Rearranging formulas

A rectangle has an area of 48 cm² and is 6 cm wide. How long is it? You already know the formula you need: area is length times width. But there the area sits alone on one side, while the length is what you are after. That is what rearranging is for: you rebuild the formula so the letter you want stands alone, and after that you only have to substitute.

A formula is just an equation

In the formula $A = a \cdot b$, the left side holds exactly as much as the right, like a balance scale in equilibrium. So the same rules apply as when solving equations: whatever you do on one side, you do on the other. If you want $a$ on its own, the $b$ it is multiplied by is in the way. Multiplying is undone by dividing, so you divide both sides by $b$: on the left the $b$ cancels, on the right $A$ divided by $b$ remains. The only difference from tasks with numbers is that here you have letters, and you calculate with them exactly as if they were numbers.

Alone really means alone

A formula is only rearranged once the letter you want appears exactly once, stands completely alone on one side, and has nothing happening to it any more: no times, no divided by, no plus. With the perimeter of a rectangle, $U = 2 \cdot (a + b)$, dividing by 2 does not finish the job, because that leaves $\frac{U}{2} = a + b$ and the $b$ is still stuck to the $a$. Only when you subtract $b$ on both sides do you get $a = \frac{U}{2} - b$ and can put numbers in. And do not worry if the letter you want ends up on the right: $8 = a$ means exactly the same as $a = 8$. One rule always applies when dividing: you may only divide by something that is not zero. With side lengths, times or speeds that is guaranteed, because otherwise the formula itself would make no sense.

Clearing up in reverse order

Often more than one operation is attached to the letter you want. Then you clear up backwards: whatever came last goes first. In $y = 2x + 7$, $x$ is first multiplied by 2 and then 7 is added. So you first subtract 7 on both sides, which gives $y - 7 = 2x$, and then divide both sides by 2: $x = \frac{y - 7}{2}$. Two small notes on this: a number written directly in front of a letter means multiplication, so $2x$ is the same as $2 \cdot x$. And if a fraction bar is in the way, you first multiply both sides by whatever sits under the bar, and the fraction is gone.

Speed, distance and time

The best known formula of this kind is the one for speed: speed is distance divided by time, written $v = \frac{s}{t}$. If you want to know how far you get, you need it rearranged for $s$: multiply both sides by $t$, the $t$ in the fraction cancels and $s = v \cdot t$ is left. If instead you are after the travel time, you then divide both sides by $v$ and get $t = \frac{s}{v}$. An example: 120 km at a steady 60 km/h means 120 divided by 60, so 2 hours. It is always the same three quantities, just rearranged to fit what you need right now.