Ram and Mohan are friends. Each has some money. If Ram gives 30 to Mohan, then Mohan will have twice the money left with Ram. But if Mohan gives 10 to Ram, then Ram will have thrice as much as is left with Mohan, How much money does each have ?
(a) 62, 34
(b) 6, 2
(c)170, 124
(d)43, 26

Options

A. a. 62, 34
B. b. 6, 2
C. c. 170, 124
D. d. 43, 26

Answer

To solve the problem, we will define the variables for the amounts of money Ram and Mohan have:

Let:
- \( R \) = amount of money Ram has
- \( M \) = amount of money Mohan has

### Step 1: Set up the equations based on the problem statement

1. **First Condition**: If Ram gives 30 to Mohan, then Mohan will have twice the money left with Ram.
- After Ram gives 30 to Mohan:
- Ram's money = \( R - 30 \)
- Mohan's money = \( M + 30 \)
- According to the condition:
\[ M + 30 = 2(R - 30) \]
- Simplifying this equation:
\[ M + 30 = 2R - 60 \]
\[ M - 2R = -90 \quad \text{(Equation 1)} \]

2. **Second Condition**: If Mohan gives 10 to Ram, then Ram will have thrice as much as is left with Mohan.
- After Mohan gives 10 to Ram:
- Ram's money = \( R + 10 \)
- Mohan's money = \( M - 10 \)
- According to the condition:
\[ R + 10 = 3(M - 10) \]
- Simplifying this equation:
\[ R + 10 = 3M - 30 \]
\[ R - 3M = -40 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

We now have the following system of equations:
1. \( M - 2R = -90 \) (Equation 1)
2. \( R - 3M = -40 \) (Equation 2)

We can rearrange Equation 1 to express \( M \) in terms of \( R \):
\[ M = 2R - 90 \]

Now, substitute \( M \) into Equation 2:
\[ R - 3(2R - 90) = -40 \]
Expanding this gives:
\[ R - 6R + 270 = -40 \]
Combining like terms:
\[ -5R + 270 = -40 \]
Subtracting 270 from both sides:
\[ -5R = -310 \]
Dividing by -5:
\[ R = 62 \]

### Step 3: Find the value of \( M \)

Now substitute \( R = 62 \) back into the equation for \( M \):
\[ M = 2(62) - 90 \]
Calculating this gives:
\[ M = 124 - 90 = 34 \]

### Conclusion

Thus, Ram has \( 62 \) and Mohan has \( 34 \).

**Final Answer**: Ram = 62, Mohan = 34

---