Created by Mr. Lafferty Maths Dept. Direct Proportion Direct Proportion Inverse Proportion Direct Proportion (Variation) Graph Inverse Proportion (Variation) Graph www.mathsrevision.com Direct Variation Inverse Variation Joint Variation 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions 9 8 6 www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion www.mathsrevision.com Direct Proportion 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 Cakes Pence 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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion Same ratio means in proportion Example : Which of these pairs are in proportion. (a) 3 driving lessons for £60 : 5 for £90 www.mathsrevision.com (b) 5 cakes for £3 : 1 cake for 60p (c) 7 golf balls for £4.20 : 10 for £6 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Direct Proportion Direct Proportion Which graph is a direct proportion graph ? x y x y x y www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Direct Proportion Direct Proportion Now try Ex 1.1 Ch7 (page 125) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Proportion www.mathsrevision.com Notice xxy = 80 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 X 1 2 4 8 y 80 40 20 10 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Proportion www.mathsrevision.com 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 y x Men Hours 3  8 (less) 1  3 x 8 = 24 hours 4  24 ÷ 4 = 6 hours 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Proportion www.mathsrevision.com Inverse Proportion Are we expecting more or less Example : It takes 10 men 12 months to build a house. How long should it take 8 men. Men Months Easier method Workers months 10  12 8 10  12 www.mathsrevision.com 1  12 x 10 = 120 y x 8  120 ÷ 8 = 15 months (more) 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Proportion www.mathsrevision.com Inverse Proportion Are we expecting more or less Example : At 9 m/s a journey takes 32 minutes. How long should it take at 12 m/s. Speed Time Easier method Speed minutes 9  32 12 9  32 mins www.mathsrevision.com 1  32 x 9 = 288 mins y x 12  288 ÷ 12 = 24 mins (less) 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Inverse Proportion Inverse Proportion Exercise 2.1 Ch7 (page 127) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion Notice C ÷ P = 20 Hence 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 ? 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion KeyPoint Two quantities which are in Direct Proportion Graphs KeyPoint Two quantities which are in Direct Proportion always lie on a straight line passing through the origin. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 ? W 1 2 3 4 D 6 9 12 www.mathsrevision.com We plot the points (1,3) , (2,6) , (3,9) , (4,12) 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion D www.mathsrevision.com D and W are in W Direct Proportion Graphs D 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 19-Apr-17 Created by Mr. Lafferty Maths Dept. W 1 2 3 4

Direct Proportion D D α W D = kW Constant k = 6 ÷ 2 = 3 Direct Proportion Graphs D 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 19-Apr-17 Created by Mr. Lafferty Maths Dept. W 1 2 3 4

Created by Mr. Lafferty Maths Dept. 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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Direct Proportion Direct Proportion Now try Ex 3.1 Ch7 (page 129) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion S 10 20 30 40 D 5 45 80 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 S 10 20 30 40 D 5 45 80 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Does D vary directly as speed S2 ? Explain your answer D S2 Direct Proportion Direct Proportion Graphs The table shows S2 and D Fill in the missing S2 values. S2 S 10 20 30 40 D 5 45 80 100 400 900 1600 www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Proportion Find a formula connecting D and S2. D α S2 D = kS2 Direct Proportion Graphs Find a formula connecting D and S2. D α S2 D = kS2 D = 5 S2 = 100 www.mathsrevision.com Constant k = 5 ÷ 100 = 0.05 Formula is : D= 0.05S2 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Direct Proportion Direct Proportion Now try Ex 3.2 Ch7 (page 131) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions 9 8 6 www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Proportion Notice W x P = £1800 Hence 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. Winners W 1 2 3 4 5 Prize P £1800 £900 £600 £450 £360 www.mathsrevision.com We can construct a graph to represent this data. What type of graph do we expect ? 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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

Inverse Proportion KeyPoint Two quantities which are in Inverse Proportion Graphs KeyPoint Two quantities which are in Inverse Proportion always lie on a decrease curve www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 ? N 1 2 3 4 5 V 1200 600 400 300 240 www.mathsrevision.com We plot the points (1,1200) , (2,600) etc... 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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

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

Created by Mr. Lafferty Maths Dept. 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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Inverse Proportion Inverse Proportion Now try Ex 4.1 Ch7 (page 129) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 19-Apr-17 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. Since y is directly proportional to x the formula is of the form y x www.mathsrevision.com k is a constant y = kx 20 = k(4) y = 20 x =4 k = 20 ÷ 4 = 5 y = 5x 19-Apr-17

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. Since d is directly proportional to P the formula is of the form www.mathsrevision.com d P k is a constant d = kP 3 = k(2) d = 3 P = 2 k = 3 ÷ 2 = 1.5 d = 1.5P

Created by Mr. Lafferty Maths Dept. Direct Variation How much will I get for £20 d P d = 1.5P d = 1.5 x 20 = 30 dollars www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 x2 y = kx2 40 = k(2)2 y = 40 x = 2 k = 40 ÷ 4 = 10 y = 10x2 19-Apr-17

Created by Mr. Lafferty Maths Dept. 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 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Direct Variation www.mathsrevision.com 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 √P C = 48 P = 36 k = 48 ÷ 6 = 8 19-Apr-17

Created by Mr. Lafferty Maths Dept. Direct Variation Harder Direct Variation How much will 100 pages cost. C √P www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Direct Variation Harder Direct Variation Ex 5.1 & 5.2 Ch7 (page 135) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

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 variation formula. 2. Calculate the constant k from information given and write down formula. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Variation Given that y is inverse proportional to x, and when y = 40, x = 4. Find a formula connecting y and x. Since y is inverse proportional to x the formula is of the form y x 1 www.mathsrevision.com k is a constant y = 40 x =4 k = 40 x 4 = 160 19-Apr-17

Inverse Variation Speed (S) varies inversely as the Time (T) When the speed is 6 kmph the Time is 2 hours Find a formula connecting S and T. Since S is inversely proportional to T the formula is of the form S T 1 www.mathsrevision.com k is a constant S = 6 T = 2 k = 6 x 2 = 12

Created by Mr. Lafferty Maths Dept. Inverse Variation Find the time when the speed is 24mph. S T 1 S = 24 T = ? www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

y x2 Inverse Variation Harder Inverse variation Given that y is inversely proportional to the square of x, and when y = 100, x = 2. Find a formula connecting y and x . Since y is inversely proportional to x squared the formula is of the form y x2 1 www.mathsrevision.com k is a constant y = 100 x = 2 k = 100 x 22 = 400 19-Apr-17

Created by Mr. Lafferty Maths Dept. Inverse Variation Harder Inverse variation Calculate y when x = 5 y x2 1 www.mathsrevision.com y = ? x = 5 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Inverse Variation www.mathsrevision.com y r3 Inverse Variation Harder Inverse variation The number (n) of ball bearings that can be made from a fixed amount of molten metal varies inversely as the cube of the radius (r). When r = 2mm ; n = 168 Find a formula connecting n and r. Since n is inversely proportional to the cube of r the formula is of the form n r3 1 www.mathsrevision.com k is a constant n = 100 r = 2 k = 168 x 23 = 1344 19-Apr-17

Created by Mr. Lafferty Maths Dept. Inverse Variation Harder Inverse variation How many ball bearings radius 4mm can be made from the this amount of metal. n r3 1 r = 4 www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Inverse Variation Ex 6.1 & 6.2 Ch7 (page 137) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Joint Variation Learning Intention Success Criteria 1. Understand the process for calculating joint variation formula. 1. To explain how to work out Joint Variation formula. 2. Calculate the constant k from information given and write down formula. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Joint Variation Given that y is inverse proportional to x, directly to z and y = 10 when x = 2, z = 4. Find a formula connecting y, x and z. Since y is inversely proportional to x and directly to z the formula is of the form www.mathsrevision.com k is a constant y = 10 x = 2 z = 4 k = 10 x 2 ÷ 4 = 5 19-Apr-17

Inverse Variation T varies directly as N and inversely as S Find a formula connecting T, N and S given T = 144 when N = 24 S = 50 Since T is directly proportional to N and inversely to S the formula is of the form www.mathsrevision.com k is a constant T = 144 N = 24 S = 50 k = 144 x 50 ÷ 24= 300

Created by Mr. Lafferty Maths Dept. Joint Variation Find T when N = 30 and S = 40. www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Joint Variation www.mathsrevision.com The height of a cone varies directly as its volume (V cm3) and inversely as the square of the radius of the base (r cm). When r = 3cm and V = 47.1 cm , h = 5 cm. Find a formula connecting h , V and r. Since h is directly proportional to V and inversely proportional to r squared, the formula is of the form www.mathsrevision.com V = 100 r = 2 h = 5 k = 5 x 9 ÷ 47.1 = 0.96 19-Apr-17

Created by Mr. Lafferty Maths Dept. Joint Variation Calculate h, when r = 4cm and V = 75cm3 h = ? r = 4 V = 75 www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Joint Variation y varies directly as √x and inversely as z3. When y = 40, x = 25 and z = 3. Find a formula connecting y, x and z. Since y is directly as √x and inversely as cube of z the formula is of the form www.mathsrevision.com y = 40 x = 25 z = 3 k = 40 x 33 ÷ √25 = 216 19-Apr-17

Created by Mr. Lafferty Maths Dept. Joint Variation Harder Inverse variation Calculate x, when y = 81 and z = 2 x = ? y = 81 z = 2 www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.

Created by Mr. Lafferty Maths Dept. Joint Variation Ex 7.1 & 7.2 Ch7 (page 140) www.mathsrevision.com 19-Apr-17 Created by Mr. Lafferty Maths Dept.