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9.4 Deriving and using formulae

9.4 Deriving and using formulae

A formula is a mathematical rule that shows the relationship between two or more variables.
For example, a formula that is o$en used in physics is: v = u + at.

In this formula, v is the subject of the formula. It is written on its own, on the le$ -hand side.
Depending on the information you are given and the variable that you want to #nd, you

may need to rearrange the formula. !is is called changing the subject of the formula.
For example, if you know the values of v, a and t in the formula above and you want to
work out the value of u, you would rearrange the equation like this. v = u + at

!is makes u the subject of the formula. u + at = v
u = v − at
If you know the values of v, u and a in the formula above and you want to work out
the value of t, you would rearrange the equation like this. v = u + at
u + at = v
!is makes t the subject of the formula.

at = v − u
−
t= vu
a
Worked example 9.4

a Write a formula for the total pay, P dollars, Li earns when he works H hours at R dollars per hour.
b Use the formula in part a to work out P when H = 8 4 1 hours and R = $7.80 an hour.
c Rearrange the formula in part a to make H the subject.
d Use the formula in part c to work out H when P = $81.70 and R = $8.60 per hour.

a P = HR Pay (P ) = number of hours (H ) × rate of pay (R ). Remember to write H × R as HR.
b P = 8.25 × 7.80 Substitute H = 8.25 and R = 7.80 into the formula.
= $64.35 Work out the answer and remember the units ($).
c P = HR To make H the subject, divide both sides of the formula by R.
R R
H= P Now rewrite the formula with H as the subject.
R
.
81 70
d H = Substitute P = 81.70 and R = 8.60 into the formula.
860
.
= 9.5 hours Work out the answer and remember the units (hours).

) Exercise 9.4

1 a Write a formula for the number of seconds, S, in any number of minutes, M.
b Use your formula in part a to work out S when M = 15.
c Rearrange your formula in part a to make M the subject.
d Use you formula in part c to work out M when S = 1350.

2 Use the formula F = ma to work out the value of: In parts c and d you must
a F when m = 12 and a = 5 b F when m = 26 and a = −3 start by changing the
c m when F = 30 and a = 2.5 d a when F = −14 and m = 8. subject of the formula.

9 Expressions and formulae 89