GCSE Revision 101 Maths Proportional Variation Daniel

GCSE Revision 101 Maths Proportional Variation © Daniel Holloway

The Basics If two quantities are in direct proportion, as one value increases, the other will increase by the same percentage With inverse proportion, one value increases as the other decreases

Direct Proportion An example of direct proportion is buying produce. If you were to buy one sandwich at 45 p, you would have to pay 90 p for two. We can show this direct proportionality with an equation: cost of sandwiches = selling price per sandwich x number purchases

Direct Proportion This can also be written as y = kx where k is the price per apple. Whenever k is used in proportion – this means the constant value When y is directly proportional to x we can write y ∝ x

Direct Proportion Worked Example If y is directly proportional to x when x = 12, y = 3 Find the value of x when y = 8 We know that y is proportional to x: y = kx Substitute the values into the formula 3 = k x 12 k = 3/12 = ¼ To find the value of x when y = 8: 8 = ¼x x = 8 x 4 = 32

Powers y can be directly proportional to the powers of x (i. e. x 2, x 3), which also forms an equation: y = kx 2 y ∝ x 3 y is directly proportional to the cube of x if y = 1 when x = 2 what is the value of y when x = 4?

Inverse Proportion An example of inversely proportional quantities is the changing lengths and widths of rectangles with the same area. As the length of the rectangle doubles, the width has to be halved in order for the area to remain the same. 3 cm 6 cm 2 1 cm 6 cm 2

Inverse Proportion Worked Example y is inversely proportional to x. When y = 3, x = 12 a) Express k in terms of x and y b) Find the value of x when y = 8 a Because y is inversely proportional to x, we say: y ∝ 1 /x which we change to y = k/x Therefore yx = k b Substitute the values of y = 3 and x = 12 in the expression: 3 x 12 = 36 so k = 36 To find the value of x when y = 8, substitute y = 8 and k = 36 into yx = k 8 x = 36 x = 4. 5

Test Yourself 1) y is inversely proportional to the square of x and when y = 9, x = 2 a Find y in terms of x b Find (+)x when y = 144 2) The resistance, R to the motion of a sailing boat is directly proportional to the square of its speed, s If R = 300 when s = 12, find the value of s when R = 150

Test Yourself 3) The wavelength, w metres, is inversely proportional to the frequency, f k. Hz, of the waves a A radio wavelength of 1000 metres has a frequency of 300 k. Hz The frequency is doubled to 600 k. Hz What is the new wavelength? b Calculate the frequency when the wavelength is 842 metres c Radio 1 has a frequency value which is equal to its wavelength value Calculate the frequency and wavelength of Radio 1 in k. Hz and metres 4) y is inversely proportional to the square root of x When y = 6, x = 4 a What is the value of y when x = 9? b What is the value of x when y = 10?