Question

Игорь и Паша красят забор за 10 часов. Паша и Володя красят этот же забор за 15 часов, а Володя и Игорь - за 24 часа. За сколько минут мальчики покрасят забор, работая втроём. Помогите пожалуйста.

Answer 1

Найдём время, за которое все команды мальчиков покрасили бы некоторое целое число заборов, т.е. наименьшее общее кратное (НОК).
Для чисел 10, 15 и 24 это 120. Тогда за 120 часов И и П покрасили бы 12 заборов, П и В - 8 заборов, И и В - 5 заборов. То есть, работая вместе одновременно, два Игоря, два Паши и два Володи за 120 часов покрасили бы 25 таких заборов. Но так как нам нужны результаты только одного Игоря, одного Паши и одного Володи, то делим этот результат на 2. Тогда, трое этих мальчиков за 120 часов покрасили бы 12,5 заборов. Теперь посчитаем, за сколько они бы управились с одним забором.
12,5 - 120
1 - х
Считаем пропорцией
х=120/12,5= 9,6. За 9,6 часов трое мальков покрасят забор.

Answer 2

Problem Analysis

We are given information about the time it takes for different pairs of boys to paint a fence. Igor and Pasha can paint the fence in 10 hours, Pasha and Volodya can paint it in 15 hours, and Volodya and Igor can paint it in 24 hours. We need to determine how long it will take for all three boys to paint the fence 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 assume that the work to be done is painting the fence, we can calculate the work rates for each pair of boys.

Let's denote the work rate of Igor and Pasha as R1, the work rate of Pasha and Volodya as R2, and the work rate of Volodya and Igor as R3.

From the given information, we know that Igor and Pasha can paint the fence in 10 hours, so their work rate is 1/10 of the fence per hour. Similarly, the work rates for Pasha and Volodya and Volodya and Igor can be calculated as 1/15 and 1/24 of the fence per hour, respectively.

To find the work rate when all three boys work together, we can add up their individual work rates. Let's denote the work rate when all three boys work together as R.

R = R1 + R2 + R3

Now, we can calculate the time it will take for all three boys to paint the fence by using the formula:

Time = 1 / R

Let's calculate the time it will take for all three boys to paint the fence.

Calculation

The work rates for each pair of boys are as follows: - Igor and Pasha: R1 = 1/10 - Pasha and Volodya: R2 = 1/15 - Volodya and Igor: R3 = 1/24

The work rate when all three boys work together is: R = R1 + R2 + R3 = 1/10 + 1/15 + 1/24

Now, let's calculate the time it will take for all three boys to paint the fence: Time = 1 / R

Answer

It will take the boys approximately 6.67 hours to paint the fence together.

Explanation

When all three boys work together, their combined work rate is R = 1/10 + 1/15 + 1/24. By calculating this value, we find that R ≈ 0.1667.

To find the time it will take for all three boys to paint the fence, we use the formula Time = 1 / R. Plugging in the value of R, we get Time ≈ 1 / 0.1667 ≈ 6.67 hours.

Therefore, it will take the boys approximately 6.67 hours to paint the fence together.