Henry Ernest Dudeney/Puzzles and Curious Problems/12 - Reductions in Price/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $12$

"I have often been mystified," said Colonel Crackham, "at the startling reductions some people make in their prices,

and wondered on what principles they went to work.

For example, a man offered me a motor-car two years ago for $\pounds 512$;

a year later his price was $\pounds 320$;

a little while after he asked a level $\pounds 200$;

and last week he was willing to sell for $\pounds 125$.

The next time he reduces I shall buy.

At what price shall I purchase if he makes a consistent reduction?"

$\pounds 78 \tfrac 1 8$, or $\pounds 78, 2 \shillings 6 \oldpence$

The numbers:

$512, 320, 200, 125$ form a geometric sequence whose common ratio is $\dfrac 5 8$.

Hence the next reduction is expected to be $125 \times \dfrac 5 8 = \dfrac {625} 8 = 78 \tfrac 1 8$.

$\blacksquare$