Question

Двое рабочих разгружают грузовик за 2 часа. Первый рабочий в одиночку может выполнить эту работу на 3 часа раньше,чем второй. За сколько часов выполнит эту работу каждый рабочий по отдельности?

Answer 1

Problem Analysis

We have two workers who are unloading a truck. They can complete the job together in 2 hours. The first worker can complete the job alone 3 hours earlier than the second worker. We need to determine how long it would take each worker to complete the job individually.

Solution

Let's assume that the second worker takes x hours to complete the job alone. Since the first worker can complete the job 3 hours earlier than the second worker, the first worker would take (x - 3) hours to complete the job alone.

To find the time it takes for two workers to complete the job together, we can use the formula:

1 / (time taken by worker 1) + 1 / (time taken by worker 2) = 1 / (time taken by both workers)

Substituting the values, we have:

1 / (x - 3) + 1 / x = 1 / 2

Now, let's solve this equation to find the values of x and (x - 3).

Calculation

To solve the equation, we can multiply all terms by the least common multiple (LCM) of the denominators to eliminate the fractions. In this case, the LCM is 2x(x - 3).

Multiplying both sides of the equation by 2x(x - 3), we get:

2x + 2(x - 3) = x(x - 3)

Simplifying the equation:

2x + 2x - 6 = x^2 - 3x

Combining like terms:

4x - 6 = x^2 - 3x

Rearranging the equation:

x^2 - 7x + 6 = 0

Factoring the quadratic equation:

x^2 - 6x - x + 6 = 0

x(x - 6) - 1(x - 6) = 0

(x - 1)(x - 6) = 0

So, we have two possible solutions for x: x = 1 and x = 6.

Since the first worker takes (x - 3) hours to complete the job alone, the possible values for the first worker's time are (1 - 3) = -2 and (6 - 3) = 3.

However, we cannot have negative time, so the first worker takes 3 hours to complete the job alone, and the second worker takes 6 hours to complete the job alone.

Answer

Therefore, the first worker can complete the job alone in 3 hours, and the second worker can complete the job alone in 6 hours.

Note: The negative value (-2) for the first worker's time is not valid in this context, as time cannot be negative.