# The intermediary of the fruit

An intermediary buys fruit from farmers and then sells it to market stalls. He receives the orders of the merchants and looks for the farmer to sell it to him so that he earns 4% of the total value of each transaction. Charges 2% commission to the farmer and another 2% to the merchant.

However, he realized that there was the possibility of improving the benefit by trickling the scales of his balance in such a way that there was a difference of 100g for every real kilo in the weighing. Thus, when buying from the farmer the balance marked 100g less for each real heavy kilo and when selling to the merchant, modifying the arms of the balance, it weighed 100g more for each kilo of real merchandise.

In a certain transaction he won € 37 in total,

How much did the farmer pay for the merchandise?

#### Solution

For every kilo of real merchandise that the intermediary buys, only the cost of 0.9Kg is paid to the farmer, so the theoretical amount purchased is increased by 100/9%, which is approximately 11.11%. (from 0.9 kg to 1 kg). However, at the time of selling to the merchant, each real Kilo purchased is sold as if it were 1.1Kg, so the amount is now increased by an additional 10% (from 1Kg to 1.1Kg).

In the end, the total gain for the trick of the balance is 1.1Kg - 0.9Kg = 0.2Kg of merchandise for every real kilo, so it achieves the equivalent of the cost of 200/9% of free merchandise in each transaction

The intermediary, when calculating the 2% commission that corresponds to pay the farmer, must do so on the amount that the farmer believes he has sold or is about 0.9Kg and on the other hand, for every kilo of real merchandise he sells the intermediary to the merchant, he believes that he is buying 1.1Kg and pays 2% commission on this amount although he only receives 1Kg.

If we define the following variables:
P = price per kilo of fruit
Q = OFFICIAL KILS (FALSE) paid to the farmer
Q1 = REAL Kilos bought from the farmer
Q2 = REAL Kilos sold to the merchant
Q3 = OFFICIAL Kilos (FALSE) sold to the merchant

We have that all these quantities can be expressed from Q:
Q1 = Q + Q * (100/9) / 100 = Q * (1 + 1/9) = Q * 10/9
Q2 = Q1 = Q * 10/9
Q3 = Q2 + Q2 * 10/100 = Q2 (1 + 1/10) = Q2 (11/10) = (Q * 10/9) * (11/10) = Q * 11/9

The price paid to the farmer will be PQ.

The total gain, which we know is 37, will be the sum of the following concepts:
- 2% paid by the farmer = PQ * 2/100
- The cost of the difference between Official Kilos bought from the farmer minus the Official Kilos sold to the merchant = P (Q * 11/9 - Q) = PQ * 2/9 (200/9% of PQ)
- 2% paid by the merchant = P (Q * 11/9) * 2/100 = PQ * 22/900

Then the total gain can be expressed as follows:
PQ * 2/100 + PQ * 2/9 + PQ * 22/900 = PQ (2/100 + 2/9 + 22/900) = 37

Where can we get the price paid to the peasant that is PQ = 37 / (2/100 + 2/9 + 22/900) = 37 / (24/90) = 150

The price paid to the peasant was 150.