Multiples of Terms in Equal Ratios/Modern Proof

Theorem

Let $a, b, c, d$ be quantities.

Let $a : b = c : d$ where $a : b$ denotes the ratio between $a$ and $b$.

Then for any numbers $m$ and $n$:

$m a : n b = m c : n d$

In the words of Euclid:

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.

(The Elements: Book $\text{V}$: Proposition $4$)

Proof

From the definition of a ratio, we have:

\(\ds a : b\)

\(=\)

\(\ds c : d\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac a b\)

\(=\)

\(\ds \frac c d\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac m n \frac a b\)

\(=\)

\(\ds \frac m n \frac c d\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac {m a} {n b}\)

\(=\)

\(\ds \frac {m c} {n d}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds m a : n b\)

\(=\)

\(\ds m c : n d\)

$\blacksquare$