Let's denote the present ages of Kunal and Sagar as $K$ and $S$, respectively.

Step 1: Form the equation based on the given ratio six years ago.

Six years ago, the ratio of Kunal's age to Sagar's age was 6:5. This gives us the following equation: $\frac{K - 6}{S - 6} = \frac{6}{5}$ Multiplying both sides by $5(S - 6)$ to eliminate the fraction: $5(K - 6) = 6(S - 6)$ Expanding both sides: $5K - 30 = 6S - 36$ Simplifying: $5K - 6S = -6 \quad \text{(Equation 1)}$

Step 2: Form the equation based on the given ratio four years hence.

Four years from now, the ratio of Kunal's age to Sagar's age will be 11:10. This gives us the following equation: $\frac{K + 4}{S + 4} = \frac{11}{10}$ Multiplying both sides by $10(S + 4)$ to eliminate the fraction: $10(K + 4) = 11(S + 4)$ Expanding both sides: $10K + 40 = 11S + 44$ Simplifying: $10K - 11S = 4 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations.

We now have two equations:

1. $5K - 6S = -6$
2. $10K - 11S = 4$

We can solve these using the elimination method. First, let's multiply Equation 1 by 2 to align the coefficients of $K$: $10K - 12S = -12$

Now subtract Equation 2 from this new equation: $(10K - 12S) - (10K - 11S) = -12 - 4$ Simplifying: $-12S + 11S = -16$ $-S = -16$ $S = 16$

Step 4: Find Kunal's age.

Substitute $S = 16$ into Equation 1: $5K - 6(16) = -6$ $5K - 96 = -6$ $5K = 90$ $K = 18$

Final Answer:

Sagar's present age is $\boxed{16}$.

Would you like more details or have any questions?

Here are 5 related questions to extend your learning:

1. If the ratio of their ages five years from now is 7:6, what will be Kunal’s age at that time?
2. How would you solve the problem if the ratio six years ago was reversed, i.e., 5:6?
3. What is the sum of Kunal’s and Sagar’s present ages?
4. If Kunal is currently 18 years old, what was his age 10 years ago?
5. How does solving systems of equations help in solving age problems?

Tip: When solving age-related problems, always align your time references (e.g., past, present, future) with the corresponding changes in age.