Diophantus of Alexandria/Arithmetica/Book 1/Problem 15

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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
\(\ds \) \(=\) \(\ds 98\)

$\blacksquare$


Sources