Henry Ernest Dudeney/Puzzles and Curious Problems/154 - Marching an Army/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $154$

Marching an Army
A body of soldiers was marching in regular column, with $5$ men more in depth than in front.
When the enemy came in sight the front was increased by $845$ men,
and the whole was drawn up in $5$ lines.
How many men were there in all?


Solution

There were $4550$ men in the army.


Proof

Let $m$ be the total number of men.

Let $n$ be the number of men in front while marching.

We have:

\(\text {(1)}: \quad\) \(\ds m\) \(=\) \(\ds n \paren {n + 5}\) $5$ men more in depth than in front
\(\text {(2)}: \quad\) \(\ds m\) \(=\) \(\ds 5 \paren {n + 845}\) the front was increased by $845$ men ... in $5$ lines.
\(\ds \leadsto \ \ \) \(\ds n \paren {n + 5}\) \(=\) \(\ds 5 \paren {n + 845}\) eliminating $m$
\(\ds \leadsto \ \ \) \(\ds n^2\) \(=\) \(\ds 4225\) simplifying
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 65\)
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds 65 \times 70 = 4550\)

$\blacksquare$


Sources

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