Ganga, Jamuna and Saraswati can do a piece of work, working together, in 1 day. Ganga is thrice efficient as Jamuna and Jamuna takes twice the number of days as Saraswati takes to do it alone. What is the difference between the number of days taken by Ganga and Saraswati?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, let's denote the efficiencies and the number of days taken by each person to complete the work.

### Step 1: Define Variables
Let:
- The efficiency of Jamuna = \( k \) (units of work per day)
- The efficiency of Ganga = \( 3k \) (since Ganga is thrice as efficient as Jamuna)
- The efficiency of Saraswati = \( x \)

### Step 2: Relate Jamuna and Saraswati
According to the problem, Jamuna takes twice the number of days as Saraswati to complete the work. If we denote the number of days taken by Saraswati to complete the work alone as \( d \), then:
- Jamuna takes \( 2d \) days to complete the work.

The efficiency can be expressed as:
- Efficiency of Jamuna = \( \frac{1}{2d} \)
- Efficiency of Saraswati = \( \frac{1}{d} \)

From the above, we know:
\[ k = \frac{1}{2d} \quad \text{(1)} \]
\[ x = \frac{1}{d} \quad \text{(2)} \]

### Step 3: Relate Efficiencies
Since we know that Jamuna's efficiency \( k \) is also related to Saraswati's efficiency \( x \):
\[ k = \frac{1}{2} x \quad \text{(from (2))} \]

### Step 4: Substitute and Solve
Now, substituting \( k \) from (1) into the equation \( k = \frac{1}{2} x \):
\[ \frac{1}{2d} = \frac{1}{2} \left(\frac{1}{d}\right) \]
This confirms that both expressions for \( k \) and \( x \) are consistent.

### Step 5: Total Efficiency
Now, the total efficiency when Ganga, Jamuna, and Saraswati work together is:
\[ \text{Total Efficiency} = 3k + k + x = 3k + k + 2k = 6k \]
Since they can complete the work in 1 day, the total efficiency is equal to 1 unit of work:
\[ 6k = 1 \quad \Rightarrow \quad k = \frac{1}{6} \]

### Step 6: Find Individual Efficiencies
Now substituting \( k \) back:
- Efficiency of Jamuna = \( k = \frac{1}{6} \)
- Efficiency of Ganga = \( 3k = \frac{3}{6} = \frac{1}{2} \)
- Efficiency of Saraswati = \( x = 2k = \frac{2}{6} = \frac{1}{3} \)

### Step 7: Calculate Days Taken
Now, we can find the number of days taken by each:
- Days taken by Ganga = \( \frac{1}{\text{Efficiency of Ganga}} = \frac{1}{\frac{1}{2}} = 2 \) days
- Days taken by Saraswati = \( \frac{1}{\text{Efficiency of Saraswati}} = \frac{1}{\frac{1}{3}} = 3 \) days

### Step 8: Find the Difference
Finally, the difference between the number of days taken by Ganga and Saraswati is:
\[ \text{Difference} = 3 - 2 = 1 \text{ day} \]

### Final Answer
The difference between the number of days taken by Ganga and Saraswati is **1 day**.

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