Two plumbers received a job. At first, one of the plumbers worked alone for 1 hour, and then they worked together for the next 4 hours. After this 40% of the job was finished. How long would it take each plumber to do the whole job by himself if it is known that it would take the first plumber 5 more hours to finish the job than it would take the second plumber?

If the fast plumber takes x hours, then the slow one takes x+5 hours.

1/(x+5) + 4(1/x + 1/(x+5)) = 2/5
x = 20

So, the 1st plumber takes 25 hours.

Check:
In 1 hour, the 1st plumber finishes 1/25 of the job.

During each of the next 4 hours, the two together finish 1/25 + 1/20 = 9/100 of the job.

1/25 + 36/100 = 40/100 = 40%

1st Plumber = 25 hrs, 2nd Plumber = 20 hrs.

To solve this question, let's break it down step by step:

Step 1: Calculate the total amount of work done when one plumber works alone for 1 hour and then both plumbers work together for 4 hours.

Since the first plumber worked alone for 1 hour, they completed 1/x of the job, where x is the number of hours it takes the first plumber to complete the job alone.
In the next 4 hours, while working together, they completed 4/y of the job, where y is the number of hours it takes both plumbers to complete the job when working together.

According to the question, 40% of the job was finished. So, we can write the equation:
1/x + 4/y = 0.40

Step 2: Express the relationship between the two plumbers' working times.

We know that the first plumber takes 5 more hours than the second plumber to complete the job alone. Therefore, we can write another equation:
x = y + 5

Step 3: Solve the system of equations.

Now, we can substitute the expression for x from the second equation into the first equation:
1/(y+5) + 4/y = 0.40

Multiply through by a common denominator of y(y+5) to clear the fractions:
y + 5 + 4(y+5) = 0.40y(y+5)

Simplify the equation:
y + 5 + 4y + 20 = 0.40y^2 + 2y + 0.40y + 2

Combine like terms:
5y + 25 = 0.40y^2 + 2.40y + 2

Rearrange the equation to set it equal to zero:
0.40y^2 - 2.60y - 23 = 0

Step 4: Solve the quadratic equation.

To solve the quadratic equation, you can use factoring, completing the square, or using the quadratic formula.

In this case, let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)

For the equation 0.40y^2 - 2.60y - 23 = 0, the values of a, b, and c are:
a = 0.40
b = -2.60
c = -23

Substituting these values into the quadratic formula:
y = (-(-2.60) ± √((-2.60)^2 - 4(0.40)(-23))) / (2 * 0.40)

Simplifying the equation further:
y = (2.60 ± √(6.76 + 36.8)) / 0.80
y = (2.60 ± √43.56) / 0.80

Taking the square root:
y = (2.60 ± 6.60) / 0.80

Simplifying further:
y = (2.60 + 6.60) / 0.80 or y = (2.60 - 6.60) / 0.80

Calculating the values:
y = 9.20 / 0.80 or y = -4.00 / 0.80
y = 11.50 or y = -5.00

Since time cannot be negative, we can ignore the negative solution.

So, the value of y is 11.50, which means both plumbers working together can complete the job in 11.50 hours.

Step 5: Calculate the value of x.

From the second equation, we know that x = y + 5. Substituting the value of y:
x = 11.50 + 5
x = 16.50

Therefore, it would take the first plumber 16.50 hours to complete the job alone, and the second plumber would take 11.50 hours to complete the job alone.

37