Two large and 1 small pumps can fill a swimming pool in 4 hours. One large and 3 small pumps can also fill the same swimming pool in 4 hours. How many hours will it take 4 large and 4 small pumps to fill the swimming pool.(We assume that all large pumps are similar and all small pumps are also similar.)
Question
Two large and 1 small pumps can fill a swimming pool in 4 hours. One large and 3 small pumps can also fill the same swimming pool in 4 hours. How many hours will it take 4 large and 4 small pumps to fill the swimming pool.(We assume that all large pumps are similar and all small pumps are also similar.)
Solution
Sure, let's solve this problem step by step.
Step 1: Let's denote the rate at which a large pump can fill the pool as L and the rate for a small pump as S. These rates are in pools/hour.
Step 2: From the problem, we know that the rate of 2 large pumps plus 1 small pump equals the rate of filling 1 pool in 4 hours. We can write this as an equation: 2L + S = 1/4.
Step 3: We also know that the rate of 1 large pump plus 3 small pumps equals the rate of filling 1 pool in 4 hours. This gives us another equation: L + 3S = 1/4.
Step 4: Now we have a system of two equations, and we can solve it. Subtract the second equation from the first to get: L - 2S = 0. This means that the rate of one large pump is twice the rate of one small pump.
Step 5: Substitute L = 2S into the first equation to get: 2(2S) + S = 1/4, which simplifies to 5S = 1/4. Solving for S, we find that S = 1/20.
Step 6: Since L = 2S, we find that L = 2/20 = 1/10.
Step 7: Now we know the rates for the large and small pumps, we can find out how long it would take 4 large and 4 small pumps to fill the pool. The combined rate of these pumps is 4L + 4S = 4(1/10) + 4(1/20) = 2/5.
Step 8: To find the time it takes to fill the pool, we divide the total volume (1 pool) by the combined rate: 1 / (2/5) = 2.5 hours.
So, it would take 2.5 hours for 4 large and 4 small pumps to fill the swimming pool.
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