Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 3) Share 35 in the ratio 3 : 4. How much is the smaller share?
Answer
15, because 35 ÷ 7 = 5 and 3 × 5 = 15. - (Year 7, Unit 5, lesson 3) Which is bigger, 2/3 or 6/7?
Answer
6/7. 2/3 = 14/21 and 6/7 = 18/21. - (Year 7, Unit 5, lesson 1) Is 3/4 equal to 9/12? How do you know?
Answer
Yes: 3 × 3 = 9 and 4 × 3 = 12, so both are multiplied by the same number.
Words used in this lesson
- proportion
- two ratios that are equal. For example: 3 : 5 = 6 : 10.
- scale factor
- the number you multiply or divide both parts by. For example: 6 : 10 is 3 : 5 with scale factor 2.
- cross product
- the first part of one ratio times the second part of the other. For example: 3 × 10 and 5 × 6.
- equal ratios
- ratios that say the same thing in different numbers. For example: 4 : 7 and 8 : 14.
- enlargement
- a copy that is bigger by a scale factor. For example: a photograph printed larger.
Bigger batch, same taste
Zainab mixes paint from 2 tins of white and 3 tins of blue. She needs more, so she adds 4 tins of white and 4 tins of blue to get 6 : 7.
Predict first: will the new paint be the same shade? What would Zainab have needed to do to both numbers to keep it? Then use the bars below with a different ratio, 3 : 5, to test your idea.
Multiplying 3 : 5 by 2 gives 6 : 10: 6 × 5 = 30 and 10 × 3 = 30, so it is equal to 3 : 5. Adding 2 to both parts gives 5 : 7: 5 × 5 = 25 but 7 × 3 = 21, so it is not.
Show me the steps
- Multiply by 2: 6 : 10, and 6 × 5 = 30 = 10 × 3, equal. Add 2: 5 : 7, and 5 × 5 = 25 but 7 × 3 = 21, not equal.
- Multiply by 4: 12 : 20 is equal to 3 : 5. Add 0: nothing changes, so 3 : 5 stays 3 : 5. For 3 : 5, only adding nothing keeps the ratio.
- Multiply by 1: 3 : 5. Add 5: 8 : 10, and 8 × 5 = 40 but 10 × 3 = 30, not equal.
- Multiply by 6: 18 : 30, and 18 × 5 = 90 = 30 × 3, equal. Add 6: 9 : 11, and 9 × 5 = 45 but 11 × 3 = 33, not equal.
Equal ratios come from multiplying or dividing both parts by the same number, called the scale factor: 3 : 5 = 6 : 10 = 9 : 15. Adding the same amount to both parts changes the ratio, unless the two parts are equal, as in 5 : 5.
Two equal ratios written a : b = c : d are a proportion. To find a missing number, find the scale factor between the two known numbers in the same position, then use it on the other. Or use the multiplier inside the ratio: in 2 : 5 the second number is 2.5 times the first, so in 18 : ? it is 18 × 2.5 = 45.
A quick check: a : b = c : d when a × d = b × c. For 3 : 5 and 6 : 10, 3 × 10 = 30 and 5 × 6 = 30. It works because if c = k × a and d = k × b, both products are k × a × b.
Back to the paint: 6 : 7 is not 2 : 3, because 2 × 7 = 14 but 3 × 6 = 18. The new mix has more white for its blue, so it is lighter. To keep the shade she needed to double both, 4 tins of white and 6 of blue, which is 2 more white and 3 more blue.
A common mistake: a smoothie uses 1 cup of yoghurt to 2 cups of fruit. For a bigger jug, “4 cups of yoghurt is 3 more, so 2 + 3 = 5 cups of fruit.” That makes 4 : 5, which has far more yoghurt for its fruit. Multiplying the yoghurt by 4 means multiplying the fruit by 4: 8 cups.
Worked example: find the scale factor
Complete 2 : 5 = 14 : ?
- Compare the first numbers: 14 ÷ 2 = 7, so the scale factor is 7.
- Use it on the second number: 5 × 7 = 35.
- Check: 2 × 35 = 70 and 5 × 14 = 70. 2 : 5 = 14 : 35.
Are 6 : 8 and 21 : 28 equal?
- Divide 6 : 8 by 2: it is 3 : 4.
- Divide 21 : 28 by 7: it is 3 : 4.
- Both simplify to 3 : 4. Check: 6 × 28 = 168 and 8 × 21 = 168. Yes, they are equal.
Your go. The first step is given. Complete 5 : 9 = 40 : ?
- Compare the first numbers: 40 ÷ 5 = 8, so the scale factor is 8.
- Now use it on the second number, then check.
Check the answer
9 × 8 = 72, so 5 : 9 = 40 : 72. Check: 5 × 72 = 360 and 9 × 40 = 360.
Check before you go on
1. Which ratio is equal to 4 : 7?
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. Complete the equal ratios: 3 : 8 = 12 : ?
3. Complete the equal ratios: 36 : 24 = 3 : ?
Level 2 · Building
4. A pancake mix uses 2 eggs for every 150 ml of milk. How many millilitres of milk go with 10 eggs?
5. On a map, 3 cm stands for 4 km on the ground. A road measures 21 cm on the map. How many kilometres is the road?
6. A photograph is 8 cm wide and 6 cm tall. An enlargement keeps the same shape and is 20 cm wide. How many centimetres tall is it?
Level 3 · Stretch
7. The quantities a, b and c satisfy a : b = 2 : 3 and b : c = 6 : 5. If a is 20, what is c?
8. Write 3.5 : 6 using the smallest whole numbers. What is the first number?
Something to think about
Mix A is 2 parts concentrate to 3 parts water. Mix B is 4 parts concentrate to 5 parts water. Ayo says they taste the same, because each has 1 more part of water than concentrate. Is she right? Which is stronger, and how can you tell?
One way to think about it
No. Each mix has 1 more part of water, but a ratio is kept by multiplying, not by the gap. Make the water the same: 2 : 3 = 10 : 15 and 4 : 5 = 12 : 15. For the same 15 parts of water, B has 12 parts of concentrate and A has 10, so B is stronger. The cross products agree: 2 × 5 = 10 and 3 × 4 = 12, which are not equal.
Exit ticket
9. A recipe uses 4 cups of flour to 3 cups of sugar. Which new amounts keep the same ratio?