Subject = WAI-COME Maths Std= 8th Name:Abhishek Vilas Arkas Teacher's name: takale.m.y.

UNIT NO 9. Variation and Proportion

9. Variation and Proportion Revision A table showing the distance travelled by a car and the time taken for it is given below. Answer the following questions. Time (hours) 2 3 4 5 Distance (km) 120 180 240 300

Time (hours) 2 3 4 5 Distance (km) 120 180 240 300 1) What is the change you see in the distance covered by a car as the time of travel increases?

Time (hours) 2 3 4 5 Distance (km) 120 180 240 300 1) As the time of travel of the car increses the distance covered by it also increses.

Time (hours) 2 3 4 5 Distance (km) 120 180 240 300 2) What do you see on calculating the ratio ‘distance to time’ in every pair ?

Time (hours) 2 3 4 5 Distance (km) 120 180 240 300 2) 120/2=60, 180/3=60, 240/4=60, 300/5=60. The ratio ‘distance:time’ in every pair is the same, that is, it is a constant.

Time (hours) 2 3 4 5 Distance (km) 120 180 240 300 3) What kind of variation is there between the two quantities time and distance travelled ?

Time (hours) 2 3 4 5 Distance (km) 120 180 240 300 3) In this example,there is direct variation between the quantities time and distance

EXERCISE Observe the groupof numberes 15,20,30,40 and answer the following questions What is the simplest form of the ratio 15:20 ? What is the simplest form of the ratio 30:40 ? What can you infer about the group of numbers 15,20,30,40,from the simplest form of the two ratios ?

ANSWER The simplest forms of the two ratios 15:20 and 30:40 are equal Hence , the numbers 15,20,30,40 are in proportion.

Direct variation and direct proportion If, of two related variables y and x , = k (k = a constant), then there is a direct variation between y and x. ‘There is a direct variation between y and x’ Is written in symboles as : ‘yα x. (It is also read as ‘y varies directely as x.’) (The symbol ‘α ’ is the letter ‘alpha’ from the Greek alphabeat.)

Direct variation and direct proportion if = k ( a constant ) then y α x or if y α x then = k (k a constant ) = k is the equation of direct variation. If =k (a constant) then y x or if y x then =k (a constant) The constant ‘k’ is called the constant of proportionality.

Example 1 there is direct variation between ‘y’ and ‘x’. When y=15 & x=10. 1. find the proportionality constant. Ans.- there is direct variation between y and x. hence, the ratio of corresponding values of y and x is constant. = k(a constant) It is given that when y=15 and x=10 = K = THE CONSTANT OF PROPORTIONLITY IS

INVERSE VARIATION, INVERSE PROPORTION 1) What change do you see in the amount of prize money each child wins as the number of children increases ? Number of children 6 9 12 15 18 Amount of Prize 90 60 45 36 30

2) What do you notice if you calaulate the product of the two numbers in each pair of “Number of children” and “ Amount of prize money” ? Number of children 6 9 12 15 18 Amount of Prize 90 60 45 36 30

3) What kind of variation is there between the number of children & the corresponding prize money per child? Number of children 6 9 12 15 18 Amount of Prize 90 60 45 36 30

1) Check if your answer are as follows. 1)As the number of children increases, the amount of prize money per child decreases. Number of children 6 9 12 15 18 Amount of Prize 90 60 45 36 30

Number of children 6 9 12 15 18 Amount of Prize 90 60 45 36 30 The product of the number of children and amount of prize money in each pair is a constant.

Number of children 6 9 12 15 18 Amount of Prize 90 60 45 36 30 3) There is inverse variation between the number of children and the amount of prize money.

Example 1. x .When x=9, y=6. Find the constant of proportionality. 1) x x ‘ y = k (a constant) when x= 9, y= 6. 9X6=k 54=k Constant of proportionality= 54

If x ‘ y= k (a constant) then y Or, if y then x ‘ y = k (a constant) In the above statements, ‘k’ is the constant of proportionality and ‘x ‘ y=k’ is the equation of inverse varition.

Expreesing inverse variation in symbols- Using y and x for the interdependent variables, we can express instantces of inverse proportion as show below. If y ‘ x =k (a constant) then y varies inversely as x. ‘ There is inverse variation between y and x is expressed using symobles as y ‘ Thus, If x ‘ y= k (a constant) then y Or, if y then x ‘ y = k (a constant)