X is twice as good as a workman as Y. X finished a piece of work in 3 hours less than Y. In how many hours could they have finished that piece of work together ?

Options

A. 3
B. 2
C. 4
D. 5

Answer

To solve the problem step by step, we will first define the work rates of workers X and Y and then find out how long it would take for them to complete the work together.

### Step 1: Define the work rates of X and Y
Let the time taken by worker Y to complete the work be \( t \) hours. Since X is twice as good as Y, X will take half the time of Y to complete the work. Therefore, the time taken by X is:
\[ t_X = \frac{t}{2} \]

### Step 2: Set up the equation based on the information given
According to the problem, X finishes the work in 3 hours less than Y. Therefore, we can write:
\[ t_Y - t_X = 3 \]
Substituting \( t_X \) from Step 1, we have:
\[ t - \frac{t}{2} = 3 \]

### Step 3: Solve for \( t \)
Now, simplify the equation:
\[ \frac{t}{2} = 3 \]
Multiplying both sides by 2 gives:
\[ t = 6 \]
So, Y takes 6 hours to complete the work, and X takes:
\[ t_X = \frac{6}{2} = 3 \text{ hours} \]

### Step 4: Calculate the work rates of X and Y
The work rate of X (work done per hour) is:
\[ \text{Rate of X} = \frac{1}{t_X} = \frac{1}{3} \text{ (work per hour)} \]
The work rate of Y is:
\[ \text{Rate of Y} = \frac{1}{t_Y} = \frac{1}{6} \text{ (work per hour)} \]

### Step 5: Find the combined work rate of X and Y
Now, we can find their combined work rate:
\[ \text{Combined Rate} = \text{Rate of X} + \text{Rate of Y} = \frac{1}{3} + \frac{1}{6} \]
To add these fractions, we need a common denominator:
\[ \frac{1}{3} = \frac{2}{6} \]
Thus,
\[ \text{Combined Rate} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \text{ (work per hour)} \]

### Step 6: Calculate the time taken to complete the work together
If their combined work rate is \( \frac{1}{2} \) work per hour, the time taken to complete 1 whole work is:
\[ \text{Time} = \frac{1 \text{ work}}{\frac{1}{2} \text{ work per hour}} = 2 \text{ hours} \]

### Final Answer
X and Y together can finish the work in **2 hours**.

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