Rearranging a Formula

Sometimes the way a formula is written, the item you are trying to find does not stand alone.  You must rearrange or transpose the formula to solve for the unknown.

When rearranging a formulas, think of a formula as a balanced scale – the quantity on the left has to equal the quantity on the right. Whatever we do to one side of the equal sign we must do to the other – add, subtract, multiply or divide a number on one side you must do the same thing to the other side to keep the sides equal.  This includes other operations such as squaring and taking the square root – as long as whatever you do to one side you do to the other.

How do you decide whether to add, subtract, multiply or divide?  First look at what you are trying to find and ask yourself: “What has been done to it?  For example you are given this formula but are asked to find the Volume:

Detention time = (Volume) / (Flow Rate)

Since Volume is divided by Flow Rate to get Volume to stand alone, you must undo the division by doing the opposite or multiplication.  But remember, when working with a formula, whatever you do to one side you must do to the other side of the equal (=) sign:

Flow Rate X Detention Time = (Volume) / (Flow Rate X Flow Rate)

Flow rate will cancel out on the right side of the equation, leaving Volume to stand alone.  Then we can work the the problem that now looks like this:

Flow Rate X Detention Time = Volume

Suppose you want to find the Flow Rate instead of the volume:

Detention Time = (Volume) / (Flow Rate)

Since Flow Rate is on the bottom (the denominator), multiply both sides by Flow Rate so it is on the top (in the numerator):

Flow Rate X Detention Time = (Volume) / (Flow Rate X  Flow Rate)

But, Flow Rate still doesn’t stand alone so it has be multiplied by Detention Time.  To undo multiplication, you must divide both sides by Detention Time, then cancel:

(Flow Rate) X [(Detention Time) / (Detention Time)] = (Volume) / (Detention Time)

Flow Rate = (Volume) / (Detention Time)

Another example – if you want to find Velocity, but are given the follow formula:

Flow Rate = Velocity X Area

Since the Velocity has been multiplied by the area, you undo that by dividing both sides of the equation by area so Velocity stands alone:

(Flow Rate) / (Area) = Velocity X [(Area) / (Area)]

(Flow Rate) / (Area) = Velocity