Question

Николай и Юрий красит стену за 9 часов. Юрий Алексей этот же забор за 18 часов красят,а Алексей и Николай за 12 часов. За какое время мальчики покрасят забор, работая втроем?

Answer 1

Н + Ю = 9 часов
Ю + А = 18 часов
А + Н = 12 часов
Н + Ю + А - ?

Пусть единичка (1), это объем работы, которые нужно выполнить за 1 час работы, тогда:

1) 1 : 9 = $\frac{1}{9}$ (заб./час) - производительность Николая и Юрия.
2) 1 : 18 = $\frac{1}{18}$ (заб./час) - производительность Юрия и Алексея
3) 1 : 12 = $\frac{1}{12}$ (заб./час) - производительность Алексея и Николая

Из полученных мною результатов, мы должны получить совместную производительность ребят:

1) ($\frac{1}{9}$ + $\frac{1}{18}$ + $\frac{1}{12}$) : 2  = $\frac{1}{8}$ (заб./час) - совместная производительность мальчиков.

2) 1 : $\frac{1}{8}$ = 8 (часов) - потребуется на покраску забора ребятам вместе.

Ответ: 8 часов

Answer 2

Problem Analysis

We are given the information that Nikolai and Yuri can paint a wall in 9 hours, Yuri and Alexey can paint the same wall in 18 hours, and Alexey and Nikolai can paint the wall in 12 hours. We need to determine how long it will take for all three boys to paint the wall together.

Solution

To solve this problem, we can use the concept of work rates. The work rate is defined as the amount of work done per unit of time. If we let the work rate of Nikolai be N, the work rate of Yuri be Y, and the work rate of Alexey be A, then we can set up the following equations:

1. Nikolai and Yuri can paint the wall in 9 hours, so their combined work rate is 1/9 of the wall per hour: N + Y = 1/9. 2. Yuri and Alexey can paint the wall in 18 hours, so their combined work rate is 1/18 of the wall per hour: Y + A = 1/18. 3. Alexey and Nikolai can paint the wall in 12 hours, so their combined work rate is 1/12 of the wall per hour: A + N = 1/12.

We can solve this system of equations to find the individual work rates of each boy. Once we have the individual work rates, we can determine how long it will take for all three boys to paint the wall together.

Solution Steps

1. Solve the system of equations to find the individual work rates of each boy. 2. Add the individual work rates to find the combined work rate of all three boys. 3. Use the combined work rate to determine how long it will take for all three boys to paint the wall together.

Solution Details

Let's solve the system of equations to find the individual work rates of each boy.

From equation 1, we have N + Y = 1/9. (Source

From equation 2, we have Y + A = 1/18. (Source

From equation 3, we have A + N = 1/12. (Source

To solve this system of equations, we can use substitution or elimination. Let's use elimination:

Adding equation 1 and equation 3, we get 2N + 2Y + 2A = 1/9 + 1/12.

Simplifying the right side, we have 2N + 2Y + 2A = 4/36 + 3/36 = 7/36.

Dividing both sides by 2, we get N + Y + A = 7/72.

Now, let's subtract equation 2 from this new equation:

(N + Y + A) - (Y + A) = 7/72 - 1/18.

Simplifying the right side, we have N = 7/72 - 4/72 = 3/72 = 1/24.

Now that we have the value of N, we can substitute it back into equation 1 to find the value of Y:

1/24 + Y = 1/9.

Subtracting 1/24 from both sides, we get Y = 1/9 - 1/24 = 8/72 - 3/72 = 5/72.

Finally, we can substitute the values of N and Y back into equation 2 to find the value of A:

5/72 + A = 1/18.

Subtracting 5/72 from both sides, we get A = 1/18 - 5/72 = 4/72 = 1/18.

Now that we have the individual work rates, we can determine the combined work rate of all three boys:

N + Y + A = 1/24 + 5/72 + 1/18 = 3/72 + 5/72 + 4/72 = 12/72 = 1/6.

The combined work rate is 1/6 of the wall per hour.

To determine how long it will take for all three boys to paint the wall together, we can use the formula:

Time = 1 / Combined work rate.

Substituting the value of the combined work rate, we have:

Time = 1 / (1/6) = 6 hours.

Therefore, it will take the three boys 6 hours to paint the wall together.

Answer

The three boys will be able to paint the wall together in 6 hours. (Source

Answer 3

Problem Analysis

We are given the information that Nikolai and Yuri can paint a wall in 9 hours, Yuri and Alexey can paint the same wall in 18 hours, and Alexey and Nikolai can paint the wall in 12 hours. We need to determine how long it will take for all three boys to paint the wall together.

Solution

To solve this problem, we can use the concept of work rates. The work rate is defined as the amount of work done per unit of time. If we let the work rate of Nikolai be N, the work rate of Yuri be Y, and the work rate of Alexey be A, then we can set up the following equations:

1. Nikolai and Yuri can paint the wall in 9 hours, so their combined work rate is 1/9 of the wall per hour: N + Y = 1/9. 2. Yuri and Alexey can paint the wall in 18 hours, so their combined work rate is 1/18 of the wall per hour: Y + A = 1/18. 3. Alexey and Nikolai can paint the wall in 12 hours, so their combined work rate is 1/12 of the wall per hour: A + N = 1/12.

We can solve this system of equations to find the individual work rates of each boy. Once we have the individual work rates, we can determine how long it will take for all three boys to paint the wall together.

Solution Steps

1. Solve the system of equations to find the individual work rates of each boy. 2. Add the individual work rates to find the combined work rate of all three boys. 3. Use the combined work rate to determine how long it will take for all three boys to paint the wall together.

Solution Details

Let's solve the system of equations:

1. From equation 1, we have N + Y = 1/9. 2. From equation 2, we have Y + A = 1/18. 3. From equation 3, we have A + N = 1/12.

To solve this system of equations, we can use substitution or elimination. Let's use elimination:

Adding equations 1 and 3, we get: (N + Y) + (A + N) = 1/9 + 1/12 2N + Y + A = 4/36 + 3/36 2N + Y + A = 7/36

Subtracting equation 2 from the above equation, we get: (2N + Y + A) - (Y + A) = 7/36 - 1/18 2N = 7/36 - 2/36 2N = 5/36

Dividing both sides by 2, we get: N = 5/72

Now, let's substitute the value of N into equation 1 to find the value of Y: 5/72 + Y = 1/9 Y = 1/9 - 5/72 Y = 8/72 - 5/72 Y = 3/72 Y = 1/24

Finally, let's substitute the values of N and Y into equation 2 to find the value of A: 1/24 + A = 1/18 A = 1/18 - 1/24 A = 4/72 - 3/72 A = 1/72

Now that we have the individual work rates, we can find the combined work rate of all three boys: Combined work rate = N + Y + A Combined work rate = 5/72 + 1/24 + 1/72 Combined work rate = 10/72 + 3/72 + 1/72 Combined work rate = 14/72 Combined work rate = 7/36

To determine how long it will take for all three boys to paint the wall together, we can use the formula: Time = 1 / Combined work rate Time = 1 / (7/36) Time = 36/7

Therefore, it will take the three boys 36/7 hours to paint the wall together.

Answer

The three boys will take approximately 5 hours and 8 minutes to paint the wall together.