Proportion

1. X Proportion Experiment and theory In science, medicine, economics, business and many kinds of research it is often necessary to find whether or not two quantities are related. A theory is put forward, and then tested by experiment. Mathematical models in the form of graphs and equations are used to test the relationship between quantities. Here are three famous examples from science.

2. X In the 19th Century George Ohm investigated the relationship between current I and voltage V in an electrical circuit. I V

3. X In the 17th Century Isaac Newton studied the connection between Force F on a body and the acceleration a it produced. F a

4. X Another 17th Century scientist Robert Hooke studied the connection between the tension T in a spring and its extension E. E T

5. y y = mx x X The work of all three scientists can be summed up in the graph below. x and y behave in the same way. If we double x then y will double also. Where k is a constant.

6. 1 2 3 4 No. Packets of crisps (N) 30 60 90 120 Cost (Cp) Direct Proportion (Or variation) There are many situations where one quantity is directly proportional to another.

7. C 120 100 80 60 40 20 Cost in pence 1 2 3 4 N Number of packets The points lie on a straight line through the origin. Hence, the cost (C) is proportional to the number of packets (N)

8. k is the gradient of the line. From the graph we see that the gradient is 30. Direct proportion, or direct variation can be checked by: (i) Drawing a graph (straight line through the origin) (ii) Calculating the gradient ratios. (they should be the same) Page 87 Exercise 1

9. Making and Using proportion Models The Silver Star leaves for the fishing grounds at 0200. By 0250 she has sailed 15km. Her skipper knows that the distance from port (D km) is proportional to the sailing time (T minutes). • Find a formula connecting T and D • Use the formula to estimate how long the boat will take to travel 84km. (in 50 minutes the boat travel 15km)

10. Page 89 Exercise 2A

11. A new car is being tested. The distance (d metres) it travels is proportional to the square of the time (t seconds).. In 3 seconds it travels 18metres. • Find a formula connecting d and t. • Use the formula to estimate how far it will travel in 5 seconds. (in 3 seconds it travels 18 m)

12. Page 90 Exercise 2B

13. Inverse Proportion Six men build a wall in 10 days. How long will it take to build the same wall with 12 men? 5 days. In this example we doubled the number of men but did not double the number of days. This is not an example of Direct proportion but an example of Inverse proportion. The graph of quantities related by inverse proportion looks like

14. If V is Proportional to N then, If V is Inversely Proportional to N then,

15. Two electrical charges attract one another with a force (F units) which varies inversely as the square of the distance (d units) between them. F = 4 when d = 6. Find F when d = 10. We know that F = 4 when d = 6. When d = 10,

16. Page 93 Exercise 3

17. Joint Variation In an typing experiment the time (T minutes) taken to type a certain number of pages (N pages) was recorded so that the speed (S words per minute) of the typist could be determined. It was found that T varied directly as N and inversely as S. Vivian types a 24 page document in 2 hours 24 minutes at a speed of 50 words per minute. How long would she take for 30 pages at 40 words per minute?

18. Vivian types a 24 page document in 2 hours 24 minutes at a speed of 50 words per minute. How long would she take for 30 pages at 40 words per minute?

19. Page 94 Exercise 4A Page 95 Exercise 4B