Henry Ernest Dudeney/Puzzles and Curious Problems/10 - Mental Arithmetic/Solution

by : $10$

 * Mental Arithmetic

Solution
Buying a box of $100$ is actually $5 \oldpence$ more than buying the cigars individually.

One presumes that the box itself must have an intrinsic worth of $5 \oldpence$.

The rule for calculating the price of $100$ of something costing $n \oldpence$ is:
 * reduce $n \oldpence$ to farthings to get $4 n$ farthings
 * double this amount to get $8 n$.
 * add $8 n \shillings$ to $4 n \oldpence$ to get the price of $100$.

Proof
We have:
 * $100 \times 7 \tfrac 3 4 = 700 + 3 \times 25 = 775$

But:
 * $775 \oldpence = 64 \shillings 7 \oldpence$

which is $5 \oldpence$ less than buying the whole box.

One presumes that the box itself may have an intrinsic worth of $5 \oldpence$.

Let $n$ be the number of (old) pennies an item costs.

Hence its cost in farthings is $4 n$.

Then $100$ of them cost $100 n \oldpence$

This is $400 n$ farthings.

This is:
 * $\dfrac {400 n} {48} \shillings = {\dfrac {384 n} {48} \shillings} + 16 n \times {\tfrac 1 4 \oldpence}$

That is:
 * $100 n \oldpence = {8 n \shillings} + {4 n \oldpence}$