How changing one quantity affects other connected quantities. Variation How changing one quantity affects other connected quantities.

The constant is called the constant of variation is a mathematical relationship between two or more variables that can be expressed by an equation in which one variable is equal to a constant times the other(s). The constant is called the constant of variation

“ .. When you double the number of tickets Direct Variation Two quantities, (for example, number of cinema tickets and total cost) are said to be in DIRECT Proportion, if : “ .. When you double the number of tickets you double the cost.” If tickets cost £6 Number of tickets 2 4 6 8 10 Total cost £12 £24 £36 £48 £60

Example : Which of these pairs vary directly? (a) 3 driving lessons for £60 : 5 for £90 (b) 5 cakes for £3 : 1 cake for 60p (c) 7 golf balls for £4.20 : 10 for £6 (d) 4 day break for £160 : 10 day for £300

Which graph shows direct variation ? x y x y x y

Hence direct variation Notice C ÷ P = 40 Hence direct variation The table below shows the cost of packets of “Biscuits”. No. of Packets (P) 1 2 3 4 5 6 Cost C pence 40 80 120 160 200 240 We can construct a graph to represent this data. What type of graph do we expect ?

A straight line through the origin with gradient 40, the cost per packet. O Cost, C (pence) Number of Packets (P) C P C = 40P The constant of variation is 40 A straight line through ORIGIN indicates DIRECT VARIATION

Remember Two quantities which vary directly, can always be represented by a straight line passing through the origin. The gradient of the line is the CONSTANT of VARIATION

W 1 2 3 4 D 6 9 12 Plot the points in the table below. Show that they vary directly. Find the formula connecting D and W ? W 1 2 3 4 D 6 9 12 We plot the points (1 , 3) , (2 , 6) , (3 , 9) , (4 , 12)

D Plotting the points (1,3) , (2,6) , (3,9) , (4,12) 11 10 Since we have a straight line passing through the origin D and W vary directly. 9 8 7 6 5 4 The gradient is 3 so the equation is D = 3W 3 2 1 The constant of variation is 3 W 1 2 3 4

Find the formula connecting I and V ? The table shows the results of an experiment to examine the relationship between voltage, V and current, I Find the formula connecting I and V ? I 0∙1 0∙2 0∙3 0∙4 V 2 4 6 8 We plot the points (0∙1 , 2) , (0∙2 , 4) , (0∙3 , 6) , (0∙4 , 8)

A straight line through the origin, so quantities vary directly. 0∙1 0∙2 0∙3 0∙4 0∙5 2 4 6 8 10 A straight line through the origin, so quantities vary directly. m = v h = 2 0∙1 = 20 Formula is V = 20I 0∙1 2 The constant of variation is 20

The symbol  is shorthand for ‘varies as’ We are now going to look at the algebra of Variation The symbol  is shorthand for ‘varies as’

y = kx Substituting gives 20 = k × 4 k = 5 y = 5x Given that y varies directly as x, and y = 20 when x = 4, find a formula connecting y and x. We write y  x We know that y = 20 when x = 4 y = kx Substituting gives The constant of variation 20 = k × 4 k = 5 So formula becomes y = 5x

d = kP Substituting gives 8 = k × 5 k = 1∙6 d = 1∙6 P The number of dollars (d) varies directly as the number of £’s (P). You get 8 dollars for £5. Find a formula connecting d and P. We write d  P We know that d = 8 when P = 5 d = kP Substituting gives The constant of variation 8 = k × 5 k = 1∙6 So formula becomes d = 1∙6 P

If W varies directly with F and when W = 24 , F = 6 If W varies directly with F and when W = 24 , F = 6 . Find the value W when F = 10. W  F W = k F When F = 10 W = ? When W = 24, F = 6 W = 4 x 10 24 = k x 6 W = 40 k = 4 W = 4 F

Find a formula connecting y and x . Find the value of y when x = 5 Given that y is directly proportional to the square of x, and when y = 40, x = 2. Find a formula connecting y and x . Find the value of y when x = 5 y x2 y  x2 y = 10 x2 y = k x2 When y = ?, x = 5 When y = 40, x = 2 y = 10 × 52 40 = k x 22 y = 250 k = 10

Given 36 pages cost 48p to produce. 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, and the cost of producing 100 page magazine. C √P C  √P C = 8 √P C = k √P When C = ?, P = 100 When P = 36, C = 48 C = 8 × √100 48 = k × √36 C = 80 p k = 8

If g varies directly with the square of h and g = 100 when h = 5 . Find the value h when g = 64. g  h2 g = 4 h2 g = k h2 When g = 64, h = ? When g = 100, h = 5 64 = 4 h2 100 = k x 52 k = 4 16 = h2 h = 4

1. If y varies directly as the cube of x and y = 40 when x = 2 find a formula connecting x and y. Find y when x = 3·5. 2. P varies directly as the square root of t and P = 20 when t = 4. Find the value of P when t = 7.

3. If y varies directly as the square of x and y = 4 when x = 5 find a formula connecting x and y. Find y when x = 6. 4. The resistance V at which a train can travel safely round a curve of radius r, varies directly as the square root of r. If V = 80 when r = 25 metres, find a formula connecting V and r. Find the value of V when r = 32 metres.

5. The bend b in a wire varies directly as the cube of the weight w attached to the midpoint of the wire. If b = 40 mm when a weight of 3 kg is attached, find the bend when a weight of 2 kg is attached. 6. The number n of particles emitted by a radioactive material on heating in a furnace varies as the square of the mass m of the material present. If n = 5400 when m = 30 grams, find n when m = 24 grams.

Inverse Variation Inverse means the reverse process Direct variation → one quantity doubles, the other doubles Inverse variation → one quantity doubles, the other ? halves

What Is Inverse Variation ? Consider the problem below: A farmer has enough cattle feed to feed 64 cows for 2 days. (a) How long would the same food last 32 cows ? 4 days (b) Complete the table below : Cows (c) 1 2 4 8 16 32 64 Days (d) 128 64 32 16 8 4 What do you notice about the pairs of numbers? 2 × 64 = 4 × 32 = 8 × 16 = ………… c × d = 128, a constant

Graphs of Inverse Proportion. Cows (c) 1 2 4 8 16 32 64 Days (d) 128 64 32 16 8 4 We are going to draw a graph of the table. Choose your scale carefully and allow each access to go at least up to 65. Estimate the position of the points (2,64) (4,32) etc as accurately as you can.

10 20 30 40 50 60 70 Cows Days (2,64) (4,32) (8,16) (16,8) (32,4) (64,2)

The graph is a typical inverse proportion graph : Cows Days As the number of cows increases the number of days decreases If we decrease the number of cows we will increase the number of days feed.

160 markers take 3 hours to complete marking their examination scripts. (a) Complete the table below : Markers 5 10 20 40 80 160 Time 3 96 48 24 12 6 (b) Draw a graph of the table.

20 40 60 80 100 120 140 160 180 Markers Time

The symbol  is shorthand for ‘varies as’ We are now going to look at the algebra of Inverse Variation The symbol  is shorthand for ‘varies as’

y = Substituting gives 20 = k = 80 Given that y varies inversely as x, and y = 20 when x = 4, find a formula connecting y and x. We write y  1 x We know that y = 20 when x = 4 y = k x Substituting gives The constant of variation 20 = k 4 k = 80 So formula becomes

It takes 12 workers 27 hours to leaflet the estate. The number of hours (H) required to deliver leaflets in a housing estate varies inversely with the number of workers (W) employed. It takes 12 workers 27 hours to leaflet the estate. Find a formula connecting W and H and the time it would take 20 workers to deliver the leaflets. We know that H = 27 when W = 12 We write H  1 W H = k W When W = 20, H= ? 27 = k 12 k = 324

If d varies inversely with w and when d = 3 , w = 9 If d varies inversely with w and when d = 3 , w = 9 . Find the value d when w = 3. When w = 3, d = ? When d = 3 , w = 9 d = 9 k = 27

If r varies inversely with the square root of f and r = 32 when f = 16, find f when r = 32. When r = 32 , f = ? When r = 32 , f = 16 k = 128 f = 16

1. If y varies inversely as the square of x and y = 9 when x = 2 find a formula connecting x and y. Find y when x = 10. 2. y varies inversely as the square root of x and y = 5 when x = 16. Find the value of y when x = 80.

3. y varies inversely as the cube of x and y = 10 when x = 2 3. y varies inversely as the cube of x and y = 10 when x = 2. Find a formula connecting x and y. Find y when x = 2·5. 4. The height h of a cone varies inversely as the square of the radius r of the base of the cone. If h = 8 when r = 5, find a formula connecting r and h. Find the height when the radius of the base is 3. 4. The height h of a cone varies inversely as the square of the radius r of the base of the cone. If h = 8 when r = 5, find a formula connecting r and h. Find the height when the radius of the base is 3.

5. The pressure P of a gas varies inversely as the volume V of the gas 5. The pressure P of a gas varies inversely as the volume V of the gas. If P = 500 when V = 1·4, find a formula connecting P and V. Find P when V = 2. 6. The intensity I of a light varies inversely as the square root of the distance d of the light from a screen. If I = 8 when d = 4, find I when d = 5.

When one quantity varies with more than one other quantity Joint Variation When one quantity varies with more than one other quantity

Joint Variation Quantities which vary directly are multiplied Quantities which vary inversely divide, the denominator(s) of fractions

If a varies directly with b and the square of c, we write If p varies inversely as q and r we write

If P varies directly with the square root of Q and inversely as the square of R, we write Direct variables on top, Inverse on bottom

If t varies jointly with m and b and t = 80 when m = 2 and b = 5, find t when m = 5 and b = 8 . When m = 5 , b = 8 , t = ? When t = 80 , m = 2 and b = 5

c varies directly with the square of m and inversely with w c varies directly with the square of m and inversely with w. c = 9 when m = 6 and w = 2 . Find c when m = 10 and w = 4 . When c = 9 , m = 6 and w = 2 When m = 10 , w = 4 and c = ? c =12∙5 k = ½

Multiples of Quantities Rather than have purely arithmetic examples you are often asked to consider the effect on one quantity of doubling, trebling, halving or changing by a given percentage the other quantity or quantities.

What is the effect on the exposure when f is doubled. The exposure E seconds required for a film varies directly as the square of the stop f used. It is found that E = 1/100 when f = 8. Find an equation connecting E and f and use it to find E when f = 16. What is the effect on the exposure when f is doubled. When f = doubled Doubling f multiplies E by 4 When f = 16, E = ?

When x is doubled, y is multiplied by 4 If y varies directly as the square of x and x is doubled, find the effect on y. When x is doubled When x is doubled, y is multiplied by 4

When x is multiplied by 4, y is doubled y varies directly as the square root of x and x is multiplied by a factor of 4, find the effect on y. When x is multiplied by 4 When x is multiplied by 4, y is doubled

When x is halved y is divided by 8 If y varies directly as the cube of x and x is halved, find the effect on y. When x is halved When x is halved y is divided by 8

v doubled and r halved gives A passenger on a bus experiences a sideways force when the bus goes round a corner. This force ( F units) varies as the mass ( m kg ),and the square of the speed of the bus ( v km/h ) and inversely as the radius of the corner( r metres ). What is the effect on F when the speed is doubled and the radius halved? v doubled and r halved gives F is multiplied by 8

The electrical resistance R ohms, of a wire varies directly as its length, L metres, and inversely as the square of its diameter, D millimeters. What is the effect on R if L is increased by 10% and D is reduced by 10%. L increased by 10% 1∙1L D decreased by 10% 0∙9D R is multiplied by 1∙36 R is increased by 36%

The lift L produced by the wing of an aircraft varies directly as its area A and inversely as the square of the airspeed v. If, for the same wing, the airspeed is increased by 10%, find the corresponding percentage decrease in the lift. v increased by 10% 1∙1v L is multiplied by 0∙83 L is decreased by 17%

Past Paper Questions

The time,T minutes ,taken for a stadium to empty varies directly as the number of spectators , S, and inversely as the number of open exits, E. Write down a relationship connecting T, S and E. It takes 12 minutes for a stadium to empty when there are 20 000 spectators and 20 open exits.

(b) How long does it take the stadium to empty when there are 36 000 spectators and 24 open exits ?

The number of letters, N , which can be typed on a sheet of paper varies inversely as the square of the size, S , of the letters used. (a) Write down a relationship connecting N and S . (b) The size of the letters used is doubled. What effect does this have on the number of letters which can be typed on the sheet of paper ? Doubling the size of letters divides the number of letters that can be typed by 4. Letter size = 2S

A frictional force is necessary for a car to round a bend A frictional force is necessary for a car to round a bend. The frictional force , F kilonewtons , varies directly as the square of the car’s speed , V metres per second, and inversely as the radius of the bend, R metres. (a) Write down a relationship between F, V and R. (b) What is the effect on F of halving V and doubling R? F is divided by 8

Boat is not beyond horizon The distance, d kilometres, to the horizon, when viewed from a cliff top, varies directly as the square root of the height, h metres, of the cliff top above sea level. From a cliff top 16 metres above sea level, the distance to the horizon is14 kilometres. A boat is 20 kilometres from a cliff whose top is 40 metres above sea level. Is the boat beyond the horizon? When h = 40, d = ? When h = 16, d = 14 Horizon is 22∙1 km away Boat is not beyond horizon

Write down a formula connecting t and d. (a) The air temperature, t º Celsius, varies inversely as the square of the distance, d metres, from a furnace. Write down a formula connecting t and d. (b) At a distance of 2 metres from the furnace, the air temperature is 50 ºC. Calculate the air temperature at a distance of 5 metres from the furnace. When d = 5, t = ? When d = 2, t = 50 Temperature is 4oC