DAY 74

Comment

It is tempting to think the two percentage changes cancel each other out, but they don't, quite. (This becomes more obvious if you take an extreme example: say Ken charges 99% more and sells 99% fewer.) (On the other hand, if Ken charges, say, twice as much but sells only half as many, the two changes do cancel each other out ... .)

One approach is to consider a specific case.
For example, imagine Linda charges 100p per kilo of apples and sells 100 kilos (so she takes £100).
How much would Ken charge and how many kilos would he sell?

Answer

Ken's takings are (1 %) less than Linda's.

Readers' methods

(We have had an excellent response to our flyer asking for methods for this item.
This selection gives an idea of the range of methods received):

Danny J, Powys:
Ken's price is 110% of Linda's but he sells only 90% as much.
90% of 110% = 0.9 x 110% = 99%.
So Ken's takings are only 99% of Linda's.

David B, Qatar:
If Linda sells L at M, Ken sells 0.9L x 1.1M = 0.99 LM.
Therefore Ken makes less than Linda.

Chantelle C, St Mark's School:
Say Linda sells 50 apples at 20p each: 50 x £0.20 = £10
Then Ken sells 45 apples at 22p each: 45 x £0.22 = £9.90

Liz P, High Wycombe HP15:
If Linda sells 100 apples at c pence each,
then Ken sells 90 apples at 1.1c pence each.
Linda's takings: 100c pence.
Ken's takings: 90 x 1.1c pence, which is 99c pence.

Derek P, Hertford SG13:
Generalisation:
Ken charges 100x % more than Linda and sells 100x % less (x = 0.1 in original case).
Ken's takings = (1 + x)(1 ­ x) times Linda's takings = (1 ­ x^2) times Linda's takings.
So Ken's takings are less than Linda's, whether x is positive or negative.

A Davis, Bristol BS6:
Cost: K/L = 110/100.
Quantity: K/L = 90/100.
So Takings: K/L = 99/100. so Ken loses by 1 %.

Phil S, Staines TW18:
Say Linda sells at 100x pence and sells 100y apples.
Then Ken sells at 110x pence but only sells 90y apples.
Linda's takings are 100x * 100y = 10000xy.
Ken's takings are 110x * 90y = 9900xy.

Tom A, Leicester LE3:
Suppose ...
100 apples at 10p each makes £10 for Linda, but
90 apples at 11p each makes £9.90 for Ken.

Preety P, Reading:
If Linda charges £x and sells y, her takings will be £xy.
Ken charges 10 % more which is x + x/10 = 11x/10.
He sells 10 % fewer which is y ­ y/10 = 9y/10.
Therefore Ken's takings are £(11/10)(9/10)xy = £99xy/100.

William WW, Birmingham B13:
Suppose, in general, that Ken charges X % more and sells X % less.
Write x for X/100.
Then K's takings as a fraction of L's = (1 + x)(1 ­ x) = 1 ­ x^2.
Since x^2 is positive, this is less than 1, so K's takings are smaller.

Sophie & Chantelle, Hounslow TW3:
If Linda sells apples at 20p each and if she sells 50 it gives you a total of £10.
Ken sells his apples at 22p each and 45 of these would come to £9.90.
Ken's takings are 10p less than Linda's.

Mrs Webster, Halifax HX4:
Say that Linda sells 1 apple for £1.00, so Ken sells 1 for £1.10.
If Linda sells 100 apples, then Ken only sells 90.
Then Linda would make £100 and Ken would only make £99.