Why do we multiply to cancel out?

Question

Say we have $\frac{g}{4} = 3.2$ why do we multiply each side by 4 to get g by itself? I don’t get why it doesn’t stay $\frac{g}{4}$ to be honest

Answer 1

Because $\frac{g}{4}$ is the same thing as $g\cdot\frac14$, when you multiply the left side by $4$ you get this:

$$\frac{g}4\cdot4=(\underbrace{g\cdot\frac14}_{\frac{g}{4}})\cdot4=g\cdot(\frac14\cdot4) = g\cdot (1) = g$$

Answer 2

Consider the equality

$$\frac{g}{a} = b $$

We can also write that as

$$\frac1a \cdot g = b $$

If we want to isolate $g $, we just multiply both sides of the equality by the inverse of $\frac1a $, which is $a $:

$$a \cdot \frac1a \cdot g = a \cdot b $$

Answer 3

Usually, you'd want to know what $g$ is. If $g/4$ is the thing you actually care about, then you'd leave it alone. Or, if $g/2$ was what you were interested in, you might just multiply by $2$ instead.

What manipulations you do to an equation depends on what you are trying to learn from it.

Answer 4

If $g$ is the price of a pizza, then your equation could describe the cost of a quarter of that pizza (a slice).

That slice costs $3$ dollars and $20$ cents.

How much does a whole pizza cost? (It's an expensive 100% Italian bufalina)