Six years ago, the ratio of age of Kunal to Sagar was 6: 5 and four years hence ratio of age of Kunal to Sagar will be 11 : 10. Find the present age of Sagar?

Options

A. A)15 years
B. B)16 years
C. C)14 years
D. D)10 years

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let Kunal's present age be \( K \) years and Sagar's present age be \( S \) years.

### Step 2: Set up the equations based on the given ratios
From the information provided:
- Six years ago, the ratio of Kunal's age to Sagar's age was 6:5. This can be expressed as:
\[ \frac{K - 6}{S - 6} = \frac{6}{5} \]
Cross-multiplying gives us:
\[ 5(K - 6) = 6(S - 6) \]
Expanding this, we get:
\[ 5K - 30 = 6S - 36 \]
Rearranging gives us our first equation:
\[ 5K - 6S = -6 \quad \text{(Equation 1)} \]

- Four years hence, the ratio of Kunal's age to Sagar's age will be 11:10. This can be expressed as:
\[ \frac{K + 4}{S + 4} = \frac{11}{10} \]
Cross-multiplying gives us:
\[ 10(K + 4) = 11(S + 4) \]
Expanding this, we get:
\[ 10K + 40 = 11S + 44 \]
Rearranging gives us our second equation:
\[ 10K - 11S = 4 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( 5K - 6S = -6 \)
2. \( 10K - 11S = 4 \)

We can solve these equations simultaneously.

From Equation 1, we can express \( K \) in terms of \( S \):
\[ 5K = 6S - 6 \implies K = \frac{6S - 6}{5} \]

Substituting \( K \) into Equation 2:
\[ 10\left(\frac{6S - 6}{5}\right) - 11S = 4 \]
Multiplying through by 5 to eliminate the fraction:
\[ 10(6S - 6) - 55S = 20 \]
Expanding gives:
\[ 60S - 60 - 55S = 20 \]
Combining like terms:
\[ 5S - 60 = 20 \]
Adding 60 to both sides:
\[ 5S = 80 \]
Dividing by 5:
\[ S = 16 \]

### Step 4: Find Kunal's present age
Now substituting \( S = 16 \) back into the expression for \( K \):
\[ K = \frac{6(16) - 6}{5} = \frac{96 - 6}{5} = \frac{90}{5} = 18 \]

### Conclusion
The present age of Sagar is \( \boxed{16} \) years.

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