10 years ago the age of Karishna was `1/3` rd of the age of Babita. 14 years hence the ratio of ages of Karishma and Babita will be 5:9. Find the ratio of their present ages:

Options

A. `13:29`
B. `11:27`
C. `29:17`
D. `13:25`

Answer

To solve the problem step by step, we will follow the information given in the question about the ages of Karishma and Babita.

### Step 1: Define Variables
Let the age of Karishma 10 years ago be \( x \). Therefore, the age of Babita 10 years ago will be \( 3x \) (since Karishma's age was \( \frac{1}{3} \) of Babita's age).

### Step 2: Express Present Ages
- Present age of Karishma = \( x + 10 \)
- Present age of Babita = \( 3x + 10 \)

### Step 3: Set Up the Future Age Ratio
According to the problem, 14 years from now, the ratio of their ages will be \( 5:9 \). Therefore, we can express this as:
\[ \frac{(x + 10 + 14)}{(3x + 10 + 14)} = \frac{5}{9} \]

### Step 4: Simplify the Equation
This simplifies to:
\[ \frac{(x + 24)}{(3x + 24)} = \frac{5}{9} \]

### Step 5: Cross Multiply
Cross multiplying gives us:
\[ 9(x + 24) = 5(3x + 24) \]

### Step 6: Expand Both Sides
Expanding both sides results in:
\[ 9x + 216 = 15x + 120 \]

### Step 7: Rearrange the Equation
Rearranging the equation to isolate \( x \):
\[ 9x - 15x = 120 - 216 \]
\[ -6x = -96 \]
\[ x = 16 \]

### Step 8: Calculate Present Ages
Now that we have \( x \), we can find the present ages:
- Present age of Karishma = \( x + 10 = 16 + 10 = 26 \)
- Present age of Babita = \( 3x + 10 = 3(16) + 10 = 48 + 10 = 58 \)

### Step 9: Find the Ratio of Present Ages
Now, we find the ratio of their present ages:
\[ \text{Ratio} = \frac{26}{58} = \frac{13}{29} \]

### Final Answer
The ratio of their present ages is \( 13:29 \).

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