Unit 5 Section 6 : Equations in Context

In this section we determine the solutions to a variety of problems by forming and solving suitable linear equations.

Example 1

Apples cost 55p per kg. Alan buys a bag of apples that costs £1.65. If the bag contains x kg of apples,

(a)

write down an equation involving x,

It is easier to work in pence.

x × 55 = 165
55x = 165
(b)

solve the equation.

x =
x = 3

Example 2

Three consecutive whole numbers add up to 36. Determine the three numbers.

If x = first number,
then x + 1 = second number,
and x + 2 = third number.

Adding these gives:

x + (x + 1) + (x + 2) = 36
3x + 3 = 36
3x = 33
x =
x = 11

and the three numbers are 11, 12 and 13.

Example 3

A taxi driver charges £2.00 plus £1.10 per mile for all journeys.

(a)

Write down the cost, in pence, for travelling m miles.

Basic cost + 110 × number of miles = 200 + 110m pence

(b)

The charge for a journey is £3.65. Write down an equation and use this to determine the distance travelled.

200 + 110m = 365
110m = 365 - 200
110m = 165
m =
m = 1.5

So the distance travelled is 1.5 miles.

Exercises

Question 1
The cost of a ticket for a football match is £9.
(a)

Write down an expression for the cost of n tickets.

(b)

Solve an equation to determine how many tickets could be bought with £108.

tickets
9n = 108
n = 108 ÷ 9
n = 12
Question 2
The cost of hiring a van is £20 per day, plus 50p for each mile travelled.
(a)

Write down an expression for the cost, c, in pounds, of travelling m miles in one day in a hired van.

c =
(b)

Write down an expression for the cost in pounds of travelling m miles during a two-day hire period.

c =
(c)

James hires a van for 2 days. He has to pay a total of £68.50. Write down an equation and solve it to determine how far he travelled.

68.50 =
m = (miles)
Question 3
Two consecutive odd numbers are x and x + 2. When these numbers are added together they total 100. Write down and solve an equation to obtain the value of x.
= 100
x =
Question 4
A removals firm charges £4 per mile plus a fixed charge of £25. Use an equation to determine the distance travelled if the bill is £39.
39 =
m = (miles)
Question 5
The price of petrol is given in pence per litre. To convert this to £ per gallon, use the flow chart given below.
(a)

Convert a price of 80p per litre to £ per gallon.

(b)

If the price is x pence per litre, write down the cost in £ per gallon.

(c)

Convert a price of £4.14 per gallon to pence per litre.

Question 6
A rectangle has length 10 m and width x m.
(a)

Write down a formula for the area of the rectangle.

m²
(b)

Use an equation to determine x if the area is 16 m².

= 16
x = (m)
(c)

Write down a formula for the perimeter of the rectangle.

(d)

Use an equation to determine x, if the perimeter is 39 m.

= 39
x = (m)
Question 7
A repairman charges £40 for the first hour of his time and £15 for each hour after that.
(a)

Write down a formula for the cost of a repair that takes n hours.

pounds
(b)

Use an equation to determine the time for a repair, if the cost is £52.50.

= 52.50
x = hour minutes
15n + 25 = 52.5
15n = 27.5
n =
So the repair took 1 hour 50 minutes.
Question 8
At a bank a charge of £2 is made for changing British Pounds (£) into Norwegian Krone (NOK). The charge is deducted first and then 9 NOK are issued for every £1 left.
(a)

Write down a formula for the number of NOK issued in exchange for £x.

Number of NOK =
(b)

Use an equation to determine how many £ you would need to change to get 900 NOK.

= 900
x = (pounds)
9(x - 2) = 900
x - 2 = 100
x = 102
So you need £102 to get 900 NOK.
Question 9
(a)

Write down a formula for the perimeter of the shape shown.

x
(b)

Calculate x if the perimeter is 2.76 m.

x = cm
(c)

Write down a formula for the area of the shape.

x²
(d)

Calculate x if the area is 8.64 m² .

x = m
6x² = 8.64
x² = 1.44
x = 1.2 (m)
Question 10
(a)

Write down a formula for the perimeter of the shape shown.

(m)
(b)

If the perimeter is 23 m, determine the length x.

x = m
Question 11
The simplified graph shows the flight details of an aeroplane travelling from London to Madrid, via Brussels.
(a)

What is the aeroplane's average speed from London to Brussels?

(b)

How can you tell from the graph, without calculating, that the aeroplane's average speed from Brussels to Madrid is greater than its average speed from London to Brussels?

Because the third section (Brussels to Madrid) has a than the first section (London to Brussels).
(c)

A different aeroplane flies from Madrid to London, via Brussels. The flight details are shown below.

Madriddepart1800
Brusselsarrive2000
depart2112
Londonarrive2218

On a copy of the graph, show the aeroplane's journey from Madrid to London, via Brussels. (Do not change the labels on the graph.)
Assume constant speed for each part of the journey.

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(d)

At what time are the two aeroplanes the same distance from London?