It takes Todd 3 hours longer than Tom to paint a 4x5 feet bedroom. Working together, both can complete the job in two hours.How long does it take each one to complete the painting job working alone?
Let it takes Todd X hours; Tom Y hours.
X=Y+3
For one hour, Todd paints 1/X part of the bedroom, Tom - 1/Y; together - 1/X+1/Y
1/(1/X+1/Y)=2
2(X+Y)=XY
2(2Y+3)=(Y+3)Y
4Y+6=Y^2+3Y
Y^2-Y-6=0 => Y=3, X=6
To solve this problem, we need to use the concept of work rate. The idea is that if two people can complete a job together in a certain amount of time, their combined work rate is equal to the work rate of each individual added together.
Let's assume that Tom's work rate is "x" and Todd's work rate is "y". We know that Tom takes 3 hours less than Todd to complete the painting job. This means that Todd's work rate is slower, so we can express it as Todd's work rate being "x - 3".
Now, given that both Tom and Todd can complete the job together in 2 hours, we can set up the following equation based on their work rates:
(1/x) + (1/(x-3)) = 1/2
To solve this equation, we can multiply both sides by the least common denominator, which would be 2x(x-3):
2(x-3) + 2x = x(x-3)
Simplifying this equation will yield:
2x - 6 + 2x = x^2 - 3x
Combining like terms, we have:
4x - 6 = x^2 - 3x
Rearranging this equation will yield:
x^2 - 7x + 6 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we get:
(x - 6)(x - 1) = 0
So, x = 6 or x = 1.
Since x represents Tom's work rate, we can interpret x = 6 as Tom needing 6 hours to complete the job alone, and x = 1 as an extraneous solution.
Now, we know Todd's work rate is x - 3, so Todd would take x - 3 = 6 - 3 = 3 hours longer than Tom to complete the job alone. Therefore, Todd would take 9 hours to complete the job alone.
In conclusion, Tom takes 6 hours to complete the painting job working alone, while Todd takes 9 hours.