Diophantus of Alexandria/Arithmetica/Book 1/Problem 1/100, 40

Problem

To divide a given number into two having a given difference.

Given number $100$, given difference $40$.

The required numbers are $30$ and $70$.

Let $N$ be the given number.

Let $d$ be the given difference.

Let $x$ be the smaller of the two numbers into which $N$ is to be divided.

From the exposition of Problem $1$:

$\text {(1)}: \quad$

$\ds x$

$=$

$\ds \dfrac {N - d} 2$

Here we have:

$\ds N$

$=$

$\ds 100$

$\ds d$

$=$

$\ds 40$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds \dfrac {100 - 40} 2$

substituting for $N$ and $d$ from $(1)$

$\ds $

$=$

$\ds 30$

simplifying

$\blacksquare$