Two years ago, Aadhya was three times as old as his son and two years hence, twice her age will be equal to five times that of her son. Find Aadhya's present age.

Options

A. 38 years
B. 36 years
C. 34 years
D. 42 years

Explanation

To solve the problem step by step, let's denote Aadhya's present age as \( A \) and her son's present age as \( S \).

### Step 1: Formulate the first equation based on the information given for two years ago.
Two years ago, Aadhya's age was \( A - 2 \) and her son's age was \( S - 2 \). According to the problem, two years ago Aadhya was three times as old as her son. Thus, we can write the equation:

\[ A - 2 = 3(S - 2) \]

### Step 2: Simplify the first equation.
Expanding the equation gives:

\[ A - 2 = 3S - 6 \]

Rearranging this, we get:

\[ A = 3S - 4 \quad \text{(Equation 1)} \]

### Step 3: Formulate the second equation based on the information given for two years hence.
Two years hence, Aadhya's age will be \( A + 2 \) and her son's age will be \( S + 2 \). According to the problem, twice Aadhya's age will be equal to five times her son's age. Thus, we can write the equation:

\[ 2(A + 2) = 5(S + 2) \]

### Step 4: Simplify the second equation.
Expanding this equation gives:

\[ 2A + 4 = 5S + 10 \]

Rearranging this, we get:

\[ 2A - 5S = 6 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2.
Now we can substitute \( A \) from Equation 1 into Equation 2:

\[ 2(3S - 4) - 5S = 6 \]

### Step 6: Simplify and solve for \( S \).
Expanding gives:

\[ 6S - 8 - 5S = 6 \]

Combining like terms results in:

\[ S - 8 = 6 \]

Adding 8 to both sides gives:

\[ S = 14 \]

### Step 7: Substitute \( S \) back to find \( A \).
Now we can substitute \( S \) back into Equation 1 to find \( A \):

\[ A = 3(14) - 4 \]

Calculating this gives:

\[ A = 42 - 4 = 38 \]

### Final Answer:
Aadhya's present age is \( \boxed{38} \).

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