Sam takes 6 hours longer than Susan to wax a floor. Working together they can wax the floor in 4 hours. How long will it take each of them working alone to wax the floor?
1/x + 1/(x+6) = 1/4
Let's denote Susan's waxing rate as x floors per hour.
Since Sam takes 6 hours longer than Susan to wax the floor, his waxing rate would be x floors per hour, divided by (x + 6) hours per floor.
When they work together, their combined waxing rate is 1 floor per 4 hours.
Thus, we can set up the equation:
1/x + 1/(x + 6) = 1/4
To solve for x, we need to multiply both sides of the equation by 4x(x + 6), which gives us:
4(x + 6) + 4x = x(x + 6)
Expanding the equation:
4x + 24 + 4x = x^2 + 6x
Combining like terms:
8x + 24 = x^2 + 6x
Rearranging the equation:
0 = x^2 - 2x - 24
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, the equation factors as:
0 = (x - 6)(x + 4)
Setting each factor to zero:
x - 6 = 0 or x + 4 = 0
Solving for x:
x = 6 or x = -4
Since the time taken cannot be negative, Susan's waxing rate is x = 6 floors per hour, and Sam's waxing rate is x + 6 = 12 floors per hour.
Therefore, it would take Susan alone 1/x = 1/6 hours to wax the floor, or approximately 10 minutes.
And it would take Sam alone 1/(x + 6) = 1/12 hours to wax the floor, or approximately 5 minutes.
To solve this problem, we can set up a system of equations based on the given information. Let's denote the time it takes Susan to wax the floor as "S" and the time it takes Sam to wax the floor as "S + 6".
We know that the rate at which they work together is additive, meaning that the sum of their rates will give us the rate at which they complete the task together. The rate at which Susan works is 1/S, and the rate at which Sam works is 1/(S + 6).
We also know that if they work together, they can complete the task in 4 hours. So, their combined rate is 1/4.
Using this information, we can set up the equation:
1/S + 1/(S + 6) = 1/4
To solve this equation, we can multiply every term by the least common denominator (4S(S + 6)):
4(S + 6) + 4S = S(S + 6)
Solving this equation will give us the time it takes Susan to wax the floor. Once we have that value, we can find Sam's time by adding 6 hours to Susan's time.
Let's solve the equation:
4(S + 6) + 4S = S(S + 6)
4S + 24 + 4S = S^2 + 6S
8S + 24 = S^2 + 6S
S^2 - 2S - 24 = 0
Factoring this quadratic equation:
(S - 6)(S + 4) = 0
So, either (S - 6) = 0 or (S + 4) = 0.
If (S - 6) = 0, then S = 6. This means it takes Susan 6 hours to wax the floor.
If (S + 4) = 0, then S = -4. However, since time cannot be negative, we can disregard this solution.
Therefore, Susan takes 6 hours to wax the floor. Sam takes 6 + 6 = 12 hours to wax the floor.