The Table Below Shows the Cost of Different Numbers of Pints of Milk

Proportion

Direct Proportion

The table below shows the cost of different numbers of pints of milk.

pints of milk, x / 1 / 2 / 3 / 4 / 5 / 6
cost in pence, y / 60

When x doubles, so does y. If we divide x by 3, we must divide y by 3 as well. We say that y and x are directly proportional. We write this as:

Draw a graph of y against x.

The graph of y against x is a straight-line graph passing through the origin.

Now, straight-line graphs have equations that look like. In our graph, the y-intercept is zero, and so the equation is

And, just to be awkward, we use k rather than m, leaving us with.

k is known as the ‘constant of proportionality’. Clearly k is the gradient of the line, which is 30, and so the formula is

Check the formula is correct by looking at the table of values.

What does the 30 actually represent?

To summarise…

Whatever we multiply or divide x by, we must do the same to y.
The graph is a straight line through the origin.
The formula is where k is the constant of proportionality.
There are many ways of describing this kind of relationship: y is directly proportional to x, y varies directly as x, or just .

1.The table shows the cost of tickets to a concert.

number of tickets, n / 3 / 6 / 15
cost in pounds, c / 13.50 / 27 / 67.50

a)Explain how you know that c is directly proportional to n.

b)Copy and complete: ‘, therefore …’

c)Find a formula connecting c and n.

d)Use your formula to find the cost of 19 tickets.

e)A school party went to the concert. The total cost of the tickets was £153. How many were in the school party?

2.The distance a bicycle travels varies directly as the number of revolutions of the front wheel. The bicycle travels 24 metres when the front wheel turns 10 times.

a)Write down a formula connecting the distance d and the number of revolutions r.

b)How far does the bicycle travel when the front wheel turns 33 times?

c)How many turns of the wheel are needed to travel a kilometre?

3.y is directly proportional to x. When Find:

a)the value of y whenx = 7

b)the value of x wheny = 42

4.The cost of a bar of chocolate is directly proportional to its mass. Find the missing numbers in the table below.

mass (g) / 50 / 130 / y
cost / 35p / x / £2

Other Forms of Direct Proportion

A sweet shop sells sticks of rock, all of the same thickness. The 10 cm long stick weighs 50 g. Write down the masses of the 15 cm and 20 cm sticks in the diagram below.

Write down a formula connecting the mass m of the stick and the length x.

The sweet shop also sells square slabs of toffee, all of the same thickness. The 1 cm long slab weighs 20 g. Write the masses of the 2 cm and 3 cm slabs on the diagram below, and plot a graph of the mass m against the length x.

How does your graph tell you that the mass m of the slab is not proportional to the length x?

Instead, let’s try plotting m against . Fill in the table, then plot the graph.

How does your graph tell you that m is proportional to ? Why is m is proportional to ?

We write this as:

What is the value of k? And what does it represent?

If the toffee came in cubes, what would the formula connecting m and x look like?

1.The amount of paint required to paint a model car is proportional to the square of the length of the car. A 20cm long car needs 30ml of paint.

a)How much paint does a 15cm long car need?

b)What is the longest car that can be painted with 50ml of paint?

2.The volume of a hot air balloon is proportional to the cube of its height. A balloon of height 5m has a volume of 240m3.

a)Find the volume of a balloon of height of 10m

b)Find the height of a balloon of volume 100m3.

3.If a stone is dropped from a building, the time taken t for the stone to reach the ground is proportional to the square root of the height h of the building.

A stone takes 4.5 seconds to reach the ground after being dropped from a building 100 feet high.

a)A stone is dropped from a building 50 feet high. How long does it take to reach the ground?

b)Janet wants to know how high York Minster is. She climbs to the top and drops a stone. It takes seven seconds to reach the ground. How high is York Minster?

Note: please do not drop stones from tall buildings. It’s incredibly dangerous.

4.y is directly proportional to the cube root of x. Find the missing values.

x / 3 / 8
y / 12 / 30

Inverse Proportion

The table below shows the time taken to travel a journey of 60 miles at various speeds.

speed, s (mph) / 10 / 20 / 30 / 40 / 50 / 60
time, t (hours)

Notice that when s doubles, t halves; and if we divide s by 3, we multiply t by three. We say that s and t are inversely proportional, because whatever we do to one, we do the opposite (inverse) to the other.

Draw a graph of t against s.

It’s not obvious from the graph what the formula connecting t and s is.

But, remember that:

In this particular case:

And so with inverse proportion we have instead of We can also write this as

To summarise, remember these facts about inverse proportion.

Whatever we multiply one quantity by, we divide the other by the same number.
The formula is of the form where k is the constant of proportionality.
The relationship can also be described as ‘y varies inversely as x’.

1.The number of letters that will fit on a line is inversely proportional to the font size. At font size 12, 66 letters will fit onto a line.

a)How many letters will fit on a line at font size 14?

b)What must the font size be to fit 100 letters on a line?

2.The loudness of a factory machine is inversely proportional to the square of the distance from it. At 120 metres, the loudness is 75 decibels.

a)How loud is the machine at 200 metres?

b)The maximum safe noise level is 90 decibels. How close can you get to the machine and still be within safe noise levels?

Note: Question 2 is an example of the ‘inverse square law’. It works for sound, light, smell, gravity, magnetism and so on.

3.y is inversely proportional to the cube of x. Find the missing values.

x / 2 / 5
y / 7 / 2

4.y is inversely proportional to the square root of x. Find the missing values.

x / 2 / 9
y / 7 / 2

Summary

As all proportion questions are pretty much the same, here’s a recap of the method.

Turn the sentence into algebra. If, say, y is directly proportional to then write and if y is inversely proportional to write

Substitute the given pair of values of y and x, and work out the value of k.

Rewrite your formula, but replace k by its value.

I’ve put this one in bold as it’s the one students most often miss out.

Use your formula to answer any other parts of the question.

And Finally…

Here’s an example of a more complicated proportion question. There is an elegant way of solving the last part, if you can spot it.

The power generated by a wind turbine (in kilowatts) is proportional to the cube of the wind speed (in metres per second).

When the wind speed is 5m/s, the power generated is 400 kW.

a)Find the power generated when the wind speed is 8m/s.

b)Find the wind speed needed to generate 5000 kW.

c)The power generated when the wind speed is x m/s is twice the power generated when the wind speed is 4 m/s. Find x.

Proportion Answers

Direct Proportion

The table below shows the cost of different numbers of pints of milk.

pints of milk, x / 1 / 2 / 3 / 4 / 5 / 6
cost in pence, y / 30 / 60 / 90 / 120 / 150 / 180

When x doubles, so does y. If we divide x by 3, we must divide y by 3 as well. We say that y and x are directly proportional. We write this as:

Draw a graph of y against x.

The graph of y against x is a straight-line graph passing through the origin.

Now, straight-line graphs have equations that look like. In our graph, the y-intercept is zero, and so the equation is

And, just to be awkward, we use k rather than m, leaving us with.

k is known as the ‘constant of proportionality’. Clearly k is the gradient of the line, which is 30, and so the formula is

Check the formula is correct by looking at the table of values.

What does the 30 actually represent?

To summarise…

Whatever we multiply or divide x by, we must do the same to y.
The graph is a straight line through the origin.
The formula is where k is the constant of proportionality.
There are many ways of describing this kind of relationship: y is directly proportional to x, y varies directly as x, or just .

1.The table shows the cost of tickets to a concert.

number of tickets, n / 3 / 6 / 15
cost in pounds, c / 13.50 / 27 / 67.50

a)Explain how you know that c is directly proportional to n. Both ×2, then both ×5.

b)Copy and complete: ‘, therefore …’ c = kn

c)Find a formula connecting c and n.

d)Use your formula to find the cost of 19 tickets. £85.50

e)A school party went to the concert. The total cost of the tickets was £153. How many were in the school party? 34

2.The distance a bicycle travels varies directly as the number of revolutions of the front wheel. The bicycle travels 24 metres when the front wheel turns 10 times.

a)Write down a formula connecting the distance d and the number of revolutions r.

b)How far does the bicycle travel when the front wheel turns 33 times? 79.2m

c)How many turns of the wheel are needed to travel a kilometre? 416.7

3.y is directly proportional to x. When Find:

a)the value of y whenx = 7 28

b)the value of x wheny = 42 10.5

4.The cost of a bar of chocolate is directly proportional to its mass. Find the missing numbers in the table below. x = 91p, y = 285.7

mass (g) / 50 / 130 / y
cost / 35p / x / £2

Other Forms of Direct Proportion

A sweet shop sells sticks of rock, all of the same thickness. The 10 cm long stick weighs 50 g. Write down the masses of the 15 cm and 20 cm sticks in the diagram below.

Write down a formula connecting the mass m of the stick and the length x.

How does your graph tell you that the mass m of the slab is not proportional to the length x?

Not a straight line.

Instead, let’s try plotting m against . Fill in the table, then plot the graph.

/ 1 / 4 / 9
m / 20 / 80 / 180

How does your graph tell you that m is proportional to ? Why is m is proportional to ?

Straight-line graph

The mass depends on the area of the slab, not the length.

We write this as:

What is the value of k? And what does it represent?

k = 20

It represents the mass of one square centimetre of toffee.

If the toffee came in cubes, what would the formula connecting m and x look like?

1.The amount of paint required to paint a model car is proportional to the square of the length of the car. A 20cm long car needs 30ml of paint.

a)How much paint does a 15cm long car need? 16.875ml

b)What is the longest car that can be painted with 50ml of paint? 25.8 cm

2.The volume of a hot air balloon is proportional to the cube of its height. A balloon of height 5m has a volume of 240m3.

a)Find the volume of a balloon of height of 10m. 1920m3

b)Find the height of a balloon of volume 100m3. 3.73 m

3.If a stone is dropped from a building, the time taken, t, for the stone to reach the ground is proportional to the square root of the height, h, of the building.

A stone takes 4.5 seconds to reach the ground after being dropped from a building 100 feet high.

a)A stone is dropped from a building 50 feet high. How long does it take to reach the ground? 3.18 seconds

b)Janet wants to know how high York Minster is. She climbs to the top and drops a stone. It takes seven seconds to reach the ground. How high is York Minster? 242 metres

Note: please do not drop stones from tall buildings. It’s incredibly dangerous.

4.y is directly proportional to the cube root of x. Find the missing values.

x / 3 / 8 / 125
y / 8.65 / 12 / 30

Inverse Proportion

The table below shows the time taken to travel a journey of 60 miles at various speeds.

speed, s (mph) / 10 / 20 / 30 / 40 / 50 / 60
time, t (hours) / 6 / 3 / 2 / 1.5 / 1.2 / 1

Draw a graph of t against s.

It’s not obvious from the graph what the formula connecting t and s is.

But, remember that:

In this particular case:

And so with inverse proportion we have instead of We can also write this as

To summarise, remember these facts about inverse proportion.

Whatever we multiply one quantity by, we divide the other by the same number.
The formula is of the form where k is the constant of proportionality.
The relationship can also be described as ‘y varies inversely as x’.

1.The number of letters that will fit on a line is inversely proportional to the font size. At font size 12, 66 letters will fit onto a line.

a)How many letters will fit on a line at font size 14? 56

b)What must the font size be to fit 100 letters on a line? 7.92

2.The loudness of a factory machine is inversely proportional to the square of the distance from it. At 120 metres, the loudness is 75 decibels.

a)How loud is the machine at 200 metres? 27 decibels

b)The maximum safe noise level is 90 decibels. How close can you get to the machine and still be within safe noise levels? 109.5 metres

Note: Question 2 is an example of the ‘inverse square law’. It works for sound, light, smell, gravity, magnetism and so on.

3.y is inversely proportional to the cube of x. Find the missing values.

x / 2 / 5 / 7.59
y / 109.375 / 7 / 2

4.y is inversely proportional to the square root of x. Find the missing values.

x / 2 / 9 / 110.25
y / 14.85 / 7 / 2

Summary

As all proportion questions are pretty much the same, here’s a recap of the method.

Turn the sentence into algebra. If, say, y is directly proportional to then write and if y is inversely proportional to write

Substitute the given pair of values of y and x, and work out the value of k.

Rewrite your formula, but replace k by its value.

I’ve put this one in bold as it’s the one students most often miss out.

Use your formula to answer any other parts of the question.

And Finally…

Here’s an example of a more complicated proportion question. There is an elegant way of solving the last part, if you can spot it.

The power generated by a wind turbine (in kilowatts) is proportional to the cube of the wind speed (in metres per second).

When the wind speed is 5m/s, the power generated is 400 kW.

d)Find the power generated when the wind speed is 8m/s. 1638.4 kW

e)Find the wind speed needed to generate 5000 kW. 11.6 m/s

f)The power generated when the wind speed is x m/s is twice the power generated when the wind speed is 4 m/s. Find x. 5.04 m/s

Quick(ish) way for part c)…