This problem is about figuring out how many times Kumar hit the target and how many times he missed during a shooting game at a fun fair. We are given the total number of attempts, the payout for hitting, the penalty for missing, and his total winnings.

Understanding the Fun Fair Shooting Problem

Let's break down the information given in the problem:

• Total number of shots attempted by Kumar = 25
• Money received for each hit = 1 Rs
• Money paid for each miss = 50 paise = 0.5 Rs
• Total money won by Kumar = 10 Rs

We need to find out the number of times Kumar hit the target.

Setting Up Equations for the Shooting Attempts

Let's use variables to represent the unknowns:

• Let \(h\) be the number of times Kumar hit the target.
• Let \(m\) be the number of times Kumar missed the target.

Since the total number of shots was 25, the sum of hits and misses must be 25. This gives us our first equation:

\(h + m = 25\) (Equation 1)

Now let's consider the money won. For each hit, he gained 1 Rs, so his total earnings from hits are \(h \times 1 = h\) Rs. For each miss, he lost 0.5 Rs, so his total loss from misses is \(m \times 0.5 = 0.5m\) Rs. His total winnings are the total earnings from hits minus the total loss from misses.

\(\text{Total Winnings} = (\text{Number of Hits} \times \text{Gain per Hit}) - (\text{Number of Misses} \times \text{Loss per Miss})\)

\(10 = (h \times 1) - (m \times 0.5)\)

This gives us our second equation:

\(h - 0.5m = 10\) (Equation 2)

Solving the System of Equations

We now have a system of two linear equations with two variables:

\(h + m = 25\)

\(h - 0.5m = 10\)

We can solve this system using substitution or elimination. Let's use substitution. From Equation 1, we can express \(m\) in terms of \(h\):

\(m = 25 - h\)

Now, substitute this expression for \(m\) into Equation 2:

\(h - 0.5 \times (25 - h) = 10\)

Now, we solve for \(h\):

\(h - 0.5 \times 25 + 0.5 \times h = 10\)

\(h - 12.5 + 0.5h = 10\)

Combine the terms with \(h\):

\(1h + 0.5h - 12.5 = 10\)

\(1.5h - 12.5 = 10\)

Add 12.5 to both sides of the equation:

\(1.5h = 10 + 12.5\)

\(1.5h = 22.5\)

Divide by 1.5 to find the value of \(h\):

\(h = \frac{22.5}{1.5}\)

\(h = \frac{225}{15}\)

\(h = 15\)

So, Kumar hit the target 15 times.

Verifying the Solution

If Kumar hit the target 15 times (\(h = 15\)), then the number of misses is:

\(m = 25 - h = 25 - 15 = 10\)

Now, let's check the total winnings with 15 hits and 10 misses:

\(\text{Total Winnings} = (15 \times 1) - (10 \times 0.5)\)

\(\text{Total Winnings} = 15 - 5\)

\(\text{Total Winnings} = 10\) Rs

This matches the total winnings given in the problem, so our solution is correct.

Conclusion

By setting up and solving equations based on the total attempts and the total winnings, we found the number of times Kumar hit the target.

The number of times Kumar hit the target is 15.