A bag contains a total of 324 coins consisting only of 20 paise coins and 25 paise coins. The total value of all the coins together is Rs 71. How many 20 paise coins are there in the bag?

Introduction / Context:
This question is a standard mixture and coins problem from quantitative aptitude. It involves two denominations of coins, their total count, and their combined value. Such problems test your ability to convert a verbal description into algebraic equations, to handle paise and rupees correctly, and to solve simultaneous linear equations. These are very common in competitive exams, where candidates must quickly set up equations and find the correct number of each type of coin.

Given Data / Assumptions:

• Total number of coins in the bag: 324.
• The bag contains only 20 paise coins and 25 paise coins.
• Total value of all coins = Rs 71.
• 1 rupee = 100 paise.
• We assume every 20 paise coin and every 25 paise coin is genuine and identical in value within its type.

Concept / Approach:
The core approach is to let the number of 20 paise coins and 25 paise coins be variables and then write two equations. The first equation is based on the total number of coins. The second equation is based on the total value, expressed in paise to avoid decimals. After that, we solve the two simultaneous equations to determine the number of each type of coin. Finally, we read off how many 20 paise coins are present, which is what the problem asks us to find.

Step-by-Step Solution:
Let the number of 20 paise coins be x. Let the number of 25 paise coins be y. Total number of coins: x + y = 324. Convert the total value to paise: Rs 71 = 7100 paise. Value equation in paise: 20 * x + 25 * y = 7100. From x + y = 324, express y as y = 324 - x. Substitute into the value equation: 20 * x + 25 * (324 - x) = 7100. Simplify: 20 * x + 25 * 324 - 25 * x = 7100. This gives: 20x - 25x + 8100 = 7100, so -5x + 8100 = 7100. Rearrange: -5x = 7100 - 8100 = -1000, so x = 200. Therefore, the number of 20 paise coins is 200.

Verification / Alternative check:
We can verify the numbers by computing the value again. If x = 200, then y = 324 - 200 = 124. Value of 20 paise coins = 200 * 20 = 4000 paise. Value of 25 paise coins = 124 * 25 = 3100 paise. Total value = 4000 + 3100 = 7100 paise, which equals Rs 71. This matches the given total value, so the solution is correct.

Why Other Options Are Wrong:
50 coins: If there were only 50 coins of 20 paise, the remaining 274 coins would be 25 paise; the resulting value would not match Rs 71.
100 coins: With 100 coins of 20 paise and 224 coins of 25 paise, the total value exceeds Rs 71.
150 coins: With 150 and 174 coins of each type, the total value again does not equal Rs 71.
175 coins: Similar checking shows that using 175 coins of 20 paise does not give the correct total value either. Only 200 fits both equations exactly.

Common Pitfalls:
A very common mistake is to mix rupees and paise in the same equation without converting to one unit, which leads to incorrect arithmetic. Another error is to misinterpret the total number of coins as the total value, or to incorrectly set up the second equation. Some students also forget that the total value must be exactly Rs 71, not approximately. Using paise throughout and proceeding carefully with algebra avoids these pitfalls.

Final Answer:
The bag contains 200 coins of 20 paise each.