# Money bag

Four peasants returned from the city lamenting how little they had.

• One of them said: If I found a bag of money I would take only the third part, the rest, together with the bag, I would give it to you.
• Another farmer commented: I would divide it among all of us equally.
• The third said: I would be satisfied with only one fifth of the money.
• And the last one said: It would be enough for me with the sixth part.

Suddenly, as if by magic, they find a bag full of money along the way and decide to distribute the money as stipulated. The first peasant would take the third part, the second corresponds a quarter to the third a fifth and the last one the sixth.

In the bag there were 8 bills: one of three rubles and the other of one ruble, five rubles and ten rubles. None of the farmers could take exactly their share without changing the tickets, so they decided to wait to see if they could make a change with a traveler.

Soon, a rider approaches and the peasants stop him to explain the problem and ask for change.

The rider told them: A ruble I can not change, but give me the bag with the money, I will put in it my ruble, of all the money I will give each one his share and I will keep the bag.

The farmers agreed so the rider collected all the money, gave the first farmer 1/3 of the total, the second 1/4, the third 1/5 and the quarter 1/6 of all the money and stayed with the bag.

The rider thanked them and left. The peasants were then thoughtful because despite having contributed a ruble to the bag and distribute all the money as they had agreed the rider left happy.

They saw that they had the same number of bills that were initially in the bag however, nobody had the 3-ruble bill.

• I received more than my share, said the first farmer.
• Me too, said the second, I received 25 more kopeks.

All received more money than their share and yet the rider took the three-ruble bill.

How much money did the farmers find? Y What tickets did you give to each of the peasants?

(The Kopek is a monetary unit that is equivalent to one hundredth of the ruble.)

#### Solution

The peasants did not know how to add broken ones correctly. If we add all the parts in which the peasants wanted to divide the money:

We have: that is less than the total of the bag. It turns out that all of them together wanted to receive less money than they had found. The money found, along with the ruble added by the rider, was divided into 60 parts. Of them 57/60 parts were left by the peasants and the rider was left with 3/60 parts.

We know that the rider was left with 3 rubles. Therefore, 3/60 = 1/20 of the total is 3 rubles then the total amount of money was 3 x 20 = 60 rubles.

The first farmer received 1/4 part, that is, 15 rubles, but if the rider did not put his ruble, he should receive 25 kopeks less, ie 14 rubles and 75 kopeks, which corresponds to 1/4 part of the money found. From the above it is confirmed that 14 rubles were found 75 kopeks x 4 = 59 rubles and 60 rubles resulted in the rider's money.

The rider, put in the bag 1 ruble and took 3. He won 2 rubles making a smart deal.

In the bag there were then five 10-ruble bills, one 5, one 3 and one 1 ruble bill. The rider gave the first farmer 20 rubles on two 10 bills; 15 rubles a second on a ticket of 10 and another of 5; 12 rubles to the third in a 10-ruble ticket and two of 1 ruble and the last farmer handed a 10-ruble bill. He kept the three ruble bill.