Direct Proportion

Direct Proportion Direct Proportion Inverse Proportion Direct Proportion (Variation) Graph Inverse Proportion (Variation) Graph www.mathsrevision.com Direct Variation Inverse Variation Joint Variation Created by Mr. Lafferty Maths Dept.

9 Starter Questions 8 6 www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Learning Intention Success Criteria 1. Understand the idea of Direct Proportion. • To explain the term Direct • Proportion. 2. Solve simple Direct Proportional problems. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Write down two quantities that are in direct proportion. Direct Proportion Direct Proportion Two quantities, (for example, number of cakes and total cost) are said to be in DIRECT Proportion, if : Are we expecting more or less “ .. When you double the number of cakes you double the cost.” Easier method CakesPence 6 420 5 Example : The cost of 6 cakes is £4.20. find the cost of 5 cakes. www.mathsrevision.com Cakes Cost 6  4.20 (less) 1  4.20 ÷ 6 = 0.70 5  0.70 x 5 = £3.50 Created by Mr. Lafferty Maths Dept.

Same ratio means in proportion Direct Proportion Direct Proportion Example : Which of these pairs are in proportion. (a) 3 driving lessons for £60 : 5 for £90 (b) 5 cakes for £3 : 1 cake for 60p www.mathsrevision.com (c) 7 golf balls for £4.20 : 10 for £6 Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Which graph is a direct proportion graph ? y y y www.mathsrevision.com x x x Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Now try Ex 1.1 Ch7 (page 125) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Inverse Proportion Inverse Proportion Learning Intention Success Criteria 1. Understand the idea of Inverse Proportion. • 1. To explain the term Inverse Proportion. 2. Solve simple inverse Proportion problems. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Inverse Proportion Inverse Proportion Inverse Proportion is when one quantity increases and the other decreases. The two quantities are said to be INVERSELY Proportional or (INDIRECTLY Proportional) to each other. Example : Fill in the following table given x and y are inversely proportional. www.mathsrevision.com Notice xxy = 80 Hence inverse proportion 40 20 10 Created by Mr. Lafferty Maths Dept.

y x Inverse Proportion Inverse Proportion Inverse Proportion is the when one quantity increases and the other decreases. The two quantities are said to be INVERSELY Proportional or (INDIRECTLY Proportional) to each other. Are we expecting more or less Easier method Workers Hours 3 8 4 Example : If it takes 3 men 8 hours to build a wall. How long will it take 4 men. (Less time !!) www.mathsrevision.com Men Hours 3  8 (less) 1  3 x 8 = 24 hours 4  24 ÷ 4 = 6 hours Created by Mr. Lafferty Maths Dept.

y x Inverse Proportion Inverse Proportion Example : It takes 10 men 12 months to build a house. How long should it take 8 men. Are we expecting more or less Men Months Easier method Workers months 10 12 8 10  12 www.mathsrevision.com 1  12 x 10 = 120 8  120 ÷ 8 = 15 months (more) Created by Mr. Lafferty Maths Dept.

y x Inverse Proportion Inverse Proportion Example : At 9 m/s a journey takes 32 minutes. How long should it take at 12 m/s. Are we expecting more or less Speed Time Easier method Speed minutes 9 32 12 9  32 mins www.mathsrevision.com 1  32 x 9 = 288 mins 12  288 ÷ 12 = 24 mins (less) Created by Mr. Lafferty Maths Dept.

Inverse Proportion Inverse Proportion Exercise 2.1 Ch7 (page 127) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Graphs Learning Intention Success Criteria 1. Understand that Direct Proportion Graph is a straight line. • 1. To explain how Direct Direct Proportion Graph is always a straight line. 2. Construct Direct Proportion Graphs. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Notice C ÷ P = 20 Hence direct proportion Direct Proportion Direct Proportion Graphs The table below shows the cost of packets of “Biscuits”. www.mathsrevision.com We can construct a graph to represent this data. What type of graph do we expect ? Created by Mr. Lafferty Maths Dept.

Notice that the points lie on a straight line passing through the origin So direct proportion Direct Proportion Graphs C α P C = k P k = 40 ÷ 2 = 20 C = 20 P Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Graphs KeyPoint Two quantities which are in Direct Proportion always lie on a straight line passing through the origin. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Graphs Ex: Plot the points in the table below. Show that they are in Direct Proportion. Find the formula connecting D and W ? www.mathsrevision.com We plot the points (1,3) , (2,6) , (3,9) , (4,12) Created by Mr. Lafferty Maths Dept.

D W Direct Proportion Direct Proportion Graphs 12 Plotting the points (1,3) , (2,6) , (3,9) , (4,12) 11 10 9 8 7 Since we have a straight line passing through the origin D and W are in Direct Proportion. 6 www.mathsrevision.com 5 4 3 2 1 Created by Mr. Lafferty Maths Dept. 0 1 2 3 4

D W Direct Proportion Direct Proportion Graphs 12 Finding the formula connecting D and W we have. 11 10 9 D α W 8 7 D = 6 W = 2 D = kW 6 www.mathsrevision.com 5 Constant k = 6 ÷ 2 = 3 4 3 Formula is : D= 3W 2 1 Created by Mr. Lafferty Maths Dept. 0 1 2 3 4

Direct Proportion Direct Proportion Graphs 1. Fill in table and construct graph 2. Find the constant of proportion (the k value) www.mathsrevision.com • Write down formula Created by Mr. Lafferty Maths Dept.

Does the distance D vary directly as speed S ? Explain your answer Direct Proportion Direct Proportion Graphs Q The distance it takes a car to brake depends on how fast it is going. The table shows the braking distance for various speeds. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Does D vary directly as speed S2 ? Explain your answer Direct Proportion Direct Proportion Graphs The table shows S2 and D Fill in the missing S2 values. D 900 100 1600 400 S2 www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Proportion Direct Proportion Graphs Find a formula connecting D and S2. D D α S2 S2 D = kS2 D = 5 S2 = 100 www.mathsrevision.com Constant k = 5 ÷ 100 = 0.05 Formula is : D= 0.05S2 Created by Mr. Lafferty Maths Dept.

Starter Questions 9 8 6 www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Inverse Proportion Inverse Proportion Graphs Learning Intention Success Criteria 1. Understand the shape of a Inverse Proportion Graph . • 1. To explain how the shape and construction of a Inverse Proportion Graph. 2. Construct Inverse Proportion Graph and find its formula. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Notice W x P = £1800 Hence inverse proportion Inverse Proportion Inverse Proportion Graphs The table below shows how the total prize money of £1800 is to be shared depending on how many winners. www.mathsrevision.com We can construct a graph to represent this data. What type of graph do we expect ? Created by Mr. Lafferty Maths Dept.

Inverse Proportion Notice that the points lie on a decreasing curve so inverse proportion Direct Proportion Graphs

Inverse Proportion Inverse Proportion Graphs KeyPoint Two quantities which are in Inverse Proportion always lie on a decrease curve www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Inverse Proportion Inverse Proportion Graphs Ex: Plot the points in the table below. Show that they are in Inverse Proportion. Find the formula connecting V and N ? www.mathsrevision.com We plot the points (1,1200) , (2,600) etc... Created by Mr. Lafferty Maths Dept.

Note that if we plotted V against then we would get a straight line. because v directly proportional to V N Inverse Proportion Inverse Proportion Graphs V V 1200 Plotting the points (1,1200) , (2,600) , (3,400) (4,300) , (5, 240) 1000 N 800 Since the points lie on a decreasing curve V and N are in Inverse Proportion. 600 www.mathsrevision.com 400 These graphs tell us the same thing 200 0 1 2 3 4 5

V Inverse Proportion Inverse Proportion Graphs 1200 Finding the formula connecting V and Nwe have. 1000 800 600 V = 1200 N = 1 www.mathsrevision.com 400 k = VN = 1200 x 1 = 1200 200 0 1 2 3 4 5 N

Direct Proportion Direct Proportion Graphs 1. Fill in table and construct graph 2. Find the constant of proportion (the k value) www.mathsrevision.com • Write down formula Created by Mr. Lafferty Maths Dept.

Inverse Proportion Inverse Proportion Now try Ex 4.1 Ch7 (page 129) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Variation Learning Intention Success Criteria 1. Understand the process for calculating direct variation formula. • 1. To explain how to work out direct variation formula. 2. Calculate the constant k from information given and write down formula. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Variation Given that y is directly proportional to x, and when y = 20, x = 4. Find a formula connecting y and x. y Since y is directly proportional to x the formula is of the form x www.mathsrevision.com k is a constant y = kx 20 = k(4) k = 20 ÷ 4 = 5 y = 20 x =4 y = 5x

Direct Variation The number of dollars (d) varies directly as the number of £’s (P). You get 3 dollars for £2. Find a formula connecting d and P. d Since d is directly proportional to P the formula is of the form P www.mathsrevision.com k is a constant d = kP 3 = k(2) d = 3 P = 2 k = 3 ÷ 2 = 1.5 d = 1.5P

Direct Variation • How much will I get for £20 d d = 1.5P P d = 1.5 x 20 = 30 dollars www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

y x2 Direct Variation Harder Direct Variation Given that y is directly proportional to the square of x, and when y = 40, x = 2. Find a formula connecting y and x . Since y is directly proportional to x squared the formula is of the form www.mathsrevision.com y = kx2 40 = k(2)2 y = 40 x = 2 k = 40 ÷ 4 = 10 y = 10x2

Direct Variation Harder Direct Variation • Calculate y when x = 5 y = 10x2 y x2 y = 10(5)2 = 10 x 25 = 250 www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

C √P Direct Variation Harder Direct Variation • The cost (C) of producing a football magazine • varies as the square root of the number of • pages (P). Given 36 pages cost 48p to produce. • Find a formula connecting C and P. Since C is directly proportional to “square root of” P the formula is of the form www.mathsrevision.com C = 48 P = 36 k = 48 ÷ 6 = 8

Direct Variation Harder Direct Variation • How much will 100 pages cost. C √P www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Variation Harder Direct Variation Ex 5.1 & 5.2 Ch7 (page 135) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Inverse Variation Learning Intention Success Criteria 1. Understand the process for calculating inverse variation formula. • 1. To explain how to work out inverse variationformula. 2. Calculate the constant k from information given and write down formula. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

Direct Proportion