Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 4) Complete 3 : 7 = 12 : ?
Answer
28, because 12 ÷ 3 = 4 and 7 × 4 = 28. - (Year 7, Unit 2, lesson 3) What is 10% of 240?
Answer
24, because 10% is one tenth and 240 ÷ 10 = 24. - (Year 7, Unit 2, lesson 4) 30% of a number is 12. What is the number?
Answer
40. 10% is 12 ÷ 3 = 4, and 100% is 4 × 10 = 40.
Words used in this lesson
- proportion
- two quantities where one is always the same multiple of the other (also: two equal ratios, Lesson 4). For example: cost and number of items.
- value of one
- the amount for a single item. For example: 9 pesos for one item.
- scale factor
- the number you multiply both quantities by. For example: 12 items is 3 times 4 items.
- rate
- how much of one thing for each one of another. For example: 4 pages a minute.
- best value
- the lowest price for one. For example: 6.5 taka a pen against 7.
Find the price of one
At a market stall, 3 mangoes cost 18 rupees. Anika wants 5 mangoes. She could work out how many 3s there are in 5, but a better first question is: what does one mango cost?
Predict first: how much will 5 mangoes cost? Write down how you found it. Then explore the figure, where every item costs the same price, whichever pack you start from.
4 items cost 36 pesos, so one item costs 36 ÷ 4 = 9. 6 items cost 9 × 6 = 54 pesos. One item costs 9 pesos however many are in the pack.
Show me the steps
- A pack of 4 costs 36 pesos, so one item costs 36 ÷ 4 = 9. Then 6 items cost 9 × 6 = 54 pesos.
- 12 is 3 times 4, so the cost is 36 × 3 = 108. The value of one gives the same: 9 × 12 = 108.
- A pack of 5 costs 45 pesos. 7 is 1.4 times 5, which is awkward, but one item is still 45 ÷ 5 = 9 and 7 cost 9 × 7 = 63.
- A pack of 10 costs 90 pesos, so one item is 90 ÷ 10 = 9. Asking for 1 item gives the value of one directly.
In a proportion, one quantity is always the same multiple of the other. The cost of 1 item stays the same however many items you buy. Two ways to solve a proportion problem:
(1) the value of one: divide to find one, then multiply to find the amount wanted; (2) a scale factor: multiply both amounts by the same number. They give the same answer, so choose the one that works with the numbers you have.
Sense check: more items must cost more, and the answer should be close to the amount you get by estimating.
Back to the market: one mango costs 18 ÷ 3 = 6 rupees, so 5 mangoes cost 5 × 6 = 30 rupees. The scale factor from 3 to 5 is 5/3, which is 1.666... and does not come out exactly, so it is much harder to use than the value of one.
A common mistake: using proportion where there is none. Ravi is 4 when his sister Dipa is 2, but when she is 10 he is 12, not 20. The gap between their ages stays 2 years, so adding keeps the gap and multiplying does not. Ask first: does doubling one quantity double the other?
Worked example: the value of one and a scale factor
A tailor uses 6 m of cloth for 4 shirts. How much cloth does she need for 10 shirts?
| Quantity | Known | One | Wanted |
|---|---|---|---|
| Shirts | 4 | 1 | 10 |
| Cloth (m) | 6 | 1.5 | 15 |
On both rows, divide by 4 to go from Known to One, then multiply by 10 to go from One to Wanted.
- The value of one: 6 ÷ 4 = 1.5 m for each shirt.
- 10 shirts: 1.5 × 10 = 15 m.
- Check with a scale factor: 10 ÷ 4 = 2.5, and 6 × 2.5 = 15. 15 m.
Pack A holds 6 pens for 42 taka. Pack B holds 10 pens for 65 taka. Which is better value?
- The price of one pen from pack A: 42 ÷ 6 = 7 taka.
- The price of one pen from pack B: 65 ÷ 10 = 6.5 taka.
- Compare the price for one of each. Pack B is better value, by 0.5 taka a pen.
Your go. The first step is given. A cyclist rides 18 km in 45 minutes. At the same speed, how far does she ride in 30 minutes?
- 15 minutes is 45 ÷ 3, so the distance for 15 minutes is 18 ÷ 3 = 6 km.
- Now find the distance for 30 minutes yourself, then check.
Check the answer
30 minutes is 2 lots of 15 minutes, so the distance is 2 × 6 = 12 km. Check: 45 minutes is 3 lots of 15, and 3 × 6 = 18.
Check before you go on
1. 5 oranges cost 35 rupees. How much do 8 oranges cost?
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. 6 notebooks cost 24 shillings. What does 1 notebook cost, in shillings?
3. A printer prints 12 pages in 3 minutes. How many pages does it print in 9 minutes, at the same rate?
Level 2 · Building
4. A recipe for 8 people uses 14 eggs. How many eggs are needed for 12 people?
5. Shop A sells 8 pencils for 36 pesos. Shop B sells 5 pencils for 25 pesos. How many pesos cheaper per pencil is the cheaper shop?
6. A car uses 5 litres of fuel to travel 60 km. How many kilometres can it travel on 7 litres, at the same rate?
Level 3 · Stretch
7. 6 workers pack 90 boxes in an hour, all working at the same rate. How many boxes do 4 workers pack in 3 hours?
8. A paint mix needs 3 parts blue to 5 parts white. Ines has 12 litres of blue and 25 litres of white, and uses as much as she can. How many litres of paint does she make?
Something to think about
A bag of 4 oranges costs 12 pesos and a bag of 10 oranges costs 28 pesos. Is the cost proportional to the number of oranges? Why might a shop price them this way?
One way to think about it
No. One orange costs 12 ÷ 4 = 3 pesos in the small bag but 28 ÷ 10 = 2.8 pesos in the large one. If the cost were proportional, 10 oranges would cost 10 × 3 = 30. The shop gives a discount for the bigger bag.
Exit ticket
9. Which of these is a proportional relationship?