Henry Ernest Dudeney/Puzzles and Curious Problems/21 - The Cost of a Suit/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $21$

The Cost of a Suit
Melville bought a suit.
The jacket cost as much as the trousers and waistcoat.
The jacket and two pairs of trousers would cost $\pounds 7, 17 \shillings 6 \oldpence$
The trousers and two waistcoats would cost $\pounds 4, 10 \shillings$
Can you tell me the cost of the suit?


Solution

The jacket costs $\pounds 3, 7 \shillings 6 \oldpence$

The trousers cost $\pounds 2, 5 \shillings$

The waistcoat costs $\pounds 1, 2 \shillings 6 \oldpence$

In total: $\pounds 6, 15 \shillings$


Proof

Let us do all this in shillings.

Hence:

\(\ds \pounds 7, 17 \shillings 6 \oldpence\) \(=\) \(\ds 157 \tfrac 1 2 \shillings\)
\(\ds \pounds 4, 10 \shillings\) \(=\) \(\ds 90 \shillings\)

Then we have:

\(\text {(1)}: \quad\) \(\ds J\) \(=\) \(\ds T + W\) The jacket cost as much as the trousers and waistcoat.
\(\text {(2)}: \quad\) \(\ds J + 2 T\) \(=\) \(\ds 157 \tfrac 1 2\) The jacket and two pairs of trousers would cost $\pounds 7, 17 \shillings 6 \oldpence$
\(\text {(3)}: \quad\) \(\ds T + 2 W\) \(=\) \(\ds 90\) The trousers and two waistcoats would cost $\pounds 4, 10 \shillings$
\(\ds \leadsto \ \ \) \(\ds T + W + 2 T\) \(=\) \(\ds 157 \tfrac 1 2\) substituting for $J$ from $(1)$ into $(2)$
\(\text {(4)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 6 T + 2 W\) \(=\) \(\ds 315\) simplifying and clearing the fraction
\(\ds \leadsto \ \ \) \(\ds 5 T\) \(=\) \(\ds 225\) $(4) - (3)$
\(\ds \leadsto \ \ \) \(\ds T\) \(=\) \(\ds 45\) simplifying
\(\ds \leadsto \ \ \) \(\ds 45 + 2 W\) \(=\) \(\ds 90\) substituting into $(2)$
\(\ds \leadsto \ \ \) \(\ds W\) \(=\) \(\ds 22 \tfrac 1 2\) simplifying
\(\ds \leadsto \ \ \) \(\ds J\) \(=\) \(\ds 45 + 22 \tfrac 1 2\) substituting into $(1)$
\(\ds \) \(=\) \(\ds 67 \tfrac 1 2\) simplifying

where $J$, $T$ and $W$ are used in the obvious way.

The given solution shows these prices converted back into pounds, shillings and pence.

$\blacksquare$


Sources

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