# Where has Jones lost 1 Rs?

Jack and Jones are brothers and they both sell apples for living. Jack sells 2 apples for 1 Rs and Jones sells 3 apples for 1 Rs. They each have 30 apples. Hence, Jack earns 15 Rs a day while Jones earns 10 Rs a day. This gives a total of 25 Rs.

One day Jack got sick and couldn't sell his apples. He gave his apples to Jones. Jones sold all 60 apples at price of 5 apples for 2 Rs (as 3 apples for 1 Rs + 2 apples for 1 Rs). At the end of the day, he counted the money and ended with a total of 24 Rs.

Where has Jones lost 1 Rs?

TL,DR: It is all about operator precedence

We have the following unit price for Jack apples: $1/2$.
We have the following unit price for Jones apples: $1/3$.
Thus making the calculation of the total amount collected: $$1/2 \times 30 + 1/3 \times 30 = 15 + 10 = 25$$ In that configuration, the unit price for a single apple is: $$(1/2 + 1/3) / 2 = 5/12$$

With the combined sale, we have the following unit price: $2/5$
Thus making the calculation of the total amount collected: $$2/5 \times 60 = 24$$
So far so good.

When we are comparing unit price of a single apple now.
The problem occurs because of operator precedence, $(1/2 + 1/3)/2 = 5/12 \ne 2/5$.

• Good answer. I fixed up your formatting a little and hid the conclusion in a spoiler tag. – Ian MacDonald Apr 1 '15 at 13:09
• It's not operator precedence. He doesn't know how to add fractions... – Joe Apr 1 '15 at 17:36
• I would recommend adding something to answer the specific question: Where did the extra 1 Rs go? You have answered why it didn't work, but don't actually have an equation working out to 1 Rs. – Wolfman Joe Apr 1 '15 at 18:25

The math is all great, above, but it still decries the common sense response:

If I purchase two apples from the first vendor and three apples from the second vendor, I have purchased five apples for two Rs. Therefore, five apples for two Rs should work. This is, in fact, true if you do not have a limited number of apples.

But we do.

Let's say ten people each come by the two brother's stands, and each of these ten people purchased two apples from the first vendor, and three apples from the second vendor. At the end of this sequence, we have had 10 people purchase 50 apples, at five apples for two Rs each.

HOWEVER! The second vendor is now out of apples! Only the first vendor still has apples remaining. The next two people who want five apples each must purchase only at the first vendor's stand - and the first vendor is two apples for an Rs. Therefore, they will spent 2.5 Rs for five apples, instead of 2 Rs.

The first ten people spent 2 Rs each for a total of 20 Rs. The last two people spent 2.5 Rs each for a total of 5 Rs.

However, when all 60 apples are combined for a total price of 5 apples for 2 Rs, those last two people spent 2 Rs each for a total of 4 Rs. And THAT is where the missing Rs went.

• Yes exactly --- the crux of the issue is the limited quantity of apples that prevents Jones from actually selling 60 apples as stated. Finding this contradiction in the puzzle is the key, not the mathematics behind average price as stated in other answers. – Nathan Apr 1 '15 at 17:25
• @Nathan - the mathematics behind average price as stated in the other answers is also correct. It in fact says the same thing that I said, but in highly technical terms. For instance, if the first brother had 24 apples and the second brother had 36 apples, this would change the average price of the apples to (24 / 2 + 36 / 3) / 60 = (12 + 12) / 60 = 24 / 60 = 2 / 5. Both are key - one is a key understood to mathematicians and the other is a key understood in common language. – Wolfman Joe Apr 1 '15 at 17:36
• @WolfmanJoe: No, the average price calculation does not explain anything. It just repeats what is already observed -- that the income is less. – Ben Voigt Apr 1 '15 at 23:03
• I agree the math is correct, but that's not the point. Your answer points out why the price changes, whereas the other answers merely derive that the average price changes (which is already readily obvious). One can see from the math that the average price changed, but still be confused as to why it changed when it seems like it shouldn't have changed. I have a B.S. in Math, so while I do understand the math, I don't believe it really addresses the crux of the puzzle. Of course that's just my opinion; others are free to disagree. – Nathan Apr 2 '15 at 3:21

At the beginning, we have an average price per apple of $\frac{(30 \times \frac1 2)+(30 \times \frac1 3)} {60} =\frac {5}{12}=0,416666$

Then, he changed the price to $\frac 25=0,4$

So, as the average price changed, his wage changes as well!