## What you need to know

A **formula **is a mathematical relationship between different quantities that is expressed with algebra. For example, one formula for **s**peed is **d**istance divided by **t**ime, which we express like

s=\dfrac{d}{t}.

In this case, we say s (speed) is the **subject** of the formula. This is because s sits on its own one side of the equation and doesn’t appear at all on the other. We can **change the subject** of the formula, for example by multiplying both sides by t the equation becomes

st=d

So now, d (distance) sits on its own on one side of the equation and thus is the new subject. Rearranging formulae in order to change the subject is the focus of this topic, and we do it by applying the core principle of: an equation remains unchanged as long as you do **the same thing to both sides**.

We will go through a series of (sometimes subtly) different examples to see how sometimes it can take several steps and a bit of ingenuity to get the subject that you desire.

**Example: **Rearrange the formula F=\dfrac{mv}{t} to make m the subject.

As a rule of thumb, it’s good to try to get rid of any algebraic fractions as soon as you can. This is to say that if the thing that you want to make the subject is part of a fraction (especially if it’s __in the denominator__) , then it’s generally a good idea to multiply up by the denominator of that fraction as a first step. It’s worth keeping in mind and is precisely what we’re going to do here. So, to kick off making m the subject, we will multiply both sides by t and get

Ft=mv

Then, if we divide both sides of the equation by v, we get

\dfrac{Ft}{v}=m

Now m is on its own on one side, and thus we have made it the subject of the formula.

**Example: **Rearrange the formula for the area of a circle to make r the subject.

Firstly, recall that the formula for the area of a circle is

A=\pi r^2

Now, the square in there makes the situation slightly different, and we’re going to make sure that the square is the last thing that we deal with (this is another good rule of thumb to follow). So, dividing both sides of the equation by \pi, we get

\dfrac{A}{\pi}=r^2

Now, r is on its own on one side but it’s not technically the subject, since it is being squared. So, if we now square root both sides of the equation, we get

\sqrt{\dfrac{A}{\pi}}=r

This completes the process of making r the subject of the formula.

**Example: **Rearrange the formula H=2R-gR to make R the subject.

This is different to what we’ve seen so far in that there are two instances of R – the thing we want to make the subject. This is not helpful, obviously, since we need there to be only one R, but there is a nice way out of this conundrum, and that is to factorise the terms involving an R.

Both of the terms on the right-hand side of the equation have a R in them, so if we take out a factor of R from them both, we get

H=R(2-g)

Now, there’s only one R. Much better. Then, if we divide by (2-g) (and yes, we divide by the whole thing in one go, since it’s the whole that is being multiplied by R), we get

\dfrac{H}{2-g}=R

Thus, we have made R the subject of the equation.

We’ve gone through a few typical examples, now you can have a go at the questions below for a few more.

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### Example Questions

1) Rearrange the formula A=\dfrac{1}{2}(a+b)h to make a the subject.

Firstly, we will multiply both sides of the equation by 2 to get rid of the fraction:

2A=(a+b)h

Then, if we divide both sides by h, we get

\dfrac{2A}{h}=a+b

Finally, subtracting b from both sides, we get

\dfrac{2A}{h}-b=a

Thus, we have made a the subject.

**NOTE: **there are alternative ways of answering this question. If, at the second step, you expand the bracket on the right-hand side, you get

2A=ah+bh

Then, subtracting bh from both sides gives you

2A-bh=ah

Finally, dividing both sides by h, we get

\dfrac{2A-bh}{h}=a

Thus making a the subject. Both of these methods/answers are correct and would get you full marks for this question.

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2) Rearrange the formula F=\dfrac{kq}{r^2} to make r the subject.

Any time the thing we want is on the denominator, we should always start by multiplying both sides by that denominator to get rid of the fraction. So, multiplying both sides by r^2, we get

Fr^2=kq

Next, divide both sides by F to get

r^2=\dfrac{kq}{F}

Finally, square rooting both sides gives us

r=\sqrt{\dfrac{kq}{F}}

Thus making r the subject of the formula.

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