Creating formula

Formulae are created for something that is calculated often.

For instance, plumbers often apply a call out charge plus an hourly rate to their customers. Writing a formula for the total cost of a job would be useful for a plumber so that they could quickly calculate costs for their customers more easily. Once a formula is written, the plumber would only need to input how long the job would take in hours, and come up with a total cost to quote very easily.

Example 1

A plumber has a call out fee of £40, plus an hourly rate of £18. Write a formula to calculate the cost of any job and calculate the cost of a job estimated to take 2 hours.

The total cost ( T) would be equal to the call out charge of £40 plus £18 for every hour ( h) worked.

This can be written as: T = 40 + 18h

Now, the total cost for customers can be worked out easily by substituting the number of hours the job will take.

A job estimated to take 2 hours can be calculated like this:

T = 40 + 18h

Substitute the number of hours as 2 hours:

T = 40 + 18 \times 2 = 40 + 36 = \pounds 76

Use BIDMAS to complete the multiplication before the addition.

A two hour job would cost £76.

Example 2

On a given day the plumber charges £130. How long did the plumber work for?

This time, it is not the total cost ( T) that is to be calculated but the hours ( h). In this instance the formula will need to be solved to find h:

T = 40 + 18h

Substitute the total cost as £130:

130 = 40 + 18h

Solve the equation:

First, take 40 from both sides: 90 = 18h

Next, divide both sides by 18: 5 = h

The plumber worked for 5 hours to earn £130.

Question

A worker's daily pay ( P) depends on the amount of hours worked ( h), the rate they are paid per hour ( r) and any bonus that is earned ( b).

Write a formula for the pay of the worker, and calculate how much a person who works for 8 hours at a rate of £9 per hour and who receives £5 in bonuses will earn.

Multiply the amount of hours worked by the rate of pay and then add on the bonus.

This can be shown by writing a formula.

\text{Pay} = \text{amount of hours worked} \times \text{rate of pay} + \text{bonus}

Using the letters given in the question gives: P = hr + b

The employee works 8 hours per day for an hourly rate of £9. The bonus they have earned is £5.

Substituting these values into the formula P = hr + b gives: P = 8 \times 9 + 5 = 72 + 5 = \pounds 77