1. GCSE :: Direct & Inverse Proportion
Dr J
Objectives: (a) Determine and use equations involving a constant a proportionality, for directly and inversely proportional relationships. (b) Recognise graphs for directly and inversely proportional relationships.
Last modified: 1st February 2019

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3. Starter
When Mo Farah runs at 8 m/s for some fixed period of time, he covers 240m. Fill in the gaps.
Speed
Distance Covered
8 m/s
240m
16 m/s
480m
4 m/s
120m
24 m/s
720m
If his speed doubled, the distance covered would similarly double. ? ? ?
×𝟑𝟎
What numerical relationship connected the speed and distance in each case?
There’s a constant scale factor between them (×𝟑𝟎). ?

4. 𝒅∝𝒔 𝒅=𝟑𝟎𝒔 Direct Proportion ? ? Speed (𝒔) Distance Covered (𝒅) 8 m/s
Where one quantity directly scales with another (e.g. as one doubles, the other doubles) we say they are directly proportional.
𝒅∝𝒔
Based on our discussion that there’s a ‘constant scale factor’ between the two, we can also write:
𝒅=𝟑𝟎𝒔 ? ?
“constant of proportionality”

5. Worked Examples ? ? 𝑦 is directly proportional to 𝑥. When 𝑥=3, 𝑦=7.5.1
𝑦 is directly proportional to 𝑥. When 𝑥=3, 𝑦=7.5.
Find a formula for 𝑦 in terms of 𝑥 and hence find the value of 𝑦 when 𝑥=4.4.
Fro Tip: You can replace the words “is directly proportional” with “=𝑘×”
𝑦=𝑘𝑥 7.5=𝑘×3 𝑘= =2.5
Therefore the formula is 𝒚=𝟐.𝟓𝒙
When 𝑥=4.4:
𝑦=2.5×4.4=11 ?
The formula just means your original equation, but where you’ve worked out 𝑘. 2 ?
𝒚=𝒌 𝒙 𝟐 𝟑𝟔=𝒌× 𝟑 𝟐 → 𝒌=𝟒 𝒚=𝟒× 𝟓 𝟐 =𝟏𝟎𝟎

6. Test Your Understanding
𝑟 is directly proportional to 𝑞. When 𝑞=12, 𝑟=9.
Find a formula for 𝑟 in terms of 𝑞.
Find the value of 𝑟 when 𝑞=17. a)
𝑟=𝑘𝑞 9=𝑘×12 → 𝑘=0.75 𝑟=0.75𝑞 b)
𝑟=0.75×17=12.75 ? ? B
𝑏 is directly proportional to the square root of 𝑎.
When 𝑎=9, 𝑏=15.
Find the value of 𝑏 when 𝑎=25. ?
𝑏=𝑘 𝑎 15=𝑘× → 𝑘=5 𝑏=5× 25 =25

7. Exercise 1
Given 𝑦 is directly proportional to 𝑥, find a formula for 𝑦 in terms of 𝑥:
𝑦=12 when 𝑥= →𝒚=𝟒𝒙
𝑦=8 when 𝑥= →𝒚=𝟎.𝟓𝒙
𝑏 is directly proportion to 𝑎. When 𝑎=8, 𝑏=10. Find the value of 𝑏 when 𝑎=13.
𝒃=𝟏𝟔.𝟐𝟓
𝑦 is directly proportional to the square of 𝑥. When 𝑥=6, 𝑦=27. Find 𝑦 when 𝑥=4.
𝒚=𝟏𝟐
𝑦 is directly proportional to the cube of 𝑥. When 𝑥=2, 𝑦=12. Find 𝑦 when 𝑥=3.
𝒚=𝟒𝟎.𝟓
𝑛 is directly proportional to the square root of 𝑚. When 𝑚=4, n=8. Find 𝑛 when m =9. 1
𝑏 is directly proportional to the square of 𝑎. When 𝑎=6, 𝑏=18. Find 𝑎 when 𝑏=32.
𝒂=𝟖
𝑏 is directly proportional to the cube root of 𝑎. When 𝑎=27, 𝑏=4.5. Find 𝑎 when 𝑏=7.5.
𝒂=𝟏𝟐𝟓
[Edexcel GCSE(9-1) Mock Set 1 Autumn H Q10ii]
𝑦∝ 𝑥 2    
Write a formula for 𝑦 in terms of 𝑥
𝒚∝ 𝒙 𝟐 → 𝒚=𝒌 𝒙 𝟐 𝟒𝟎𝟎=𝒌× 𝟓 𝟐 𝒌=𝟏𝟔 𝒚=𝟏𝟔 𝒙 𝟐 6 ? ? 2 7 ? ? 3 8 ? 4 𝒙  5  6  𝒚 
400 
576  ? ? 5 ?

8. Back in a Mo…
Mo is running a 5000m race. Here are his times when he runs at different speeds:
Speed (s)
Time (t)
5 m/s
1000s
20 m/s
250s
1 m/s
5000s
6.47 m/s
773s (his PB)
How are the speeds and times related? ?
This time as his speed doubles, his time halves (i.e. the opposite). Note also that the speed and time also multiply to give the same value.

9. 𝒔∝ 𝟏 𝒕 𝒔= 𝒌 𝒕 Inverse Proportion Speed (s) Time (t) 6.47 m/s
773s (his PB)
5 m/s
1000s
10 m/s
500s
We say they are indirectly or inversely proportional.
𝒔∝ 𝟏 𝒕
𝒔= 𝒌 𝒕
Notice that as 𝑡 doubles, the RHS becomes half as big.
Fro Tip: When you see the words “inversely proportional to”, replace with “=𝑘÷”

10. Quickfire First Step ? ? ? ? ? ? ? Question… First thing you’d write…
Reminder:
For “(directly) proportional to”, use “=𝒌×”
For “inversely proportional to”, use "=𝒌÷” ?
𝑦 is directly proportional to 𝑥.
𝒚=𝒌𝒙 ?
𝒚= 𝒌 𝒙
𝑦 is inversely proportional to 𝑥. ?
𝑦 is directly proportional to the square of 𝑥.
𝒚=𝒌 𝒙 𝟐 ?
𝑦 is inversely proportional to the square root of 𝑥.
𝒚= 𝒌 𝒙 ?
𝑦 is inversely proportional to the cube of 𝑥.
𝒚= 𝒌 𝒙 𝟑
𝑦 is proportional to the cube root of 𝑥. ?
𝒚=𝒌 𝟑 𝒙
𝒚= 𝒌 𝒙 𝟑
𝑦∝ 1 𝑥 3 ?

11. Worked Examples ? ? 𝑦 is inversely proportional to 𝑥. When 𝑥=6, 𝑦=8.
Find 𝑦 when 𝑥=9.
Fro Tip: When you see the words “inversely proportional to”, replace with “=𝑘÷”
𝒚= 𝒌 𝒙 𝟖= 𝒌 𝟔 → 𝒌=𝟒𝟖 𝒚= 𝟒𝟖 𝟗 =𝟓.𝟑𝟑 𝒕𝒐 𝟑𝒔𝒇 ?
𝑏 is inversely proportional to the square root of 𝑎.
When 𝑎=4, 𝑏=10.
Find 𝑏 when 𝑎=25.
𝒃= 𝒌 𝒂 𝟏𝟎= 𝒌 𝟒 → 𝒌=𝟐𝟎 𝒃= 𝟐𝟎 𝟐𝟓 =𝟒 ?

12. Test Your Understanding?
𝟒𝟎 𝑽 ?
𝟒𝟎 𝟐 =𝟐𝟎 B ?
𝟏.𝟑𝟑 𝒕𝒐 𝟑𝒔𝒇

13. Exercise 2
𝑦 is inversely proportional to 𝑥. 𝑦=3 when 𝑥=15. Find a formula for 𝑦 in terms of 𝑥.
𝒚= 𝟒𝟓 𝒙
𝑏 is inversely proportional to the square root of 𝑎. 4 1 𝒂
1.44 1 25 𝒃
3.2
3.84
0.768 ? ? ?
𝑦 is inversely proportional to the cube of 𝑥. 5 2
𝑞 is inversely proportional to the square of 𝑟. 3 5
0.740
1.5
0.324
100 ? 3 5
0.949 4
1.44 40 ? ? ?
𝑦 is inversely proportional to one more than 𝑥. 6 3
𝑦 is inversely proportional to 𝑥. 8 5
1.142 35 ? 4 6
0.190
6.5
4.643
27.3 ? ? ?
[Edexcel GCSE Jun2016-1H Q24] Given that 𝑦∝ 1 𝑥 2 , complete this table of values. 4
Edexcel IGCSE May2015-3H Q22]
𝐴, 𝑟 and 𝑇 are three variables. 𝐴 is proportional to  𝑇 2 . 𝐴 is also proportional to  𝑟 𝑇=47 when 𝑟=0.25.
Find 𝑟 when 𝑇=365. Give your answer correct to 3sf. 𝒓=𝟎.𝟗𝟖𝟎 N 𝑥 1 2 5 10 𝑦
100 25 4 ? ? ?

14. Proportion and their Graphs
Which of these graphs represent variables which are directly proportional to each other?
For proportional variables 𝑦 and 𝑥, 𝑦=𝑘𝑥. This is the equation of a straight line that goes through the origin.
hours on computer
electricity usage
years
rabbit population
Temperature (C)
Temperature (F)
Yay 
Yay  
Yay
Nay   
Nay 
Nay

15. Proportion and their Graphs
Which of these graphs represent variables which are inversely proportional to each other?
When you do curved graphs, you’ll see that 𝑦= 𝑘 𝑥 is known as a reciprocal graph, which has the shape of the second graph.
Distance from offender
Smell intensity
Distance from offender
Smell intensity 
Yay
Yay 
Nay   
Nay