Henry Ernest Dudeney/Puzzles and Curious Problems/10 - Mental Arithmetic/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $10$
- Mental Arithmetic
- 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.
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:
- $100 n \oldpence = {8 n \shillings} + {4 n \oldpence}$
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $10$. -- Mental Arithmetic