if one machine can do a piece of work in 10 hours and a second machine can do the same work in 15 hours, how many hours will it take both machines simultaneously to do the work?
A) 12.5
B) 25
C) 5
D) 6
1/10 + 1/15 = 1/x
solve for x
To find out how many hours it would take for both machines to do the work simultaneously, we need to calculate their combined work rate.
Let's assume that the work done by the first machine in one hour is represented by "x."
Therefore, the work done by the second machine in one hour would be represented by "x + y," where "y" is the additional work done by the second machine.
According to the information given, the first machine completes the work in 10 hours, which means its work rate is 1/10 or "x = 1/10."
Similarly, the second machine completes the work in 15 hours, so its work rate is 1/15 or "x + y = 1/15."
To find the combined work rate, we add the individual work rates: x + (x + y) = 1/10 + 1/15.
Combining the fractions gives us 2x + y = 3/30 + 2/30.
Simplifying further, we have 2x + y = 5/30.
Since we know that x = 1/10, we substitute it into the equation: 2(1/10) + y = 5/30.
Simplifying, we get 2/10 + y = 5/30.
Further simplification gives us 1/5 + y = 1/6.
To isolate "y" on one side, we can subtract 1/5 from both sides: y = 1/6 - 1/5.
To find a common denominator, we multiply the fractions numerically: y = 5/30 - 6/30.
Simplifying, we get y = -1/30.
The negative sign indicates that the second machine is working in the opposite direction or interfering with the first machine's work.
Therefore, to find the time it takes for both machines to complete the work together, we divide the total work by their combined work rate.
The total work rate is x + y = 1/10 + (-1/30) = 1/10 - 1/30 = 3/30 - 1/30 = 2/30 = 1/15.
Since their combined work rate is 1/15, it will take both machines 15 hours to do the work together.
Thus, the correct answer is option B) 25.