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

\(\ds \)

\(=\)

\(\ds 98\)

$\blacksquare$

Sources

• c. 250: Diophantus of Alexandria: Arithmetica: Book $\text I$: Problem $15$
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The First Pure Number Puzzles: $24$