A man has only 20 paise coins and 25 paise coins in a bag; if he has 50 coins in all and the total value of the coins is Rs 10.25, then how many 20 paise coins does he have?

Introduction / Context:
This is a classic coin problem involving two denominations and a total count. We are given the total number of coins and the total monetary value, and we must determine how many coins of one denomination are present. Such problems test basic algebra, particularly forming and solving simultaneous linear equations with two variables under logical constraints.

Given Data / Assumptions:
- Only 20 paise and 25 paise coins are present in the bag.
- Total number of coins = 50.
- Total value of all coins = Rs 10.25 = 1025 paise.
- Let number of 20 paise coins = x, number of 25 paise coins = y.
- x and y are non negative integers and x + y = 50.

Concept / Approach:
We use two equations. The first equation represents the total number of coins. The second equation represents the total value in paise. Solving these two equations simultaneously gives the values of x and y. Finally, we match the value of x with the given options to find the correct answer.

Step-by-Step Solution:
Let x = number of 20 paise coins and y = number of 25 paise coins. Total coins: x + y = 50. Total value in paise: 20x + 25y = 1025. From x + y = 50, we get y = 50 − x. Substitute in the value equation: 20x + 25(50 − x) = 1025. This gives 20x + 1250 − 25x = 1025. Simplify: −5x + 1250 = 1025, so −5x = −225, hence x = 45. Therefore, the man has 45 coins of 20 paise each.

Verification / Alternative check:
If x = 45, then y = 50 − 45 = 5. Value of 20 paise coins = 45 * 20 = 900 paise. Value of 25 paise coins = 5 * 25 = 125 paise. Total value = 900 + 125 = 1025 paise = Rs 10.25, which matches the given amount exactly.

Why Other Options Are Wrong:
If x = 42, then y = 8 and total value = 42*20 + 8*25 = 840 + 200 = 1040 paise, which is too high.
If x = 38, then y = 12 and total value = 38*20 + 12*25 = 760 + 300 = 1060 paise, incorrect.
If x = 36, then y = 14 and total value = 36*20 + 14*25 = 720 + 350 = 1070 paise, also incorrect.

Common Pitfalls:
Common errors include mixing rupees and paise without converting units, miswriting the value equation, or forgetting that x and y must sum to 50. Careful conversion of rupees to paise and methodical algebra help avoid these mistakes.

Final Answer:
The man has 45 coins of 20 paise.