An amount of Rs 782 is divided into three parts in the proportion 1/2 : 2/3 : 3/4. What is the value of the first part?

Introduction / Context:
This question involves dividing a sum of money into three parts according to a ratio given in fractional form. Instead of simple integers, the ratio uses fractions such as 1/2, 2/3 and 3/4. The main idea is to convert these fractional parts into an equivalent integer ratio and then use standard proportional division to allocate the total amount. This is a useful skill for handling more complex ratio scenarios.

Given Data / Assumptions:

• Total amount to be divided = Rs 782.
• The three parts are proportional to 1/2, 2/3 and 3/4.
• We must find the first part corresponding to 1/2.
• The entire amount is exactly divisible according to this ratio.

Concept / Approach:
To simplify calculations, we first convert 1/2, 2/3 and 3/4 into a common integer ratio by finding a common denominator. Multiplying each fraction by the least common multiple of the denominators yields equivalent whole numbers. Once we have an integer ratio, we sum the ratio parts to find the total number of shares. Each share corresponds to a fixed amount of money. Multiplying the value per share by the number of shares in the first part gives the required answer.

Step-by-Step Solution:
Fractions: 1/2, 2/3 and 3/4. Least common multiple of denominators 2, 3 and 4 is 12. Multiply each fraction by 12: 1/2 * 12 = 6, 2/3 * 12 = 8, 3/4 * 12 = 9. So the integer ratio is 6 : 8 : 9. Sum of ratio parts = 6 + 8 + 9 = 23 parts. Total amount Rs 782 corresponds to 23 parts. Value of one part = 782 / 23 = 34. The first part corresponds to 6 parts. First part = 6 * 34 = Rs 204.

Verification / Alternative check:
We can compute the other two parts to confirm consistency. The second part is 8 * 34 = Rs 272, and the third part is 9 * 34 = Rs 306. Adding them gives 204 + 272 + 306 = Rs 782, which matches the total amount. Since the distribution is exact and consistent, Rs 204 for the first part is correct.

Why Other Options Are Wrong:
Rs 408 is double the correct first part and corresponds to misreading the ratio or doubling 6 parts incorrectly. Rs 412 and Rs 220 do not respect the integer ratio when the remaining parts are computed. Rs 272 is the value of the second part, not the first. Only Rs 204 fits the correct proportional division.

Common Pitfalls:
Some learners attempt to use the fractions directly without converting to an integer ratio, which can lead to complicated arithmetic. Others may incorrectly use the numerators 1, 2 and 3 alone, ignoring denominators. Always convert fractional ratios using a common denominator so that the parts are truly proportional.

Final Answer:
The first part of the division is Rs 204.