A bag contains 50 paise, 25 paise and 10 paise coins in the ratio 5 : 9 : 4, and the total amount in the bag is Rs. 206. Find the number of 50 paise, 25 paise and 10 paise coins respectively.

Introduction / Context:
This is a standard ratio and proportion problem involving coins of different denominations. We know the ratio of the numbers of 50 paise, 25 paise and 10 paise coins, and we know the total value of all the coins together. The task is to find the exact number of each type of coin from the ratio and the total amount.

Given Data / Assumptions:
- Denominations are 50 paise, 25 paise and 10 paise. - Their counts are in the ratio 5 : 9 : 4. - Total amount in the bag is Rs. 206. - 1 rupee = 100 paise and values are 0.50, 0.25 and 0.10 rupees respectively.

Concept / Approach:
If the numbers of coins are 5x, 9x and 4x, then the total value is the sum of number of coins times the denomination of each type. Setting this expression equal to 206 gives an equation in x. Solving for x gives the exact counts. This is a direct application of ratio scaling and linear equations.

Step-by-Step Solution:
Step 1: Let the number of 50 paise coins be 5x, 25 paise coins be 9x, and 10 paise coins be 4x. Step 2: Value of one 50 paise coin is Rs. 0.50, so total value of 50 paise coins is 5x * 0.50 = 2.5x rupees. Step 3: Value of 25 paise coins is 9x * 0.25 = 2.25x rupees. Step 4: Value of 10 paise coins is 4x * 0.10 = 0.4x rupees. Step 5: Total value = 2.5x + 2.25x + 0.4x = 5.15x rupees. Step 6: Given that the total value is Rs. 206, we have 5.15x = 206. Step 7: Solve for x: x = 206 / 5.15 = 40. Step 8: Therefore number of 50 paise coins = 5x = 5 * 40 = 200. Step 9: Number of 25 paise coins = 9x = 9 * 40 = 360. Step 10: Number of 10 paise coins = 4x = 4 * 40 = 160.

Verification / Alternative check:
Compute the total value again using the found numbers. Value of 200 coins of 50 paise is 200 * 0.50 = Rs. 100. Value of 360 coins of 25 paise is 360 * 0.25 = Rs. 90. Value of 160 coins of 10 paise is 160 * 0.10 = Rs. 16. Adding up, 100 + 90 + 16 = Rs. 206, which matches the given total, so the counts are correct.

Why Other Options Are Wrong:
- 360, 160, 200 and 160, 360, 200: These permutations do not respect the original ratio order of 50 paise, 25 paise and 10 paise coins. - 200, 160, 300: This set gives a different total value and does not sum to Rs. 206.

Common Pitfalls:
Learners sometimes mix up the order of denominations and assign the ratio numbers incorrectly, or they convert paise to rupees wrongly. Another mistake is to treat 5 : 9 : 4 as direct counts without scaling them to meet the known total amount. Always assign the ratio numbers in the same order as the denominations given and then introduce a common multiplier x to solve the problem.

Final Answer:
The bag contains 200 coins of 50 paise, 360 coins of 25 paise and 160 coins of 10 paise.