Raja and Kundan together can build a wall in 20 days, Kundan and Mahesh build the same wall in 30 days and Mahesh and Raja can build the same wall in 24 days. In how many days can all the three complete the same wall while working together?

Options

A. 18
B. 16
C. 20
D. 14

Answer

To solve the problem, we need to determine how long it will take for Raja, Kundan, and Mahesh to build the wall together. We will follow these steps:

### Step 1: Determine the work done by each pair
1. **Raja and Kundan can build the wall in 20 days.**
- Work done = 1 wall
- Efficiency of Raja + Kundan = \( \frac{1 \text{ wall}}{20 \text{ days}} = \frac{1}{20} \text{ walls/day} \)

2. **Kundan and Mahesh can build the wall in 30 days.**
- Work done = 1 wall
- Efficiency of Kundan + Mahesh = \( \frac{1 \text{ wall}}{30 \text{ days}} = \frac{1}{30} \text{ walls/day} \)

3. **Mahesh and Raja can build the wall in 24 days.**
- Work done = 1 wall
- Efficiency of Mahesh + Raja = \( \frac{1 \text{ wall}}{24 \text{ days}} = \frac{1}{24} \text{ walls/day} \)

### Step 2: Set up equations for efficiency
Let:
- Efficiency of Raja = \( R \)
- Efficiency of Kundan = \( K \)
- Efficiency of Mahesh = \( M \)

From the information above, we can set up the following equations:
1. \( R + K = \frac{1}{20} \) (Equation 1)
2. \( K + M = \frac{1}{30} \) (Equation 2)
3. \( M + R = \frac{1}{24} \) (Equation 3)

### Step 3: Solve the equations
To find the individual efficiencies, we can add all three equations:
\[ (R + K) + (K + M) + (M + R) = \frac{1}{20} + \frac{1}{30} + \frac{1}{24} \]
This simplifies to:
\[ 2R + 2K + 2M = \frac{1}{20} + \frac{1}{30} + \frac{1}{24} \]
Now, we need to find the LCM of 20, 30, and 24 to combine the fractions:
- LCM(20, 30, 24) = 120

Now convert each fraction:
- \( \frac{1}{20} = \frac{6}{120} \)
- \( \frac{1}{30} = \frac{4}{120} \)
- \( \frac{1}{24} = \frac{5}{120} \)

Adding these together:
\[ \frac{6 + 4 + 5}{120} = \frac{15}{120} = \frac{1}{8} \]
Thus:
\[ 2R + 2K + 2M = \frac{1}{8} \]
Dividing by 2 gives:
\[ R + K + M = \frac{1}{16} \]

### Step 4: Calculate the time taken by all three working together
The combined efficiency of Raja, Kundan, and Mahesh is \( \frac{1}{16} \) walls per day. To find out how many days it will take them to complete 1 wall, we take the reciprocal of their combined efficiency:
\[ \text{Time} = \frac{1 \text{ wall}}{\frac{1}{16} \text{ walls/day}} = 16 \text{ days} \]

### Final Answer
Therefore, Raja, Kundan, and Mahesh can complete the wall together in **16 days**.

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