A bag contains 1 rupee coins, 50 paise coins and 10 paise coins in the ratio 3 : 2 : 3, and the total amount of money is Rs. 12.90. How many 50 paise coins are there in the bag?

Introduction / Context:
This question on ratio and proportion involves different coin denominations contributing to a total amount. It checks whether the student can translate a ratio of counts into an algebraic expression for total money and then solve for the number of coins of a particular denomination, which is a very practical skill for handling mixtures of values.

Given Data / Assumptions:

• The bag contains three types of coins: 1 rupee coins, 50 paise coins, and 10 paise coins.• The ratio of the number of these coins is 3 : 2 : 3.• The total amount of money in the bag is Rs. 12.90.• We must find the number of 50 paise coins only.

Concept / Approach:
The ratio 3 : 2 : 3 represents the relative counts of each type of coin, not their values. Let the common multiplier be x, so the counts become 3x, 2x and 3x. Then we convert these counts into rupees by multiplying each count by the respective coin value: 1 rupee for the first type, 0.50 rupee for the second type, and 0.10 rupee for the third type. Summing these values gives a total expression in terms of x, which we equate to Rs. 12.90. Solving for x allows us to find the number of each coin, especially the 50 paise coins.

Step-by-Step Solution:
Step 1: Let the number of 1 rupee coins be 3x, the number of 50 paise coins be 2x, and the number of 10 paise coins be 3x.Step 2: Convert each type into rupees.• Value from 1 rupee coins = 3x * 1 = 3x rupees.• Value from 50 paise coins = 2x * 0.50 = 1x rupee.• Value from 10 paise coins = 3x * 0.10 = 0.30x rupee.Step 3: Total money in the bag = 3x + x + 0.30x = 4.30x rupees.Step 4: We are told that this total is Rs. 12.90, so 4.30x = 12.90.Step 5: Solve for x by dividing both sides by 4.30: x = 12.90 / 4.30 = 3.Step 6: Number of 50 paise coins = 2x = 2 * 3 = 6 coins.

Verification / Alternative check:
We can verify by computing the actual total value using x = 3. The counts are: 1 rupee coins = 3 * 3 = 9; 50 paise coins = 2 * 3 = 6; 10 paise coins = 3 * 3 = 9. Total value = 9 * 1 + 6 * 0.50 + 9 * 0.10 = 9 + 3 + 0.90 = 12.90 rupees. This matches the given total amount, confirming that x has been calculated correctly and that the number of 50 paise coins is 6.

Why Other Options Are Wrong:
• 3: This would correspond to x = 1.5, which does not produce the correct total amount of Rs. 12.90.• 9: This would require a different ratio multiplier and would give a total value larger than Rs. 12.90.• 12: This would drastically increase the total number of coins and hence the total amount, making it inconsistent with Rs. 12.90.

Common Pitfalls:
Learners often confuse the ratio of numbers of coins with the ratio of values of coins. Another mistake is to forget to convert paise into rupees properly and instead mix units such as treating 50 paise as 50 rupees by mistake. Some also forget that the ratio represents a common multiplier and try to solve directly without introducing the variable x. Finally, errors may occur if students round intermediate values incorrectly, though in this case the division is exact.

Final Answer:
The number of 50 paise coins in the bag is 6.