At the end of the year 2002, Ram was half as old as his grandfather. The sum of the years in which they were born is 3854. Age of Ram at the end of year 2003 is_____

Options

A. 50 years
B. 35 years
C. 51 years
D. 36 years

Answer

To solve the problem, we need to set up a system of equations based on the information given.

### Step 1: Define Variables
Let:
- \( R \) = Age of Ram at the end of 2002
- \( G \) = Age of Ram's grandfather at the end of 2002

### Step 2: Set Up the Equations
From the problem, we know two things:
1. Ram was half as old as his grandfather at the end of 2002.
\[ R = \frac{1}{2} G \]

2. The sum of the years in which they were born is 3854. The year of birth can be calculated as:
- Year of birth of Ram = 2002 - R
- Year of birth of grandfather = 2002 - G
Therefore, we have:
\[ (2002 - R) + (2002 - G) = 3854 \]
Simplifying this gives:
\[ 4004 - R - G = 3854 \]
Rearranging it, we get:
\[ R + G = 150 \]

### Step 3: Solve the Equations
Now we have a system of equations:
1. \( R = \frac{1}{2} G \)
2. \( R + G = 150 \)

Substituting the first equation into the second:
\[ \frac{1}{2} G + G = 150 \]
Combining like terms:
\[ \frac{3}{2} G = 150 \]
Multiplying both sides by \( \frac{2}{3} \):
\[ G = 100 \]

Now substituting \( G \) back into the first equation to find \( R \):
\[ R = \frac{1}{2} \times 100 = 50 \]

### Step 4: Find Age of Ram at the End of 2003
To find Ram's age at the end of 2003, we simply add 1 year to his age at the end of 2002:
\[ \text{Age of Ram at the end of 2003} = R + 1 = 50 + 1 = 51 \]

### Final Answer
The age of Ram at the end of the year 2003 is **51 years**.

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