Sum of A's and B's age 12 years ago is 176. A's age 36 years ago is equal to B's age 12 years ago. Find the age of A after two year ?

Options

A. 114 yrs
B. 130 yrs
C. 86 yrs
D. 106 yrs

Explanation

To solve the problem step by step, let's define the variables and set up the equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = Present age of A
- \( y \) = Present age of B

### Step 2: Set Up the First Equation
According to the problem, the sum of A's and B's ages 12 years ago is 176. This can be expressed as:
\[ (x - 12) + (y - 12) = 176 \]
Simplifying this gives:
\[ x + y - 24 = 176 \]
Adding 24 to both sides:
\[ x + y = 200 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that A's age 36 years ago is equal to B's age 12 years ago. This can be expressed as:
\[ x - 36 = y - 12 \]
Rearranging this gives:
\[ x - y = 24 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( x + y = 200 \)
2. \( x - y = 24 \)

We can solve these equations simultaneously.

Adding Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 200 + 24 \]
This simplifies to:
\[ 2x = 224 \]
Dividing both sides by 2:
\[ x = 112 \]

### Step 5: Find B's Age
Now, we can find B's age using Equation 1:
\[ 112 + y = 200 \]
Subtracting 112 from both sides:
\[ y = 88 \]

### Step 6: Find A's Age After 2 Years
To find A's age after 2 years:
\[ \text{A's age after 2 years} = x + 2 = 112 + 2 = 114 \]

### Final Answer
Thus, A's age after 2 years will be **114 years**.

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