R and S can complete a certain job in 6 and 4 days respectively, while they work individually. What will be the least number of days they will take to complete the same job, if they work on alternate days?
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Wait, how can the answer be less than 4 days? R alone takes 6 days, so even if S helps, the total time should be more than 4 days, right? I'm confused.
Think about it: if S works alone, he takes 4 days. If R helps him on alternate days, the total time should be less than 4 days because R contributes on some days. So the answer can be less than 4.
But if they alternate, on some days only one person works, so the progress might be slower than if they worked together. So the time could be more than 4? Actually, if they work together continuously, time = (6*4)/(6+4) = 2.4 days. With alternating, it should be between 2.4 and 4? Wait, if R works alone on odd days and S alone on even days, the rate is slower than together, so time > 2.4. But can it exceed 4? If R works alone on some days, his rate is lower, so total time could be more than S alone? Let's calculate: In 2 days, work done = 1/6 + 1/4 = 5/12. So in 4 days, work done = 10/12 = 5/6. Remaining 1/6. On day 5, R works alone, takes 1 day to do 1/6? Actually R does 1/6 in 1 day. So total 5 days. So answer is 5 days. So it's more than 4. So my initial thought was wrong.
This is tricky because the least number of days depends on who starts. If R starts, we got 5 days. If S starts, maybe it's different? Let's check: S starts: day1 S does 1/4, day2 R does 1/6, total 5/12 in 2 days. Same pattern, after 4 days 10/12, remaining 1/6. On day5, S works, does 1/4 in a day, so 1/6 takes (1/6)/(1/4)=2/3 day. Total 4 + 2/3 = 4.67 days. So if S starts, it's 4.67 days. But the question asks for least number of days, so we should choose the smaller? But 5 vs 4.67, 4.67 is smaller. So answer is 4.67? But the choices include 4.67. However, if R starts, it's 5 days. So the least is 4.67. But wait, the question says 'they work on alternate days' without specifying who starts. So we can choose the order to minimize? Usually these problems assume they alternate, but the starting person might be fixed? Often it's ambiguous. But if we can choose, then 4.67 is the answer. But the official answer might be 5? I'm not sure.
I think the problem means they work on alternate days, but we don't get to choose who starts; it's just that they alternate. So the least number of days would be the minimum over the two possible starting orders. So 4.67 is less than 5, so answer is 4.67. But I've seen similar problems where the answer is 5, so maybe the question implies that R starts? Or maybe the least number is when they work together? No, they work on alternate days. Hmm.
The answer is 4.67 days (D). Here's why: If S starts, after 4 days work done = 2*(1/4+1/6)=5/6, remaining 1/6. On day 5, S works alone and can finish 1/6 in (1/6)/(1/4)=2/3 day, so total 4.67 days. If R starts, total 5 days. So least is 4.67.
But the question says 'least number of days' so we can choose the order that minimizes time? That seems odd. Usually they specify who starts. Without that, I'd assume they alternate starting with R or something? Actually, many problems say 'starting with R' or 'starting with S'. Here it's ambiguous.
I think the least number of days means we consider both possibilities and take the minimum. So 4.67 is correct. But if the problem intended a fixed order, it would say. So I'll go with D.
This is a classic work-rate problem but with a twist. I got 5 days initially, but then realized the starting person matters. The least is when S starts because S is faster. So 4.67 days. Good question to test attention to detail.
What it tests
Your ability to combine individual work rates, usually by working with the fraction of a job completed per unit of time.
Common trap
Adding the times instead of the rates, or forgetting to take the reciprocal when combining rates.