The ratio of Mona’s age to the age of her mother is 5 : 13. The difference of their ages is 24 years. The ratio of their ages after 3 years will be-

Options

A. 1:3
B. 2:3
C. 3:7
D. None of these

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define the Variables
Let Mona's age be represented as \(5x\) and her mother's age as \(13x\), based on the given ratio of their ages (5:13).

### Step 2: Set Up the Equation for the Age Difference
According to the problem, the difference in their ages is 24 years. Therefore, we can set up the equation:
\[ 13x - 5x = 24 \]

### Step 3: Simplify the Equation
Now, simplify the left side of the equation:
\[ 8x = 24 \]

### Step 4: Solve for \(x\)
To find the value of \(x\), divide both sides by 8:
\[ x = \frac{24}{8} = 3 \]

### Step 5: Calculate Their Current Ages
Now that we have \(x\), we can find Mona's and her mother's current ages:
- Mona's age:
\[ 5x = 5 \times 3 = 15 \text{ years} \]
- Mother's age:
\[ 13x = 13 \times 3 = 39 \text{ years} \]

### Step 6: Calculate Their Ages After 3 Years
Next, we need to find their ages after 3 years:
- Mona's age after 3 years:
\[ 15 + 3 = 18 \text{ years} \]
- Mother's age after 3 years:
\[ 39 + 3 = 42 \text{ years} \]

### Step 7: Find the Ratio of Their Ages After 3 Years
Now, we can find the ratio of their ages after 3 years:
\[ \text{Ratio} = \frac{\text{Mona's age}}{\text{Mother's age}} = \frac{18}{42} \]

### Step 8: Simplify the Ratio
To simplify the ratio, divide both the numerator and the denominator by their greatest common divisor (GCD), which is 6:
\[ \frac{18 \div 6}{42 \div 6} = \frac{3}{7} \]

### Final Answer
The ratio of Mona's age to her mother's age after 3 years is \(3:7\).

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