Kalyan is twice as old as his sister and five times younger than his father. After two years their combined age will be 58. How old is Kalyan's sister now?

Options

A. 5
B. 4
C. 6
D. 3

Answer

To solve the problem step by step, we will define the ages of Kalyan, his sister, and his father based on the information given in the question.

### Step 1: Define Variables
Let Kalyan's age be \( x \).

### Step 2: Express the Ages of Others
According to the problem:
- Kalyan's sister is half his age, so her age is \( \frac{x}{2} \).
- Kalyan's father is five times Kalyan's age, so his age is \( 5x \).

### Step 3: Set Up the Equation for Future Ages
After 2 years, their ages will be:
- Kalyan's age: \( x + 2 \)
- Sister's age: \( \frac{x}{2} + 2 \)
- Father's age: \( 5x + 2 \)

The combined age after 2 years will be 58. Therefore, we can set up the equation:
\[ (x + 2) + \left(\frac{x}{2} + 2\right) + (5x + 2) = 58 \]

### Step 4: Simplify the Equation
Now, simplify the left side of the equation:
\[ x + 2 + \frac{x}{2} + 2 + 5x + 2 = 58 \]
Combine like terms:
\[ x + 5x + \frac{x}{2} + 6 = 58 \]
This simplifies to:
\[ 6x + \frac{x}{2} + 6 = 58 \]

### Step 5: Eliminate the Fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 2(6x) + 2\left(\frac{x}{2}\right) + 2(6) = 2(58) \]
This results in:
\[ 12x + x + 12 = 116 \]
Combine like terms:
\[ 13x + 12 = 116 \]

### Step 6: Solve for \( x \)
Subtract 12 from both sides:
\[ 13x = 116 - 12 \]
\[ 13x = 104 \]
Now, divide by 13:
\[ x = \frac{104}{13} = 8 \]

### Step 7: Find Kalyan's Sister's Age
Now that we know Kalyan's age \( x = 8 \), we can find his sister's age:
\[ \text{Sister's age} = \frac{x}{2} = \frac{8}{2} = 4 \]

### Conclusion
Kalyan's sister is currently **4 years old**.

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