X working alone takes 75 days more than Y to do a work, and working together they complete it in 20 days. If a certain sum has been earmarked as wages for his work, in what ratio should it get distributed among X and Y?
Question
X working alone takes 75 days more than Y to do a work, and working together they complete it in 20 days. If a certain sum has been earmarked as wages for his work, in what ratio should it get distributed among X and Y?
Solution 1
Let's solve this step by step:
Step 1: Let's assume the time taken by Y alone to complete the work is 'd' days. So, the time taken by X alone to complete the work is 'd + 75' days.
Step 2: We know that if Y can complete the work in 'd' days, then in one day, Y can complete '1/d' of the work. Similarly, if X can complete the work in 'd + 75' days, then in one day, X can complete '1/(d + 75)' of the work.
Step 3: According to the problem, when X and Y work together, they can complete the work in 20 days. This means that in one day, they can complete '1/20' of the work.
Step 4: So, we can write the equation as: 1/d + 1/(d + 75) = 1/20.
Step 5: Solving this equation, we get d = 60. So, Y alone can complete the work in 60 days and X alone can complete the work in 60 + 75 = 135 days.
Step 6: The ratio of time taken by X and Y is 135:60 or 27:12 or 9:4. But we know that if a person takes less time to complete the work, he should get more wages. So, the wages should be distributed in the inverse ratio of time taken.
Step 7: Therefore, the ratio in which the wages should be distributed among X and Y is the inverse of 9:4, which is 4:9.
Solution 2
Let's solve this step by step:
Step 1: Let's assume the time taken by Y to complete the work alone is 'd' days. So, X takes 'd+75' days to complete the same work alone.
Step 2: Now, we know that work done is inversely proportional to the time taken. So, the work done by Y in 1 day is 1/d and the work done by X in 1 day is 1/(d+75).
Step 3: According to the problem, when X and Y work together, they complete the work in 20 days. So, the combined work done by X and Y in 1 day is 1/20.
Step 4: Therefore, we can write the equation as: 1/d + 1/(d+75) = 1/20.
Step 5: Solving this equation, we get d = 60. So, Y takes 60 days to complete the work alone and X takes 60+75 = 135 days to complete the work alone.
Step 6: Now, the ratio of the work done by X and Y in 1 day is 1/135 : 1/60 which simplifies to 60 : 135 or 4 : 9.
So, the wages should be distributed in the ratio 4:9 between X and Y.
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