X can do a piece of work in 15 days. If he is joined by Y who is 50% more efficient, in what time will X and Y together finish the work?

Options

A. 10 days
B. 6 days
C. 18 days
D. Data insufficient

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the efficiency of X
X can complete the work in 15 days. Therefore, the work done by X in one day is:
\[ \text{Efficiency of X} = \frac{1}{15} \text{ of the work per day} \]

### Step 2: Determine the efficiency of Y
Y is 50% more efficient than X. If we consider X's efficiency as 100%, then Y's efficiency will be:
\[ \text{Efficiency of Y} = 100\% + 50\% = 150\% \]
To express this in terms of work done per day, we can convert it to a fraction:
\[ \text{Efficiency of Y} = \frac{150}{100} \times \frac{1}{15} = \frac{1.5}{15} = \frac{1}{10} \text{ of the work per day} \]

### Step 3: Calculate the combined efficiency of X and Y
Now, we can find the combined efficiency of X and Y:
\[ \text{Combined Efficiency} = \text{Efficiency of X} + \text{Efficiency of Y} = \frac{1}{15} + \frac{1}{10} \]
To add these fractions, we need a common denominator. The least common multiple of 15 and 10 is 30:
\[ \frac{1}{15} = \frac{2}{30}, \quad \frac{1}{10} = \frac{3}{30} \]
So,
\[ \text{Combined Efficiency} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \text{ of the work per day} \]

### Step 4: Calculate the time taken by X and Y together to finish the work
If X and Y together can complete \(\frac{1}{6}\) of the work in one day, then the total time taken to complete the entire work is:
\[ \text{Time} = \frac{1 \text{ (whole work)}}{\frac{1}{6} \text{ (work per day)}} = 6 \text{ days} \]

### Final Answer
X and Y together will finish the work in **6 days**.

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