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

Puzzles and Curious Problems by Henry Ernest Dudeney: $10$

If a tobacconist offers a cigar at $7 \tfrac 3 4 \oldpence$,

but says we can have the box of $100$ for $65 \shillings$,

shall we save much by buying the box?

In other words, what would $100$ at $7 \tfrac 3 4 \oldpence$ cost?

By a little rule that we shall give the calculation takes only a few moments.

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 $100$.

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:

$100 n \oldpence = {8 n \shillings} + {4 n \oldpence}$

$\blacksquare$