In a shooting game a person takes 40 shots. He earns 50 paise for every successful hit and loses 10 paise for every miss. If his net gain at the end is Rs 5, how many hits did he make?

Introduction / Context:
This is a word problem involving linear equations and basic money calculations. The player earns money for each success and loses money for each failure across a fixed number of attempts. We are given the net gain and must determine how many of the attempts were successful. Problems like this are common in aptitude sections to test algebraic modeling skills using real-life style scenarios involving profit and loss per event.

Given Data / Assumptions:

• Total number of shots taken is 40.
• The player earns 50 paise (Rs 0.50) for each hit.
• The player loses 10 paise (Rs 0.10) for each miss.
• His net gain from all shots is Rs 5, which is 500 paise.
• We must find the number of hits.

Concept / Approach:
Let the number of hits be h and the number of misses be m. Since there are 40 shots in total, m = 40 − h. The total earnings from hits are 50h paise, and the total losses from misses are 10m paise. The net gain is earnings minus losses and must equal 500 paise. So we set up the equation 50h − 10m = 500 and substitute m = 40 − h, reducing it to a single equation in h. Solving this equation gives the required number of successful hits.

Step-by-Step Solution:
Step 1: Let the number of hits be h. Step 2: Since there are 40 shots in total, the number of misses is m = 40 − h. Step 3: Earnings from hits are 50h paise. Step 4: Losses from misses are 10m paise = 10(40 − h) paise. Step 5: Net gain in paise is earnings minus losses: 50h − 10(40 − h). Step 6: The net gain is given as Rs 5, which is 500 paise. Step 7: Set up the equation: 50h − 10(40 − h) = 500. Step 8: Expand: 50h − 400 + 10h = 500. Step 9: Combine like terms: 60h − 400 = 500. Step 10: Add 400 to both sides: 60h = 900. Step 11: Divide both sides by 60: h = 900 / 60 = 15. Step 12: Therefore, the player made 15 successful hits.

Verification / Alternative check:
Compute net gain using h = 15. Then m = 40 − 15 = 25. Earnings from hits: 15 * 50 paise = 750 paise. Losses from misses: 25 * 10 paise = 250 paise. Net gain = 750 − 250 = 500 paise, which is Rs 5. This matches the given net gain and confirms that 15 hits is correct.

Why Other Options Are Wrong:
Option (b) 20: Would give earnings 20 * 50 = 1000 paise and losses 20 * 10 = 200 paise, net 800 paise, not 500.
Option (c) 25: Would give earnings 25 * 50 = 1250 paise and losses 15 * 10 = 150 paise, net 1100 paise.
Option (d) 30: Would give earnings 30 * 50 = 1500 paise and losses 10 * 10 = 100 paise, net 1400 paise.

Common Pitfalls:
Common errors include miscounting the number of misses, mixing rupees and paise without converting, or wrongly adding earnings and losses instead of subtracting losses from earnings. Setting up the net gain equation carefully and keeping all values in the same units (here, paise) avoids these mistakes.

Final Answer:
The number of successful hits is 15.