Calculating the 4th number from 3 related numbers

Question

I was taught a long time ago this really useful trick to calculate an unknown value when you know 3 existing values that are related. An example would be something like:

If there are 21 peas in 3 pods how many are there in 17 pods?

This gives us 2 values that have a relation, the next value that fits a certain rule that would reveal the missing value.

I was taught to write it down in a kind of 3 out of 4 square, and you would multiply diagonally and divide up and down. But I can't quite remember the exact rule and if this little trick has a name.

Maybe this is a dumb question for an advanced math QA site, if so please tell me and I will find somewhere else to ask.

Answer 1

You might be referring to cross-multiplication.

Let's take your example. Assuming each pod has an equal number of peas, we have 21 peas in 3 pods, so in one pod we have $21/3 = 7$ peas. To find the number of peas in 17 pods, we can just multiply $7 \frac{\text{peas}}{\text{pod}}$ by $17$ pods to arrive at $17 \text{ pods} \times 7 \frac{\text{peas}}{\text{pod}} = 119 \text{ peas}.$

This is a much easier calculation when we use cross-multiplication. First we set up our proportion:

$$\frac{21 \text{ peas}}{3 \text{ pods}} = \frac{x \text{ peas}}{17 \text{ pods}}.$$

Here we have used the variable $x$ to represent our unknown: the number of peas in 17 pods. We now cross multiply (and drop our units for convenience): $$21 \times 17 = 3x.$$

Next, we can divide by 3 to isolate our variable $x$:

$$\frac{21 \times 17}{3} = x.$$

So $x = 119$, which means that there are $119$ peas in 17 pods.