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Year 9: Ratio & Proportion Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 7 th September 2014

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Starter Puzzles ? ? ? 1 2 (You can work in pairs) 3 ?

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Direct Proportion Iain wants to make a Baked Alaska. He requires 24 eggs to make 700g of the pudding. How many eggs does he require to make 1250g? We say that the number of eggs and the mass of the pudding are ‘directly proportional’. There is some underlying ‘scaling’ between the two things. Method 1: Unitary Method Method 2: Ratio Method ?

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Test Your Understanding Solve the following using both the ‘unitary method’ (i.e. find quantity for one unit) and the ‘ratio method’. The mass of 16cm 3 of Neoginium is 24g. What is the mass of 20cm 3 of the same element? Unitary Method Ratio Method If you finish: Q Q ?? ?

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Direct Proportion Two things are directly proportional if they in the same ratio. Can you suggest variables that might be directly proportional? Speed and distance travelled (if you double your speed, you double the distance travelled). Total cost and quantity purchased. Length of steel rod and weight. Why is “hours revised for maths exam” and “maths exam mark” not likely to be directly proportional? If we 5 hours revision resulted in 60%, then clearly 10 hours revision wouldn’t result in 120%! So while the two things are ‘correlated’, they are not directly proportional. Electricity cost is directly proportional to the hours of TV watched. If 2 hours are watched, a cost of 14p is incurred. If 7 hours is watched, what cost is incurred? ? ? Q

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Exercise 1 2 1 3 4 5 6 7 ? ? ? ? ? ? ? ? ? ?

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Inverse Proportion Suppose Mo runs at a speed of 8m/s for second, and takes 120 seconds to finish. If he ran double the speed at 16m/s, what happens to his time? It halves! And what do you notice about the speed multiplied by the time in the two cases? It remains the same. When two quantities are inversely (or ‘indirectly’) proportional, their product remains constant. ? ?

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Example Four bricklayers can build a certain wall in ten days. How long would it take five bricklayers to build it? Q Method 1: Constant ProductMethod 2: Unitary Method Find how many days one bricklayer would take! 4 bricklayers take 10 days. 1 bricklayer takes 40 days 5 bricklayers take 8 days ??

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Test Your Understanding Eleven taps fill a tank in three hours. How long would it take to fill the tank if only six taps are working? Q If you finish that quickly… If it takes 3 men 4 hours to dig 6 holes, how many hours does it take for 6 men to build 8 holes? Q 3 men 4 hours for 6 holes 6 men 2 hours for 6 holes 6 men 1/3 hours for 1 holes 6 men 8/3 hours for 8 holes (which is 2 hours 40 minutes) Notice that “men” and “hours” are inversely proportional but “hours” and “holes” are directly proportional. ? ?

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Exercise 2 1 2 3 4 6 7 8 5 ? ? ? ? ? ? ? ?

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