Ratio of the ages of Mahesh and Nilesh is 5 : x. Mahesh is 18 years younger to Ramesh. After nine years Ramesh will be 47 years old. If the difference between the ages of Mahesh and Nilesh is same as the age of Ramesh, what is the value of x ?

Options

A. a. 11.8
B. b. 12.9
C. c. 13.7
D. d. 14.5

Answer

To solve the problem step by step, we will break down the information given and derive the necessary equations to find the value of \( x \).

### Step 1: Understand the given information
We know:
- The ratio of the ages of Mahesh (M) and Nilesh (N) is \( 5 : x \).
- Mahesh is 18 years younger than Ramesh (R).
- After 9 years, Ramesh will be 47 years old.

### Step 2: Calculate Ramesh's current age
Since Ramesh will be 47 years old after 9 years, we can find his current age:
\[ R = 47 - 9 = 38 \text{ years} \]

### Step 3: Express Mahesh's age in terms of Ramesh's age
Since Mahesh is 18 years younger than Ramesh:
\[ M = R - 18 = 38 - 18 = 20 \text{ years} \]

### Step 4: Set up the ratio of ages
From the ratio of ages given:
\[ \frac{M}{N} = \frac{5}{x} \]
Substituting the value of Mahesh's age:
\[ \frac{20}{N} = \frac{5}{x} \]

### Step 5: Cross-multiply to find N
Cross-multiplying gives:
\[ 20x = 5N \]
Rearranging this, we find:
\[ N = \frac{20x}{5} = 4x \]

### Step 6: Use the difference in ages
The problem states that the difference between the ages of Mahesh and Nilesh is the same as the age of Ramesh:
\[ |M - N| = R \]
Substituting the values we have:
\[ |20 - 4x| = 38 \]

### Step 7: Solve the equation
This absolute value equation can be split into two cases:
1. \( 20 - 4x = 38 \)
2. \( 20 - 4x = -38 \)

**Case 1:**
\[ 20 - 4x = 38 \implies -4x = 38 - 20 \implies -4x = 18 \implies x = -\frac{18}{4} \text{ (not valid since age cannot be negative)} \]

**Case 2:**
\[ 20 - 4x = -38 \implies -4x = -38 - 20 \implies -4x = -58 \implies 4x = 58 \implies x = \frac{58}{4} = 14.5 \]

### Conclusion
Thus, the value of \( x \) is \( 14.5 \).

---