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2. What do we mean when we say two quantities are in proportion ? It means that if: one of them doubles, the other one also doubles. one of them trebles, the other one also trebles. one of them x4, the other one also x4. one of them halves, the other one also halves. one of them ÷4, the other one also ÷4. Can you give examples of directly proportional quantities from every day life?

3. © T Madas Directly proportional quantities: They increase or decrease at the same rate More formally: Two variables are directly proportional if the ratio between them remains constant.

4. What does the graph of two directly proportional quantities looks like? Cost of packets of pens 3 pens cost £2

5. © T Madas Cost of packets of pens 12108642Cost (£) 181512963Number of pens 3 pens cost £2 Let us plot the information of this table in a graph

6. © T Madas Cost of packets of pens 12108642Cost (£) 181512963Number of pens 3 pens cost £2 4 8 12 16 20 24 12 10 8 6 4 2 0 pens £

7. © T Madas 4 8 12 16 20 24 12 10 8 6 4 2 0 pens £ when graphed the points of Directly Proportional Quantities: 1.always form a straight line through the origin 2.always form the corners of similar rectangles whose opposite corner is at the origin. 3.the line is a diagonal of every rectangle

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9. 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 The data above has been obtained from a chemistry experiment and concerns two quantities, u and v. Are u and v directly proportional quantities?

10. © T Madas 0 5 10 15 20 25 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 20 15 10 5 v the quantities u and v are directly proportional u

11. © T Madas v u u v 0 5 10 15 20 25 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 20 15 10 5 17 25 What is the gradient of the line? gradient= diff in y diff in x = 25 17 ≈ 1.47 the ratio between directly proportional quantities remains constant. Work the ratio v : u from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round?

12. © T Madas u v 0 5 10 15 20 25 12 8.16 10 6.8 8 5.44 5 3.4 v u 25211916 1714.2812.9210.88 20 15 10 5 25 What is the gradient of the line? gradient= diff in y diff in x = 25 17 ≈ 1.47 the ratio between directly proportional quantities remains constant. Work the ratio v : u from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round? 17 25 = 0.68 u : v

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14. What do we mean when we say two quantities are inversely proportional ? It means that if: one of them doubles, the other one halves. one of them x3, the other one ÷3. one of them x4, the other one ÷4. one of them ÷2, the other one x2. one of them ÷10, the other one x10. Can you give an example of inversely proportional quantities from every day life?

15. © T Madas The Civic Centre is to be painted, so they call a firm of decorators. If this firm provide: 1 decorator 2 decorators 3 decorators 4 decorators 5 decorators 6 decorators 10 decorators 12 decorators 15 decorators 20 decorators 30 decorators 60 decorators 120 decorators will take 60 days for the job will take 30 days for the job will take 20 days for the job will take 15 days for the job will take 12 days for the job will take 10 days for the job will take 6 days for the job will take 5 days for the job will take 4 days for the job will take 3 days for the job will take 2 days for the job will take 1 day for the job will take ½ day for the job 1 x 60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10 10 x 6 12 x 5 15 x 4 20 x 3 30 x 2 60 x 1 120 x ½

16. © T Madas INVERSELY PROPORTIONAL QUANTITIES One increases at the same rate as the other one decreases. More formally: Two quantities are inversely proportional if their product remains constant.

17. What does the graph of two inversely proportional quantities looks like? 1 decorator takes 24 days to finish a job

18. © T Madas 1 decorator takes 24 days to finish a job 1234681224Days 2412864321No of decorators 0 5 10 15 20 25 20 15 10 5 decorators days

19. © T Madas 0 5 10 15 20 25 20 15 10 5 decorators days The graphed points of Inversely Proportional Quantities: 1.always lie on a curve like the one shown below. 2.always form the corners of rectangles of constant area whose opposite corner is at the origin. Hyperbola

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21. The data above has been obtained from the physics department and concerns two quantities, P and A. Are P and A inversely proportional quantities? 3.754.567.51011.251518 A 242015129865 P

22. © T Madas 0 5 10 15 20 25 3.754.567.5911.251518 A 2420151210865 P 20 15 10 5 P A Hyperbola

23. © T Madas 1234567 4.5 4 3.5 3 2.5 2 1.5 1 0.5 When plotted, Inversely Proportional quantities, always show as Hyperbolas.

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