units which are consistent with the other rate.

Example 6: Movie - comparing rates.
Two telecommunications companies are advertising discount call rates. Company A has an interstate call rate of 7 cents per minute. Company B is advertising a rate of $5.00 per hour. Which company has the most economical hourly rate?

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It is not always obvious that some quantities are rates. Rates commonly discussed in the news are often mentioned without their units and this can make it very difficult for the layperson to understand what they really mean.

For example, if we hear that the unemployment rate in VIC is 6.5%, what does this really mean?

According to the Australian Bureau of Statistics, AusStats: 6203.0 Labour Force, Australia, the employment rate is defined as;

Unemployment Rate:
For any group, the number of unemployed persons expressed as a percentage of the labour force in the same group.

Persons aged 15 years and over who were not employed during the reference week, and had actively looked for full-time or part-time work at any time in the four weeks up to the end of the reference week and:

were available for work in the reference week; or

were waiting to start a new job within four weeks from the end of the reference week, and could have started in the reference week if the job had been available then.

Labour force:
For any group, persons who were employed or unemployed, as defined.

Labour force status:
A classification of the civilian population aged 15 years and over into employed, unemployed or not in the labour force, as defined. The definitions conform closely to the international standard definitions adopted by the International Conferences of Labour Statisticians.

Further applications

Example: The current in a river is flowing at 7 km/h. How long will it take a log to travel 25 kms downstream?

Working Out Thinking
river flow rate = 7 km/h
the log travels 25 kms at a rate of 7 km/h

25km/(7km/h) = 3.57 hours
or 3 hours and about 34 minutes

I need to find out how long it will take for the log to travel 25 kms downstream and I know that the river is flowing at 7 km/h. So, what the question is really saying is, if a log is travelling down the river at 7 km/h, how long will it take to travel 25 km?

NOTE: If I include the units in my working out I will see that they give me a clue as to what units will be in my answer. (In this case it was fairly obvious that my answer was going to be in hours.)

Example 7: Movie
On Monday I filled up my car with petrol and paid 85.9 cents/litre. If I spent $55, how many litres did I get?

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Quick quiz

1. Express each of the following sentences as rates.
a) A bus travelled 800 kilometres in 8 hours.
b) A sprinter ran 100 metres in 10 seconds.
c) Vaughan worked 7 hours for $84.
d) A hamburger restaurant sells 330 hamburgers in 2 hours.
e) Marly kicked 96 goals in 12 games.
a) If a car travels at 60 km/h, how long will it take to travel 200 km?
b) A specialist has 14 appointments a day. If she works for 6 hours a day, how much time is scheduled for each patient?
c) The shower runs at 5 litres per minute. If Rory showers for an average of 7 minutes a day every day, how much water does he use in a week?
d) John is paid $8 an hour. If he works 17 hours this week, how much will he be paid?
John asks for a pay rise and is given an increase of $1.70 per hour. What is his new rate of pay?
a) Three cars are endurance racing. Car X travels 330 kilometres in 3 hours, car Y travels 500 kilometres in 4.5 hours and car Z travels 120 kilometres in an hour. Which car is travelling the fastest?
b) A health insurance company is investigating the claim rates of its members. It finds on average that 21-30 year olds make 16 claims in a year, 31-50 year olds make an average of 56 claims per 2 years and 50+ year olds make about 17 claims per quarter. Work out the claim rate per quarter for each group. Which group has the highest claim rate?
c) A factory makes 300 chocolate bars per minute, how many bars are made in 12 minutes? Work out how long it takes to make each bar?

To view the quiz answers, click here.



University of Melbourne