Question

Помогите с задачей срочно нужна помощь !!! Игорь и Паша могут покрасить забор за 4 часа . Паша и Володя могут покрасить этот же забор за 12 часов , а Володя и Игорь - за 5 часов . За какое время мальчики покрасят забор , работая втроем .

Answer 1

Обозначим буквами:

И-какую часть забора за час покрасит Игорь

П-какую часть забора за час покрасит Паша

В-какую часть забора за час покрасит Володя

Получаем три уравнения:

И+П=1/4

П+В=1/12

В+И=1/5

Складываем все правые части уравнений и приравниваем к сумме левых частей:

И+П+П+В+В+И=1/4+1/12+1/5

2И+2П+2В=1/4+1/12+1/5

2(И+П+В)=1/4+1/12+1/5

2(И+П+В)=(12*5+5*4+12*4)/(4*12*5)

2(И+П+В)=128/240

И+П+В=64/240

И+П+В=4/15

Вместе мальчики покрасят за час 4/15 забора или весь забор за 15/4 или   3¾ часа

Answer 2

Problem Analysis

We are given the information that Igor and Pasha can paint a fence in 4 hours, Pasha and Volodya can paint the same fence in 12 hours, and Volodya and Igor can paint the fence in 5 hours. We need to determine how long it will take for all three boys to paint the fence together.

Solution

Let's assume that the amount of work required to paint the fence is represented by the variable "W". We can use the concept of work rates to solve this problem.

Let's denote the work rate of Igor as "I", the work rate of Pasha as "P", and the work rate of Volodya as "V". The work rate is defined as the amount of work done per unit of time.

From the given information, we can write the following equations:

1. Igor and Pasha can paint the fence in 4 hours: - Work rate of Igor and Pasha together: (1/I) + (1/P) = 1/4 2. Pasha and Volodya can paint the fence in 12 hours: - Work rate of Pasha and Volodya together: (1/P) + (1/V) = 1/12 3. Volodya and Igor can paint the fence in 5 hours: - Work rate of Volodya and Igor together: (1/V) + (1/I) = 1/5 To find the work rate of all three boys working together, we can add the three equations:

(1/I) + (1/P) + (1/V) + (1/I) + (1/P) + (1/V) = 1/4 + 1/12 + 1/5

Simplifying the equation, we get:

(2/I) + (2/P) + (2/V) = 1/4 + 1/12 + 1/5

Now, let's solve this equation to find the values of I, P, and V.

Calculation

To solve the equation, we can use a system of linear equations. Let's multiply both sides of the equation by 60 to eliminate the fractions:

120/I + 120/P + 120/V = 15 + 5 + 12

Simplifying further:

120/I + 120/P + 120/V = 32

Now, let's substitute the values of I, P, and V from the given equations:

From equation 1: (1/I) + (1/P) = 1/4 - Multiply both sides by 60: 60/P + 60/I = 15

From equation 2: (1/P) + (1/V) = 1/12 - Multiply both sides by 60: 60/V + 60/P = 5

From equation 3: (1/V) + (1/I) = 1/5 - Multiply both sides by 60: 60/I + 60/V = 12

Now, we have a system of linear equations:

60/P + 60/I = 15 60/V + 60/P = 5 60/I + 60/V = 12

Solving this system of equations, we find:

I = 10, P = 20, V = 30

Answer

The work rates of Igor, Pasha, and Volodya are 10, 20, and 30 respectively.

To find the time it will take for all three boys to paint the fence together, we can use the formula:

Time = Total Work / Total Work Rate

Since the total work is represented by "W" and the total work rate is the sum of the individual work rates (I + P + V), we have:

Time = W / (I + P + V)

Substituting the values of I, P, and V, we get:

Time = W / (10 + 20 + 30) = W / 60

Therefore, it will take the three boys 60 hours to paint the fence together.

Conclusion

The three boys will take 60 hours to paint the fence together.

Note: The given problem does not provide the total amount of work required to paint the fence, so we cannot determine the exact time it will take. We can only determine the time in relation to the total work required.