# Covent Garden Problem Puzzle - Solution

### The Puzzle:

Here is a puzzle known as the Covent Garden Problem, which appeared in London half a century ago, accompanied by the somewhat surprising assertion that it had mystified the best mathematicians of England:

Mrs. Smith and Mrs. Jones had equal number of apples but Mrs. Jones had larger fruits and was selling hers at the rate of two for a penny, while Mrs. Smith sold three of hers for a penny.

Mrs. Smith was for some reason called away and asked Mrs. Jones to dispose of her stock. Upon accepting the responsibility of disposing her friend's stock, Mrs. Jones mixed them together and sold them of at the rate of five apples for two pence.

When Mrs. Smith returned the next day the apples had all been disposed of, but when they came to divide the proceeds they found that they were just seven pence short, and it is this shortage in the apple or financial market which has disturbed the mathematical equilibrium for such a long period.

Supposing that they divided the money equally, each taking one-half, the problem is to tell just how much money Mrs. Jones lost by the unfortunate partnership?

Mrs. Smith and Mrs. Jones had equal number of apples but Mrs. Jones had larger fruits and was selling hers at the rate of two for a penny, while Mrs. Smith sold three of hers for a penny.

Mrs. Smith was for some reason called away and asked Mrs. Jones to dispose of her stock. Upon accepting the responsibility of disposing her friend's stock, Mrs. Jones mixed them together and sold them of at the rate of five apples for two pence.

When Mrs. Smith returned the next day the apples had all been disposed of, but when they came to divide the proceeds they found that they were just seven pence short, and it is this shortage in the apple or financial market which has disturbed the mathematical equilibrium for such a long period.

Supposing that they divided the money equally, each taking one-half, the problem is to tell just how much money Mrs. Jones lost by the unfortunate partnership?

### Our Solution:

The mixed apples were sold of at the rate of five apples for two pence. So they must have had a multiple of five i.e. 5, 10, 15, 20, 25, 30, ..., 60, 65, ... etc apples.

But the minimum number of apples they could have together is 60; so that 30 would have been of Mrs. Smith's that would fetch her 10 (an integer) pence and the other 30 of Mrs. Jones's that would fetch her 15 (also an integer) pence.

When sold separately it would fetch them 10+15=25 pence altogether. But when sold together it would fetch them 60X2/5=24 pence i.e. a loss of one (25-24=1) pence.

Since they lost 7 pence altogether; they had altogether 60X7=420 apples that fetched them only 420X2/5=168 pence and they shared 84 pence each of them. But Mrs. Jones could sell her 420/2=210 apples for 210/2=105 pence so she lost "21 pence".

Note: to solve it algebraically:

They lost 7 pence altogether

Suppose each lady has x apples

x/2 + x/3 - 2(2x/5) = 7

15x + 10x - 24x = 210

x = 210

Note: Mrs. Johns lost 21 pence.

But without working Mrs. Smith earned 14 extra pence!

(84 pence − 210/3 pence = 14 pence).

Not very fair!

(Perhaps Mrs. Johns was not very good at math)

But the minimum number of apples they could have together is 60; so that 30 would have been of Mrs. Smith's that would fetch her 10 (an integer) pence and the other 30 of Mrs. Jones's that would fetch her 15 (also an integer) pence.

When sold separately it would fetch them 10+15=25 pence altogether. But when sold together it would fetch them 60X2/5=24 pence i.e. a loss of one (25-24=1) pence.

Since they lost 7 pence altogether; they had altogether 60X7=420 apples that fetched them only 420X2/5=168 pence and they shared 84 pence each of them. But Mrs. Jones could sell her 420/2=210 apples for 210/2=105 pence so she lost "21 pence".

Note: to solve it algebraically:

They lost 7 pence altogether

Suppose each lady has x apples

x/2 + x/3 - 2(2x/5) = 7

15x + 10x - 24x = 210

x = 210

Note: Mrs. Johns lost 21 pence.

But without working Mrs. Smith earned 14 extra pence!

(84 pence − 210/3 pence = 14 pence).

Not very fair!

(Perhaps Mrs. Johns was not very good at math)

Puzzle Author: Loyd, Sam

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