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Proportion.pptx
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- 2. Learning Objectives: • Towrite and use formulae to solve problems involving direct proportion • To write and use formulae to solve problems involving inverse proportion
- 3. Proportion Wednesday, January 11, 2023 Express relationshipsusing proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Starter Task Which exchange rate would be best for my trip to France? £1 = €1.28 £5 = €6.35 £25=€31.25 £20 = €26 How many Euros can I get for £400 at the best available price? MoneyCo. XchangeIt FXX TraveX
- 4. Proportion Wednesday, January 11, 2023 Express relationshipsusing proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Unitary Method Which exchange rate would be best for my trip to France? £1 = €1.28 £5 = €6.35 £25=€31.25 £20 = €26 How many Euros can I get for £400 at the best available price? MoneyCo. XchangeIt FXX TravEX No change £1 €1.28 €1.27 €1.25 €1.30 ÷ 5 ÷ 25 ÷ 20
- 5. Direct Proportion Wednesday, January 11,2023 Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal If two variables are directly proportional that means that if one doubles the other doubles If one is halved the other is halved The two variables are linked by a constant (k) Their ratio stays the same as they increase or decrease
- 6. Direct Proportion This relationshipcan be expressed in any of these ways: ◼ y is directly proportional to x. ◼ y varies directly with x. ◼ y varies as x. In symbols, direct proportion relationships can be written as y ∝ x. The ∝ sign can then be replaced by ‘= k’ to give the formula y = kx, where k is the constant of proportionality.
- 7. Key Words Directly/ inversely proportional square cube rootequation substitute variable constant Proportion Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by inverse proportion Find equations for variables inversely proportional to squares, cubes or roots Wednesday, January 11, 2023 I want to build a wall in my garden. The builders say it will take 216 man hours to build the wall. A “man hour” is one hour worked by one man. It will, therefore, take 1 builder 216 hours to finish the job, 2 builders would do it in half the time (108 hours) and so on…. How long will it take 3 builders? How many builders would it take to finish in 12 hours?
- 8. Time (hours) Numberof builders 1 216 2 108 3 4 6 8 9 12 18 24 Key Words Directly/ inversely proportional square cube root equation substitute variable constant Inverse Proportion Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by inverse proportion Find equations for variables inversely proportional to squares, cubes or roots Wednesday, January 11, 2023 Time (hours) Number of builders 1 216 2 108 3 72 4 54 6 36 8 27 9 24 12 18 18 12 24 9 Time (hours) Number of builders 𝑻 × 𝒏 1 216 2 108 3 72 4 54 6 36 8 27 9 24 12 18 18 12 24 9 Time (hours) Number of builders 𝑻 × 𝒏 1 216 1 × 216 2 108 2 × 108 3 72 3 × 72 4 54 4 × 54 6 36 6 × 36 8 27 8 × 27 9 24 9 × 24 12 18 12 × 18 18 12 18 × 12 24 9 24 × 9 Number of builders Time (hours) 𝑻 × 𝒏 1 216 216 2 108 216 3 72 216 4 54 216 6 36 216 8 27 216 9 24 216 12 18 216 18 12 216 24 9 216 Here is a table of values for some integers that will work: For the man hours to remain at 216 we get these values: The relationship can be seen by adding an extra column: The product of the time and the number of builders….. ……remains CONSTANT.
- 9. Key Words Directly/ inversely proportional square cube rootequation substitute variable constant Inverse Proportion Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by inverse proportion Find equations for variables inversely proportional to squares, cubes or roots Wednesday, January 11, 2023 In this problem, the number of builders and time taken for the job are INVERSELY PROPORTIONAL, meaning that as one variable increases the other variable decreases. i.e. More builders = less time, More time= less builders This can be expressed as: ‘t is inversely proportional to n’. In symbols, this is written as t ∝ 1 𝑛 . The ∝ sign can then be replaced by ‘= k’ to give the formula T = 𝑘 𝑛 where k is the constant of proportionality.
- 10. Inverse proportion • Thetemperature of a cup of coffee decreases as time increases. A graph of temperature (T) against time (t) shows an inverse relationship.
- 11. Direct Proportion Wednesday, January 11,2023 Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Example1: 𝑆 is directly proportional to 𝑇. When 𝑆 = 30 𝑇 = 5. Write an equation linking 𝑆 and 𝑇 𝑆 ∝ 𝑇 𝑆 = 𝑘𝑇 30 = 𝑘 × 5 30 ÷ 5 = 𝑘 6 = 𝑘 The final equation is 𝑆 = 6𝑇
- 12. Direct Proportion Wednesday, January 11,2023 Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Example2: 𝑆 is directly proportional to the square of 𝑇. When 𝑆 = 50 𝑇 = 5. Write an equation linking 𝑆 and 𝑇 𝑆 ∝ 𝑇2 𝑆 = 𝑘𝑇2 50 = 𝑘 × 52 50 ÷ 25 = 𝑘 2 = 𝑘 The final equation is 𝑆 = 2𝑇2
- 13. Direct Proportion Wednesday, January 11,2023 Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Example3: 𝐶 is directly proportional to the square root of 𝐷. When C = 18 𝐷 = 36. Write an equation linking 𝐶 and 𝐷 C ∝ 𝐷 C = 𝑘 𝐷 18 = 𝑘 × 36 18 ÷ 6 = 𝑘 3 = 𝑘 The final equation is C = 3 𝐷
- 15. Answers Wednesday, January 11, 2023 Express relationshipsusing proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal
- 16. Key Words Directly/ inversely proportional square cube rootequation substitute variable constant Inverse Proportion Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by inverse proportion Find equations for variables inversely proportional to squares, cubes or roots 𝑡 is inversely proportional to 𝑛. When 𝑛 = 2, 𝑡 = 108. (a) Find an equation linking 𝑡 and 𝑛. (b) When 𝑛 = 6 find the value of 𝑡 (c) When 𝑡 = 24 find the value of 𝑛 Example1: Wednesday, January 11, 2023 (a) 𝑡 ∝ 1 𝑛 𝑡 = 𝑘 𝑛 108 = 𝑘 2 108 × 2 = 𝑘 216 = 𝑘 𝑡 = 216 𝑛 Inversely proportional General equation substitution constant Final equation (b) When 𝑛 = 6 𝑡 = 216 𝑛 𝑡 = 216 6 𝑡 = 36 (c) When 𝑡 = 24 𝑡 = 216 𝑛 24 = 216 𝑛 24𝑛 = 216 𝑛 = 216 24 𝑛 = 9
- 17. Key Words Directly/ inversely proportional square cube rootequation substitute variable constant Inverse Proportion Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by inverse proportion Find equations for variables inversely proportional to squares, cubes or roots 𝑥 is inversely proportional to 𝑦2 . When 𝑥 = 5, 𝑦 = 4. (a) Find an equation linking 𝑥 and y. (b) When 𝑦 = 2 find the value of 𝑥 (c) When 𝑥 = 5 find the value of y Example2: Wednesday, January 11, 2023 (a) 𝑥 ∝ 1 𝑦2 𝑥 = 𝑘 𝑦2 5 = 𝑘 42 16 × 5 = 𝑘 80 = 𝑘 𝑥 = 80 𝑦2 Inversely proportional General equation substitution constant Final equation (b) When 𝑦 = 2 𝑥 = 80 𝑦2 𝑥 = 80 22 𝑥 = 20 (c) When 𝑥 = 5 𝑥 = 80 𝑦2 5 = 80 𝑦2 𝑦2 = 80 5 𝑦2 = 16 𝑦 = 16 𝑦 = 4
- 18. Your turn… 1 )y is inversely proportional to x. If y = 4 when x = 3, find a) the formula for y in terms of x b) y when x = 2 c) x when y = 3 d) Sketch the graph of y against x. 2) V varies inversely with the cube of w. If V = 12.5 when w = 2, find a) the formula for V in terms of w b) V when w = 1 c) w when V = 0.8
- 19. I dropped astone from the top of the Shard in London. The distance the stone falls is directly proportional to the square of the time it has been falling. After 5 seconds the stone has fallen 122.50 metres. How far will it have fallen after 8 seconds? Wednesday, January 11, 2023 Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Home Problem Solving for next time Before next lesson consider this problem: You should use direct proportion to find a solution. Look up the building online…..
- 20. Direct Proportion Wednesday, January 11,2023 Express relationships using proportional notation and the general equation for those variables Find a final equation for variables linked by direct proportion Find equations for variables directly proportional to squares, cubes or roots Key Words directly proportional square cube root equation substitute variable diagonal Review: 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it? (Assume everyone works at the same rate) Is this a Direct Proportion question? How do you know? How might we describe the link?