Formulae 2
Construct and Use Simple Formulae

A formula describes how one quantity relates to one or more other quantities. For example, a formula for the area of a rectangle describes how to find the area, given the length and width of the rectangle.

The perimeter of the rectangle would be given by the formula

P = 2l + 2w

Here again there are no multiplication signs and 2l means 2 × l and 2w means 2 × w.

Worked Examples

1
(a)

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

Show me

The perimeter is found by adding together the lengths of all the sides, so the formula will be

P = a + b + b + a + c

but as a and b are both added in twice, this can be simplified to

P = 2a + 2b + c
(b)

Find the perimeter if

a = 2 cm, b = 3 cm and c = 5 cm

Show me

If a = 2 cm, b = 3 cm and c = 5 cm,

P = 2 × 2 + 2 × 3 + 5
= 4 + 6 + 5
= 15 cm
2

When laying a patio, a landscape gardener charges a basic fee of £30 plus £12 per hour.

Find a formula for calculating the gardener's charge.

Show me

Let C = charge and n = number of hours.

The charge is made up of

a fixed £30 and £12 × the number of hours or £12n.

So the total charge in £ is given by

C = 30 + 12n

Exercises

Find a formula for the perimeter of each shape, and find the perimeter for the specified values.

(a)
P = = cm
(b)
P = = cm
(c)
P = = cm
(d)
P = = cm
(e)
P = = cm
(f)
P = = cm
(g)
P = = cm
(h)
P = = cm

Find a formula for the area of each of the shapes below and find the area for the values given.

(a)
A = = cm2
(b)
A = = cm2
(c)
A = = cm2
(d)
A = = cm2
(e)
A = = cm2
(f)
A = = cm2

Tickets for a school concert are sold at £6 for adults and £4 for children.

(a)

If p adults and q children buy tickets, write a formula for the total value, T, of the ticket sales.

T =
(b)

Find the total value of the ticket sales if p = 50 and q = 20 .

£

Rachel is one year older than Bradley. Carla is three years younger than Bradley

If Bradley is x years old, write down expressions for:

(a)

Rachel's age;

(b)

Carla's age;

(c)

the sum of all three children's ages.

A window cleaner charges a fee of £8 for visiting a house and £4 for every window that he cleans.

(a)

Write down a formula for finding the total cost C, in £, when n windows are cleaned.

C =
(b)

Find C if n = 8.

£

A taxi driver charges a fee of £5, plus £4 for every km that the taxi travels.

(a)

Find a formula for the cost C of a journey that covers x km.

C =
(b)

Find C if x = 3.

km

A gardener builds paths using paving slabs laid out in a pattern as shown, with white slabs on each side of a row of red slabs.

(a)

If n red slabs are used, how many white slabs are needed?

(b)

Another gardener puts a white slab at each end of the path as shown below.

If n red slabs are used, how many white slabs are needed?

Juice drinks cost 27 p each.

Write down a formula for the cost, C p, of n drinks.

C = p

Mr James works a basic week of 40 hours at a rate of £16 an hour. His overtime rate is £4 per hour MORE than his basic rate.

Calculate

(a)

his TOTAL wage for a basic week.

£
(b)

his wage for a week in which he worked 47 hours.

£