GCSE :: Direct & Inverse Proportion Dr J Frost (jfrost@tiffin.kingston.sch.uk) @DrFrostMaths 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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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). ?

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”

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= 7.5 3 =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

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× 9 → 𝑘=5 𝑏=5× 25 =25

Exercise 1 Given 𝑦 is directly proportional to 𝑥, find a formula for 𝑦 in terms of 𝑥: 𝑦=12 when 𝑥=3 →𝒚=𝟒𝒙 𝑦=8 when 𝑥=16 →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 2016 - 2H 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 ?

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.

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÷”

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 ?

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 ?

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

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 3 . 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 ? ? ?

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

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