A and B can do a work in 30 days. If A worked twice as fast and B worked one-fourth as efficiently as he usually does, the work would still be completed in 30 days. Find the time taken (in days) by A alone to do the work.
Question
A and B can do a work in 30 days. If A worked twice as fast and B worked one-fourth as efficiently as he usually does, the work would still be completed in 30 days. Find the time taken (in days) by A alone to do the work.
Solution
Let's solve this step by step:
Step 1: Let's assume the total work to be done is 1 unit. So, A and B together can do this 1 unit of work in 30 days. Therefore, their combined work rate is 1/30 units per day.
Step 2: Now, if A works twice as fast and B works one-fourth as efficiently, the work is still completed in 30 days. This means their new combined work rate is also 1/30 units per day.
Step 3: Let's denote the work rate of A as 'a' and that of B as 'b'. So, from step 1, we have a + b = 1/30.
Step 4: From step 2, we have 2a + b/4 = 1/30.
Step 5: We can solve these two equations to find the values of 'a' and 'b'. Multiplying the second equation by 4, we get 8a + b = 4/30. Subtracting the first equation from this, we get 7a = 3/30, so a = 3/210 = 1/70.
Step 6: So, A alone can do the work in 70 days.
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