Rs. 69 is divided among 115 family members so that each lady receives 50 paise less than each gentleman, and each gentleman receives twice as many paise as each lady. How many gentlemen are there in the family?

Introduction / Context:
This question combines basic algebra with a real life style distribution scenario. The total money is given, along with conditions relating how much each gentleman and each lady receives. We need to determine how many gentlemen are in the family. The problem is solved by converting rupees to paise, setting up equations based on the payment conditions and the total number of members, and then solving for the unknown counts.

Given Data / Assumptions:

Total money to be distributed = Rs. 69.
Total family members = 115.
Each gentleman receives twice as many paise as each lady.
Each lady receives 50 paise less than each gentleman.
We are asked to find the number of gentlemen.
All calculations are easier if we work in paise instead of rupees.

Concept / Approach:
First convert 69 rupees into paise to avoid decimals: 1 rupee = 100 paise, so 69 rupees = 6,900 paise. Let the amount received by each lady be L paise and by each gentleman be G paise. Then G is given as twice L, and at the same time, G is also 50 paise more than L. Using these relationships, we find L and G. Then we use the total amount equation in terms of the number of ladies and gentlemen and the total count of 115 to solve for how many gentlemen there are.

Step-by-Step Solution:
Step 1: Convert total amount to paise. Total money = 69 rupees = 69 * 100 = 6,900 paise. Step 2: Let each lady receive L paise, each gentleman receive G paise. Given: Each gentleman receives twice as many paise as each lady, so G = 2L. Also given: Each lady receives 50 paise less than each gentleman, so G - L = 50. Step 3: Use G = 2L in G - L = 50. 2L - L = 50. So L = 50 paise. Then G = 2L = 2 * 50 = 100 paise. Step 4: Let the number of gentlemen be g and the number of ladies be l. Total members: g + l = 115. Total money in paise: 100g + 50l = 6,900. Step 5: Use l = 115 - g in the money equation. 100g + 50(115 - g) = 6,900. 100g + 5,750 - 50g = 6,900. (100g - 50g) + 5,750 = 6,900. 50g + 5,750 = 6,900. Step 6: Subtract 5,750 from both sides. 50g = 1,150. g = 1,150 / 50 = 23. So there are 23 gentlemen in the family.

Verification / Alternative check:
If there are 23 gentlemen, then the number of ladies is 115 - 23 = 92. Total money given to gentlemen is 23 * 100 = 2,300 paise. Total money given to ladies is 92 * 50 = 4,600 paise. Combined, this is 2,300 + 4,600 = 6,900 paise, which equals 69 rupees. This matches the original total, so the distribution is consistent and the count of 23 gentlemen is correct.

Why Other Options Are Wrong:
If there were 92 gentlemen instead, the number of ladies would be 23, and the total payout would not equal 6,900 paise.
Similarly, choices like 48, 52 or 27 for the count of gentlemen do not satisfy both the total members equation and the total money equation when tested.

Common Pitfalls:
Some learners forget to convert rupees to paise and work with decimals incorrectly. Others misinterpret the phrase that each gentleman receives twice as many paise as each lady and may set up the relationships incorrectly. It is also easy to mix up the conditions about who gets more and by how much. Carefully setting up both the value equations (G = 2L and G - L = 50) and the count equations ensures a correct and straightforward solution.

Final Answer:
The number of gentlemen in the family is 23.