Henry Ernest Dudeney/Puzzles and Curious Problems/158 - Newsboys/Solution

by : $158$

 * Newsboys
 * A contest took place amongst some newspaper boys.
 * Tom Smith sold one paper more than a quarter of the whole lot they had secured;
 * Billy Jones disposed of one paper more than a quarter of the remainder;
 * Ned Smith sold one paper more than a quarter of what was left;
 * and Charlie Jones disposed of just one paper more than a quarter of the remainder.
 * At this stage it was found that the Smiths were exactly $100$ papers ahead,
 * but little Jimmy Jones, the youngest kid of the bunch, sold all that were left,
 * so that in this friendly encounter the Joneses won by how many papers do you think?

Solution
The Jones team won, by $620$ papers to $400$.

Proof
Let $n$ be the total number of papers sold

Let $S_T$, $J_B$, $S_N$, $J_C$ and $J_J$ be the number of papers sold by Tom Smith, Billy Jones, Ned Smith, Charlie Jones and Jimmy Jones respectively.

We have:

These can be expressed more usefully as:

This set of simultaneous linear equations can be expressed conveniently in matrix form as:


 * $\begin {pmatrix}

-1 & 0 &  0 &  0 &  0 &  4 \\ -1 &  0 &  0 &  0 &  4 &  1 \\ -1 &  0 &  0 &  4 &  1 &  1 \\ -1 &  0 &  4 &  1 &  1 &  1 \\ -1 &  1 &  1 &  1 &  1 &  1 \\ 0 &  0 & -1 &  1 & -1 &  1 \\ \end {pmatrix} \begin {pmatrix} n \\ J_J \\ J_C \\ S_N \\ J_B \\ S_T \end {pmatrix} = \begin {pmatrix} 4 \\ 4 \\ 4 \\ 4 \\ 0 \\ 100 \end {pmatrix}$

It remains to solve this matrix equation.

In reduced echelon form, this gives:

The result can be read off directly.

There were $1020$ newspapers.

The Smith brothers sold $256$ and $144$ between them, making $400$.

The two older Jones brothers sold $192$ and $108$ between them, making $300$.

But then Jimmy Jones sold the remaining $320$ and the Jones team won by a margin of $220$.