Problem of missing dollar

Once upon a time there were two friends Ameya and Saransh who were trying to make some money by selling oranges everyday. Orange lelo, lelo lelo. The way they sell them was like this : Ameya sell them at a price of 2 oranges for a dollar while Saransh sell them at a price of 3 oranges for a dollar. Everyday they had a stock of 30 oranges and in this way Ameya make 15$ while Saransh make only 10$. Daily they spent this price together for eating in a restaurant where their diet was fixed and so everyday they need 25$. One day Ameya got sick and he asked Saransh to sell his oranges as well, so they decided that since Ameya sell his oranges for 2 oranges a dollar and Saransh sell his oranges for 3 oranges a dollar, Saransh will have to sell 5 oranges for 2$ but after selling all the oranges Saransh got only 24$ as there will be 12 sets of 5 oranges, each set costing 2$, hence the total cost for 12 sets will be 24$ (12x2). Now Ameya and Saransh cannot eat outside and your job is to find what happened wrong in the process and what can be done to prevent this from happening in future.

Solution :

The solution is very simple, consider both Ameya and Saransh separately selling their oranges, now Ameya will have have 15 sets (30/2) and Saransh will have 10 sets (30/3).

Now each set of Saransh corresponds to a dollar and similarly each set of Ameya corresponds to a dollar, now imagine they are selling the oranges together, Ameya take his set and Saransh also take his set and they handover the whole complete set for 2$ but there is a problem in this, look Saransh only have 10 sets so by the time when all sets of Saransh will be over there still be 5 sets of oranges(hence 10 oranges) left with Ameya, now when Saransh is selling the oranges he still thinks after 10 sets, the cost of remaining 10 oranges is 2$ for 5 oranges thus he decreases the value of Ameya oranges and hence he only get 4$(2x2) because he doesn't know that the left over sets belong to purely Ameya hence the cost of left over oranges must be 2 oranges for 1$ thereby Ameya will get 5$ for those 10 oranges, So here you can see the difference of 1$.

Here is the table, how it goes :

Selling Individual:

Selling together :

Well that was an intuitive way to realize the problem and its solution. There is more formal way to solve the problem here comes the mathematics :

Case 1 : Separate selling

a) Ameya

Total oranges = 30

Cost of 2 oranges = 1$

-> Cost of 1 orange = 0.5$

-> Cost of 30 oranges = 0.5*30 = 15$

b) Saransh

Total oranges = 30

Cost of 3 oranges = 1$

-> Cost of 1 orange = 1/3$

-> Cost of 30 oranges = 1/3$ * 30 = 10$

Hence, the total cost will be 25$.

Average cost for 1 orange = (0.5 + 1/3) /2 = 5/12 = 0.4167 $

Case 2 : Selling mixed oranges

Total oranges = 60

Cost of 5 oranges = 2$

-> Cost of 1 orange = 0.4$

-> Cost of 60 oranges = 0.4$ * 60 = 24$

Hence, in this case we get 24$.

Average cost for 1 orange = 2/5 = 0.4 $

Difference in the average cost of one orange = 0.0167 $ (or 5/12 - 2/5 = 1/60 $) and hence for 60 oranges this difference corresponds to 1/60 x 60 = 1$, hence we can see the 1$ difference.

Now how to avoid the problem in future? One thing that can be done is that after selling of 10 sets i.e 50 oranges and earning 20$ (each set for 2$), the remaining 10 oranges should be sold for 2 oranges for a dollar in order to get remaining 5$.

Thanks for reading folks, have a nice day!