# Diophantus of Alexandria/Arithmetica/Book 1/Problem 15

## Problem

Two numbers are such that:
if the first receives $30$ from the second, they are in the ratio $2 : 1$
if the second receives $50$ from the first, they are in the ratio $1 : 3$.

What are these numbers?

## Solution

The first number is $98$.
The second number is $94$.

## Proof

Let $x$ and $y$ be the numbers requested.

Then we have:

 $\ds x + 30$ $=$ $\ds 2 \paren {y - 30}$ $\ds 3 \paren {x - 50}$ $=$ $\ds y + 50$ $\text {(1)}: \quad$ $\ds \leadsto \ \$ $\ds x + 90$ $=$ $\ds 2 y$ rearranging $\text {(2)}: \quad$ $\ds 3 x$ $=$ $\ds y + 200$ $\text {(3)}: \quad$ $\ds \leadsto \ \$ $\ds 3 x$ $=$ $\ds 6 y - 270$ $3 \times (1)$ and rearranging $\ds \leadsto \ \$ $\ds 0$ $=$ $\ds 5 y - 470$ $(3) - (2)$ $\ds \leadsto \ \$ $\ds y$ $=$ $\ds 94$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds 2 \times 94 - 90$ substituting for $y$ in $(1)$ and rearranging $\ds$ $=$ $\ds 98$

$\blacksquare$