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Year 9: Ratio & Proportion

Year 9: Ratio & Proportion. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 6 th September 2013. Starter Puzzles. Advice: If two ratios and are the same, then .

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Year 9: Ratio & Proportion

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  1. Year 9: Ratio & Proportion Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 6th September 2013

  2. Starter Puzzles Advice: If two ratios and are the same, then The ratio of males to females at a party is 3:5. There are twelve more females than males. How many people are at the party? Answer 3 There are 7 Aardvarks and 12 Buffalo in a classroom. The ratio of Aardvarks to Buffalo is . What is ? 1 ? ? There are two regular polygons, and . The ratio of their exterior angles is . The ratio of their interior angles is . Prove that polygon has 30 sides. (Hint: it may be helpful to start with the exterior angle of being and working from there) Num sides = 4 My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Frostitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. What’s the new ratio of black to yellow fish? Answer: 2 ? ?

  3. Direct Proportion Dr Frost records the number of hours of revision his class did for their Maths landmark, and then sees how well they do:

  4. Direct Proportion Key definition to write down: 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’. You can think of this as and having some fixed ratio (although doesn’t actually state what this ratio is).

  5. Yay or 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. 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.  Yay   Nay #2: If the runner’s speed increases by 5, the distance covered increases by 5.  Yay  Nay

  6. How to solve proportion problems 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? Now with 25% horsemeat! Step 1: Write out relationship between variables w  l So w = kl ? Step 2: Work out constant of proportionality (k) 66 = k × 10 So k = 66  10 = 6.6 ? 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 ?

  7. Example Given that is directly proportional to , fill in the missing values in this table. ? ?

  8. Example Given that is directly proportional to the square of , fill in the missing values in this table. ? ?

  9. Example Given that , fill in the missing values in this table. ? ?

  10. Exercises 1 4 ? ? ? ? 2 5 ? ? ? ? 3 6 ? ? ? ?

  11. Exercises Rayner Higher GCSE Mathematics ?

  12. Proportion and 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. electricity usage Temperature (F) rabbit population hours on computer Temperature (C) years Yay  Yay   Yay Nay    Nay Nay 

  13. Checking your understanding… Given that is proportional to the cube of , fill in the missing values in this table. ? ?

  14. Back to the Mo… Mo is running a 5000m race. Here are his times when he runs at different speeds: How are the speeds and times related?

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

  16. Example Given that is inversely proportional to , fill in the missing values in this table. ? ?

  17. This is a bear safe. It’s bare safe. Mystery Contrived Real-Life Example of Awesomeness Click to Reveal

  18. In his bear safe, he keeps a fish. To keep the fish alive and fresh, he needs to fill his safe with water. Suppose there is a fixed amount of water. Relationship between width () of safe and water level () is: ?

  19. If when the safe is 2m wide the water level is 1m, what is the width of the safe when the water level is 3.5m? Answer: 0.34m ?

  20. Checking your understanding… Given that L is inversely proportional to √x, fill in the missing values in this table. ? ?

  21. Exercises is inversely proportional to the square root of . is inversely proportional to . 1 4 ? ? ? ? is inversely proportional to the square of . is inversely proportional to the cube of . 2 5 ? ? ? ? is inversely proportional to one more than . is inversely proportional to . 3 6 ? ? ? ?

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

  23. The Wall of Proportion Destiny 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? 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 ? c  h c = kh f  w2 f = kw2 Relationship ? ? 379,000,000 = k x 235 So k = 1,612,766 3 = k x 102 So k = 3  100 = 0.03 Find k ? ? Cost of Shard = 1612766 x 310 = £500 million f = kw2 So f = 0.03 x 152 = 6.75 gallons/s Use equation

  24. The Wall of Proportion Destiny 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 200m2. When its surface area is 500m2, what is its length? 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 ? ? s  l2 s = kl2 v  w3 v = kw3 Relationship ? ? 200 = k x 502 k = 0.08 5600 = k x 103 So k = 5.6 Find k ? ? Weee!! 500 = 0.08 x l2 l2 = 6250 So l = 79.06m v = kw3 So 20100 = 5.6 x w3 w = 3√3589 = 15.31cm Use equation Super Fun Happy Slide

  25. 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? 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? The problem ? 1 d2 k d2 1 d k d I  I = p  p = Relationship ? ? 587000 = k  2 k = 1,174,000 32 = k  102 So k = 3200 Find k ? ? p = 1174000 / 7 = $168k 5 = 3200  d2 d = √640 = 25.3m Use equation

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