The product of the ages of Priya and Aneeta is 240. If twice the age of Aneeta is more than Priya's age by 4 years. What is Aneeta's age ?

Options

A. 11 years
B. 12 years
C. 13 years
D. 19 years

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( P \) = Age of Priya
- \( A \) = Age of Aneeta

### Step 2: Set Up the Equations
From the problem, we know two things:
1. The product of their ages is 240:
\[ P \times A = 240 \]
2. Twice Aneeta's age is more than Priya's age by 4 years:
\[ 2A = P + 4 \]

### Step 3: Express Priya's Age in Terms of Aneeta's Age
From the second equation, we can express \( P \) in terms of \( A \):
\[ P = 2A - 4 \]

### Step 4: Substitute into the First Equation
Now, substitute \( P \) in the first equation:
\[ (2A - 4) \times A = 240 \]

### Step 5: Expand and Rearrange the Equation
Expanding the equation gives:
\[ 2A^2 - 4A = 240 \]
Now, rearranging it to form a standard quadratic equation:
\[ 2A^2 - 4A - 240 = 0 \]

### Step 6: Simplify the Equation
To simplify, divide the entire equation by 2:
\[ A^2 - 2A - 120 = 0 \]

### Step 7: Factor the Quadratic Equation
Now, we need to factor the quadratic equation:
We are looking for two numbers that multiply to -120 and add to -2. The numbers are 10 and -12.
Thus, we can factor it as:
\[ (A - 12)(A + 10) = 0 \]

### Step 8: Solve for Aneeta's Age
Setting each factor to zero gives us:
1. \( A - 12 = 0 \) → \( A = 12 \)
2. \( A + 10 = 0 \) → \( A = -10 \) (not a valid age)

Thus, Aneeta's age is:
\[ A = 12 \]

### Conclusion
Aneeta's age is **12 years**.

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