If A and B work together,they can finish a task in 2 hours. If they do the task separately, B can finish 3 hours earlier than A can. Find the time required for B to finish the task alone.
Recall that if A takes A hours, and B takes B hours,
1/A + 1/B = 1/2
So, if B does the task in x hours, A takes x+3 hours:
1/(x+3) + 1/x = 1/2
Now just solve for x.
To solve this problem, we can set up a system of equations. Let's assume that A takes x hours to complete the task alone, and B takes y hours to complete the task alone.
From the given information:
1. If A and B work together, they can finish the task in 2 hours. This can be expressed as:
1/x + 1/y = 1/2 (Equation 1)
2. If they work separately, B can finish 3 hours earlier than A can. This can be expressed as:
y = x - 3 (Equation 2)
We now have a system of two equations that we need to solve.
To find the time required for B to finish the task alone, we need to find the value of y.
We can solve the system of equations by substitution or elimination method.
Let's solve it using the substitution method:
Step 1: Solve Equation 2 for x in terms of y:
x = y + 3
Step 2: Substitute the value of x in Equation 1:
1/(y + 3) + 1/y = 1/2
Step 3: Multiply the equation by 2y(y + 3) to eliminate the fractions:
2y + 2(y + 3) = y(y + 3)
Step 4: Simplify and move all terms to one side:
2y + 2y + 6 = y^2 + 3y
Step 5: Rearrange the equation to form a quadratic equation:
y^2 + y - 6 = 0
Step 6: Factorize the quadratic equation:
(y + 3)(y - 2) = 0
Step 7: Set each factor equal to zero and solve for y:
y + 3 = 0 or y - 2 = 0
y = -3 or y = 2
Since the number of hours cannot be negative, the value of y is 2.
Therefore, it takes B 2 hours to finish the task alone.