Question

Один купец прошел через три города, и взыскали с него в первом городе пошлины половину и треть имущества, во втором городе половину и треть [с того, что осталось], и в третьем городе снова взыскали половину и треть [с того, что у него было]; и когда он прибыл домой, у него осталось 1000 [денежных единиц]. Итак, узнай, сколько всего у него взыскали пошлины

Answer 1

Половина и треть - это пять шестых. Значит после каждого города у него осталась одна шестая от того, что было. До того, как купец посетил тот или иной город, у него было в шесть раз больше, чем после. В итоге количество денег сократилось в 6 в третьей степени раз. Т. е. в начале было 11 * 6 в третьей степени = 11 * 216 = 2376 денежков.

Answer 2

Problem Analysis

A merchant passed through three cities and was charged duties in each city. In the first city, he was charged half of his property plus one-third. In the second city, he was charged half of what remained plus one-third. In the third city, he was again charged half of what he had left plus one-third. When he arrived home, he had 1000 monetary units left. The task is to determine the total amount of duties he was charged.

Solution

To solve this problem, we need to calculate the total amount of duties charged in each city and then sum them up.

Let's assume the merchant's initial property is P monetary units.

In the first city, he was charged half of his property plus one-third. So, the amount charged in the first city is:

D1 = (1/2)P + (1/3)

After paying the duties in the first city, the merchant has (1/2)P + (2/3) monetary units left.

In the second city, he was charged half of what remained plus one-third. So, the amount charged in the second city is:

D2 = (1/2)((1/2)P + (2/3)) + (1/3)

After paying the duties in the second city, the merchant has (1/4)P + (5/6) monetary units left.

In the third city, he was charged half of what he had left plus one-third. So, the amount charged in the third city is:

D3 = (1/2)((1/4)P + (5/6)) + (1/3)

After paying the duties in the third city, the merchant has (1/8)P + (11/12) monetary units left.

According to the problem, when the merchant arrived home, he had 1000 monetary units left. So, we can set up the following equation:

(1/8)P + (11/12) = 1000

To find the value of P, we can solve this equation.

Calculation

Let's solve the equation to find the value of P:

(1/8)P + (11/12) = 1000

Multiplying both sides of the equation by 24 to eliminate the denominators:

3P + 22 = 24000

Subtracting 22 from both sides of the equation:

3P = 23978

Dividing both sides of the equation by 3:

P = 7992.67

So, the initial property of the merchant was approximately 7992.67 monetary units.

Total Duties Charged

Now that we know the initial property of the merchant, we can calculate the total amount of duties charged by summing up the amounts charged in each city:

Total Duties = D1 + D2 + D3

Substituting the values of D1, D2, and D3:

Total Duties = (1/2)P + (1/3) + (1/2)((1/2)P + (2/3)) + (1/3) + (1/2)((1/4)P + (5/6)) + (1/3)

Simplifying the expression:

Total Duties = (1/2)P + (1/3) + (1/4)P + (1/3) + (1/8)P + (1/2) + (1/3)

Total Duties = (15/8)P + (11/6)

Substituting the value of P:

Total Duties = (15/8)(7992.67) + (11/6)

Calculating the value:

Total Duties ≈ 14995.84

Therefore, the total amount of duties charged from the merchant is approximately 14995.84 monetary units.

Answer

The total amount of duties charged from the merchant is approximately 14995.84 monetary units.