Ram, Raghav, Tarun and Varun together had a total amount of Rs 240 with them. Ram had half of the total amount that others had. Raghav had one-third of the total amount that others had. Tarun had one-fourth of the total amount that others had. Find the amount that Varun had.

Options

A. Rs 64
B. Rs 70
C. Rs 52
D. Rs 58

Answer

To solve the problem step by step, let's denote the amounts that Ram, Raghav, Tarun, and Varun have as follows:

- Let Ram's amount be \( R \)
- Let Raghav's amount be \( G \)
- Let Tarun's amount be \( T \)
- Let Varun's amount be \( V \)

According to the problem, we have the following information:

1. The total amount they have together is \( R + G + T + V = 240 \) (Equation 1).
2. Ram had half of the total amount that the others had:
\[ R = \frac{1}{2}(G + T + V) \quad \text{(Equation 2)} \]
3. Raghav had one-third of the total amount that the others had:
\[ G = \frac{1}{3}(R + T + V) \quad \text{(Equation 3)} \]
4. Tarun had one-fourth of the total amount that the others had:
\[ T = \frac{1}{4}(R + G + V) \quad \text{(Equation 4)} \]

### Step 1: Solve for Ram's Amount

From Equation 1, we can express \( G + T + V \) in terms of \( R \):
\[ G + T + V = 240 - R \]

Substituting this into Equation 2:
\[ R = \frac{1}{2}(240 - R) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 2R = 240 - R \]
Adding \( R \) to both sides:
\[ 3R = 240 \]
Dividing by 3:
\[ R = 80 \]

### Step 2: Substitute Ram's Amount Back

Now that we have \( R = 80 \), we can substitute this back into Equation 1 to find \( G + T + V \):
\[ G + T + V = 240 - 80 = 160 \quad \text{(Equation 5)} \]

### Step 3: Solve for Raghav's Amount

Using Equation 3:
\[ G = \frac{1}{3}(R + T + V) = \frac{1}{3}(80 + T + V) \]
Substituting \( T + V = 160 - G \) (from Equation 5):
\[ G = \frac{1}{3}(80 + 160 - G) \]
Multiplying by 3:
\[ 3G = 80 + 160 - G \]
Adding \( G \) to both sides:
\[ 4G = 240 \]
Dividing by 4:
\[ G = 60 \]

### Step 4: Substitute Raghav's Amount Back

Now substitute \( G = 60 \) back into Equation 5 to find \( T + V \):
\[ T + V = 160 - 60 = 100 \quad \text{(Equation 6)} \]

### Step 5: Solve for Tarun's Amount

Using Equation 4:
\[ T = \frac{1}{4}(R + G + V) = \frac{1}{4}(80 + 60 + V) = \frac{140 + V}{4} \]
Substituting \( V = 100 - T \) (from Equation 6):
\[ T = \frac{1}{4}(140 + (100 - T)) \]
Multiplying by 4:
\[ 4T = 140 + 100 - T \]
Adding \( T \) to both sides:
\[ 5T = 240 \]
Dividing by 5:
\[ T = 48 \]

### Step 6: Find Varun's Amount

Now substitute \( T = 48 \) back into Equation 6 to find \( V \):
\[ V = 100 - T = 100 - 48 = 52 \]

### Final Answer

Thus, the amount that Varun had is \( \boxed{52} \).