Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Year 8, Unit 1, lesson 2) Simplify the ratio 18 : 30.
Answer
3 : 5. Divide both parts by their highest common factor, 6. - (Year 7, Unit 2, lesson 3) What is 15% of 80?
Answer
12. 10% is 8 and 5% is 4, so 8 + 4 = 12. - (Year 7, Unit 3, lesson 6) Solve 6n = 42.
Answer
n = 7, because 42 ÷ 6 = 7.
Words used in this lesson
- rate
- a comparison of two quantities in different units. For example: 36 litres in 4 minutes.
- unit rate
- the amount for one of the second quantity. For example: 9 litres per minute.
- per
- for each one. For example: rupees per kilogram.
- steady
- the same amount in each equal time. For example: a steady 9 litres a minute.
- compare
- put two things side by side to see which is greater. For example: 7 > 3.
One minute at a time
A hosepipe in Nairobi fills a 10-litre bucket in 2 minutes. A bigger hosepipe on a building site fills a 200-litre drum in 25 minutes. The drum took far longer, but it is also far bigger. Predict first: which hosepipe is faster, and what could you work out for each one that would let you say?
Start with a single tap. The figure shares the litres out equally over the minutes, so every minute is a column of the same height.
36 litres in 4 minutes: 36 ÷ 4 = 9 litres in each minute, a rate of 9 litres per minute.
Show me the steps
- 12 litres in 2 minutes: 12 ÷ 2 = 6 litres per minute.
- 36 litres in 4 minutes: 36 ÷ 4 = 9 litres per minute. Four equal minutes, 9 litres in each.
- 60 litres in 5 minutes: 60 ÷ 5 = 12 litres per minute.
- 48 litres in 5 minutes: 48 ÷ 5 = 9.6 litres per minute. A rate does not have to be a whole number.
- 24 litres in 6 minutes is 4 litres per minute, and so is 12 litres in 3 minutes: different amounts and times, the same rate.
A rate compares two quantities in different units, such as litres and minutes, or rupees and kilograms. Its unit has “per” in it: litres per minute, rupees per kilogram.
The unit rate is the amount for one of the second quantity: divide the first quantity by the second. 36 litres in 4 minutes is 36 ÷ 4 = 9 litres per minute.
Two rates can be compared when they are per the same thing. The larger unit rate is the faster, or the heavier, or the dearer, depending on what is being measured.
A ratio such as 3 : 5 compares two amounts in the same unit. A rate compares two amounts in different units, so the unit stays with the number: 9 litres per minute is not the same thing as 9.
Comparing two taps
Tap A is fixed at 45 litres in 5 minutes. Set tap B and see which one fills faster. Look at the totals, then look at the columns.
Tap A: 45 ÷ 5 = 9 litres per minute. Tap B: 56 ÷ 7 = 8 litres per minute. Tap A is faster, even though tap B delivered more litres.
Show me the steps
- Tap B 56 litres in 7 minutes: 56 ÷ 7 = 8 litres per minute, less than tap A's 9. More litres, but slower.
- Tap B 54 litres in 6 minutes: 54 ÷ 6 = 9 litres per minute. The same rate as tap A, with a different amount and a different time.
- Tap B 40 litres in 4 minutes: 40 ÷ 4 = 10 litres per minute. Fewer litres than tap A, and still the faster tap.
- Tap B 24 litres in 8 minutes: 24 ÷ 8 = 3 litres per minute, the slowest setting.
A common mistake: “It delivered more litres, so it is faster.” A tap that fills 90 litres in 15 minutes delivers more than one that fills 60 litres in 8 minutes, but 90 ÷ 15 = 6 litres per minute and 60 ÷ 8 = 7.5 litres per minute. The smaller amount was the faster tap, because it took far less time. Always compare the amount for one.
Worked example: the two hosepipes
The bucket hose fills 10 litres in 2 minutes. The drum hose fills 200 litres in 25 minutes. Which is faster?
- Find litres per minute for each. Bucket hose: 10 ÷ 2 = 5 litres per minute.
- Drum hose: 200 ÷ 25 = 8 litres per minute.
- Both rates are per minute, so compare them: 8 > 5. The drum hose is faster, by 3 litres in every minute.
Worked example: changing the “per”
A printer in a Dubai office prints 150 pages in 6 minutes. Give its rate in pages per minute and in pages per hour.
- Pages per minute: 150 ÷ 6 = 25.
- There are 60 minutes in an hour, and the printer prints 25 pages in each: 25 × 60 = 1500.
- 25 pages per minute, which is 1500 pages per hour. The same rate, written per a different unit of time.
Your go. The first step is given. A sprinkler in Lima waters 90 m² of lawn in 15 minutes. Find its rate in square metres per minute, then per hour.
- Per minute: 90 ÷ 15 = 6 square metres.
- Now change it to per hour, then check.
Check the answer
There are 60 minutes in an hour, so 6 × 60 = 360. The rate is 6 m² per minute, which is 360 m² per hour.
Check before you go on
1. A van uses 30 litres of fuel to travel 240 km. What is its fuel use as a rate?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A baker in Lagos packs 84 loaves in 6 hours at a steady pace. How many loaves per hour is that?
3. 2.5 kg of rice costs 275 rupees. What is the price in rupees per kilogram?
Level 2 · Building
4. Four typists each typed for a different length of time. Who typed the fastest, in words per minute?
5. A cyclist rides 7.5 km in 30 minutes at a steady pace. How many kilometres is that per hour?
6. Tap P fills a tank at 5 litres per minute. Tap Q fills the same tank with 42 litres in 7 minutes. Both are open together. How many litres per minute is that?
Level 3 · Stretch
7. A swimmer swims 400 m in 8 minutes. A cyclist rides 3 km in 10 minutes. How many times as fast as the swimmer is the cyclist?
8. A machine in Dhaka sews 3 shirts every 20 minutes. At that steady rate, how many shirts does it sew in 8 hours?
Something to think about
Two rates can be added when they are per the same thing. Can you add 5 litres per minute to 30 litres per hour? Say what you must do first, and give the total.
One way to think about it
Put both rates over the same time. 30 litres per hour is 30 ÷ 60 = 0.5 litres per minute, so the total is 5.5 litres per minute. Or the other way: 5 litres per minute is 5 × 60 = 300 litres per hour, so the total is 330 litres per hour. Both say the same thing.
Exit ticket
9. Bus A travels 360 km on 40 litres of fuel. Bus B travels 280 km on 35 litres. Femi says “Bus B is better on fuel, because it used fewer litres.” Which reply is right?