1. GCSE: Direct & Indirect Proportion Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 31 st August 2015

2. Dr Frost records the number of hours of revision his class did for their Maths landmark, and then sees how well they do: Hours of revision (h)Marks (m) 315 525 630 945 1155 1575 Direct Proportion

3. If variables x and y are directly proportional, then we write y  x, and y = kx where k is a constant, known as the ‘constant of proportionality’. Key definition to write down: Direct Proportion

4. When variables are proportional, it means we’re allowed to multiply/divide both variables by the same number, but we’re not allowed to add/subtract.  Nay Two variables s and d represent the speed of a runner and the distance covered within some fixed time. s and d are directly proportional. Which of the following statements are true: #1: If the runner’s speed triples, the distance covered triples. #2: If the runner’s speed increases by 5, the distance covered increases by 5.  Yay  Nay Yay Yay or Nay

5. The weight of a tube of sweets is proportional to its length. When the tube is 10cm long, the sweets weigh 66g. A longer pack is 16cm. How much does it weigh? 66 = k × 10 So k = 66  10 = 6.6 w  l So w = kl Step 1: Write out relationship between variables Step 2: Work out constant of proportionality (k) Step 3: Use our completed equation to find missing values w = 6.6 l So if l = 16cm, then w = 6.6 x 16 = 105.6g ? ? How to solve proportion problems

6. 304013 456019.5 ? ? Example

7. 466.32 81820 ? ? Example

8. 9151.94 4254.219.5 ? ? Example

9. Exercises 153550 214970 1 4107.07 32200100 2 855.23 358.487.5100 3 2536169 12.51532.5 4 2764216 91218 5 4620 6.59.127.3 6 ? ? ? ? ? ? ? ? ? ? ? ?

10. Edexcel GCSE Maths Textbook Page 551 – Exercise 34A Q1a, c, 2a, c, 5 Page 553 – Exercise 34B Q1a, 2a, 6, 7 Further Exercises

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

12. r6104.646 V42194.4419.5 Checking your understanding… ? ?

13. Speed (s)Time (t) 6.47 m/s773s (his PB) 5 m/s1000s 6 m/s833.33s 1 m/s5000s 6.3 m/s793.7s Mo is running a 5000m race. Here are his times when he runs at different speeds: How are the speeds and times related? Back to the Mo…

14. We say that variables s and t are inversely proportional (also known as ‘indirectly proportional’). Inverse (aka ‘Indirect’) Proportion ?

15. 840.889 51045 ? ? Example

16. x16500.25 L52.8340 Given that L is inversely proportional to √ x, fill in the missing values in this table. ? ? Check your understanding…

17. 1.44125 3.23.840.768 Exercises 1522.55.625 328 1 350.949 41.4440 2 851.142 5835 3 4 350.740 1.50.324100 5 460.190 6.54.64327.3 6 ? ? ? ? ? ? ? ? ? ? ? ?

18.  Inverse Proportion and graphs Which of these graphs represent variables which are inversely proportional to each other? Distance from offender Smell intensity Distance from offender Smell intensity  Yay  Nay Yay Nay

19. The Wall of Proportion Destiny The flow of water from a square pipe is proportional to its area. When the width of the pipe is 10cm, the flow is 3 gallons a second. What is the flow when the width is 15cm? The problem Relationship Find k Use equation The quoted cost of building a skyscraper is proportional to its height. The cost for Canary Wharf (235m) is £379 million. How much would the Shard (310m) cost? c  h c = kh f  w 2 f = kw 2 379,000,000 = k x 235 So k = 1,612,766 Cost of Shard = 1612766 x 310 = £500 million 3 = k x 10 2 So k = 3  100 = 0.03 f = kw 2 So f = 0.03 x 15 2 = 6.75 gallons/s ? ? ? ? ?

20. Weee!! The Wall of Proportion Destiny The value of a gold cube is proportional to its volume. When the cube is 10cm wide, its value is £5600. When its value is £20100, how wide is it? The problem Relationship Find k Use equation I am constructing a new ‘Super Fun Happy Slide’, whose surface area is proportional to its length squared. When its length is 50m, its surface area is 200m 2. When its surface area is 500m 2, what is its length? s  l 2 s = kl 2 v  w 3 v = kw 3 200 = k x 502 k = 0.08 500 = 0.08 x l 2 l 2 = 6250 So l = 79.06m 5600 = k x 10 3 So k = 5.6 v = kw 3 So 20100 = 5.6 x w 3 w = 3 √3589 = 15.31cm ? ? ?? ? Super Fun Happy Slide ?

21. The Wall of Proportion Destiny In Physics, intensity of light is inversely proportional to the distance from the source squared. At 10m from the light source, the intensity is 32cd (Candelas). What is the distance from the source when the intensity is 5cd? The problem Relationship Find k Use equation In the land of Canadia, house prices are inversely proportional to its distance from the centre of a major city, Torintino. A house 2km from the centre is valued at $587k. How much is a house worth 7km away? p  p = I  I = 587000 = k  2 k = 1,174,000 p = 1174000 / 7 = $168k 32 = k  10 2 So k = 3200 5 = 3200  d 2 d = √640 = 25.3m ? ? ?? ? 1d1d kdkd 1d21d2 kd2kd2