X, Y and Z can complete a job in 10 days, 12 days, and 15 days respectively. They started working together and after some days, Y left. The remaining work was completed by X and Z. A total of ₹24,000 were paid as wages to the three. The wages paid to X were ₹8,000 more than those paid to Y. After how many days did Y leave?
Question
X, Y and Z can complete a job in 10 days, 12 days, and 15 days respectively. They started working together and after some days, Y left. The remaining work was completed by X and Z. A total of ₹24,000 were paid as wages to the three. The wages paid to X were ₹8,000 more than those paid to Y. After how many days did Y leave?
Solution
To solve this problem, we need to first understand the rate at which each person works.
- X can complete the job in 10 days, so X's work rate is 1/10 of the job per day.
- Y can complete the job in 12 days, so Y's work rate is 1/12 of the job per day.
- Z can complete the job in 15 days, so Z's work rate is 1/15 of the job per day.
When they work together, their combined work rate is 1/10 + 1/12 + 1/15 = 0.1833 of the job per day.
We know that the total wages paid were ₹24,000, and X was paid ₹8,000 more than Y. Let's denote Y's wage as W. Then X's wage is W + ₹8,000, and Z's wage is ₹24,000 - W - (W + ₹8,000) = ₹16,000 - W.
The wages are proportional to the amount of work done, so W/(W + ₹8,000) = (1/12) / (1/10), which gives us W = ₹9,600. Therefore, X was paid ₹17,600 and Z was paid ₹6,800.
Now, let's denote the number of days Y worked as D. Then X and Z worked together for (total work / combined work rate of X and Z) - D days.
We know that the total work is 1 (as it's one job), and the combined work rate of X and Z is 1/10 + 1/15 = 0.1667 of the job per day.
So, D = (1 / 0.1833) - (1 / 0.1667) = 5.46 days.
Therefore, Y left after approximately 5 or 6 days.
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