Dear Uncle Colin,

How can I tell whether $\frac{221}{391}$ and $\frac{403}{713}$ are equivalent?

- Calculator Answer Not Considered Enough, LOL

Hi, CANCEL, and thanks for your message!

There’s a naive way to do it and a clever way.

Let’s do it naively

The naive way is to see whether $\frac{221}{391} - \frac{403}{713} = 0$.

Turning it into a single fraction gives $\frac{221 \times 713 - 391 \times 403}{391 \times 713}$. Is that top equal to zero? With a bit of long multiplication, it turns out that $221 \times 713 = 157,573$ and so does $391 \times 403$.

A ninja!

“$(400-9) \times (400+3) = 160,000 - 6\times 400 - 27 = 157,573$”

“And the other one?”

“Got to dash. People to see… off.”

But that’s a bit unsatisfying - not to say long-winded.

Let’s do it cleverly

I can see immediately that 221 and 391 factorise – they’re both close to square numbers and they’re both off by a square number!

  • 221 is $15^2 - 2^2$, so it’s $13 \times 17$.
  • 391 is $20^2 - 3^2$, so it’s $23 \times 17$.

The first fraction is $\frac{13}{23}$.

That can inform our second fraction, too: is $403$ a multiple of 13? Yes, 31 times. And 713 is $31 \times 23$, so the second fraction is also $\frac{13}{23}$.

The two are equivalent! Hope that helps,

- Uncle Colin