Rearranging Formula | Made Easy For GCSE Maths
Introduction
After reading this article you will be able to:
- To be able to apply algebraic techniques to rearrange formula
- Be able to rearrange formula if the variable appears more than once
Rearranging Formula - An Example
Let us consider the formula for the area of a triangle which is A=\frac{1}{2} b h
Now because the variable A stands on its own we say that it is the subject of the formula.
It is possible to make other variables the subject and this is done by applying the same algebraic techniques that are used to solve equations.
Rearranging Formula - Subtracting a term from both sides
Let us consider the formula
M-n+3 and we want to make n the subject.
The simplest thing to do here is to subtract three from both sides which will give us:
M-3=n+3-3And this can be simplified to M-3=n
This is the answer, but we generally write the subject of a formula on the left side, so our complete answer is:
n=M-3
Subtracting a term from both sides and performing a division
Consider the formula
L=7 q+7and we want to make q the subject
Step 1: Subtract 7 from both sides to give
L-7=7 qStep 2: Divide by 7 on both sides to give
q=\frac{L-7}{7}Rearranging formula involving a squared term
A slightly more difficult formula to rearrange is
B=3 d^2-4and we want to make d the subject
Step 1: Add 4 to both sides
Step 2: Divide both sides by 3
Step 3: To undo the squared term, we must know take the square root of both sides
\begin{aligned} & B+4=3 d^2 \\ & \frac{B+4}{3}=d^2 \\ & d= \pm \sqrt{\frac{B+4}{3}} \end{aligned}Remember that if you take the square root of anything then you have two possible answers which is indicated by the \pm
Functional Skills Maths
Maths exams are becoming more focused on applying mathematics to practical problems. Let us see how this can be done within the topic of rearranging formulas.
Example:
The price of 6 iced buns is 80 pence more than the price of 7 pies. Let the price of the iced buns be x and the price of a pie be y
1st – Create a formula using the first sentence
2nd – Rearrange this formula to make y the subject
If the cost of an iced bun is 67 pence what is the cost of a pie?
Solution:
a)
The cost of 6 iced buns is 6 x
The cost of 7 pies is 7 y
So, 6 x = 7 y + 80 [cost of 6 iced buns is 80 pence more than 7 pies]
b)
Subtract 80 from both sides and then dividing by 7 we have:
y=\frac{6 x-80}{7}c)
To find the cost of a pie substitute the cost of a iced bun into the formula above to give:
y=\frac{6(74)-80}{7}=46So the cost of a pie is 46 pence each.
- Rearranging formula if the variable appears more than once
So far we have only seen formulas where the subject appears only once. In the event of variables appearing more than once it is quite often a case of having to factorise as the following example will demonstrate.
Example:
Make b the subject in the following formula
a b+p=c b+q1st: we need to get all the b terms together by subtracting cb from both sides
2nd: subtract p from both sides
3rd: on the left hand side, b, is common so we can factorise
4th: now we can just divide both sides by the brackets and we have b as the subject:
\begin{aligned} & a b-c b+p-q \\ & a b-c b=q-p \\ & b(a-c)=q-p \\ & b=\frac{q-p}{a-c} \end{aligned}Example:
Make n the subject from the formula:
3=\frac{a n+c}{m n+e}1st: multiply both sides by the denominator
2nd: expand brackets
3rd: collect n terms on the left hand side and subtracting 3e from both sides
4th: on the left, n is common so we can factorise
5th: we can now divide both sides by the brackets to give n as the subject of the formula
\begin{gathered} 3(m n+e)=a n+c \\ 3 m n+3 e=a n+c \\ 3 m n-a n=c-3 e \\ n(3 m-a)=c-3 e \\ n=\frac{c-3 e}{3 m-a} \end{gathered}If you can solve equations then you can rearrange formulas. The patterns that exist within solving maths equations also exist here. What you do need to be careful about is when you have the same variable on the opposite sides. Here you are effectively collecting like terms and then factorising.