# Diophantus of Alexandria/Arithmetica/Book 1/Problem 8

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## Problem

What number must be added to $100$ and to $20$ (the same number added to each) so that the sums are in the ratio $3 : 1$?

## Solution

$20$

## Proof

Let $x$ be the number added.

Then we have:

 $\ds 100 + x$ $=$ $\ds 3 \paren {20 + x}$ $\ds \leadsto \ \$ $\ds 100 - 60$ $=$ $\ds 2 x$ rearranging $\ds \leadsto \ \$ $\ds x$ $=$ $\ds 20$ simplifying

$\blacksquare$