Rs. 782 is divided into three parts in the ratio 12 : 23 : 34. What is the amount of the first part?

Introduction / Context:
This problem is a straightforward application of ratio and proportion where a fixed sum of money is divided into three parts according to a given ratio. The question asks specifically for the first part. These questions are very common in bank exams, SSC, and other competitive tests, and they reinforce the idea that ratios represent relative shares of a total quantity.

Given Data / Assumptions:

• Total amount = Rs. 782.
• Ratio of the three parts = 12 : 23 : 34.
• The first part corresponds to the term 12 in the ratio.
• All three parts are non negative and add up exactly to Rs. 782.

Concept / Approach:
If a sum S is divided into parts in the ratio a : b : c, then the first part is (a / (a + b + c)) * S, the second part is (b / (a + b + c)) * S and the third part is (c / (a + b + c)) * S. Here, a = 12, b = 23 and c = 34. Once we compute the sum of the ratio terms, we can easily compute the value of one part by simple multiplication and division.

Step-by-Step Solution:
Given ratio of parts = 12 : 23 : 34.Compute the total of ratio terms: 12 + 23 + 34 = 69.Total amount S = Rs. 782.Value of one ratio unit = 782 / 69.Compute 782 / 69 = 11.34 approximately, but we want the exact first part.First part = 12 * (782 / 69) = (12 * 782) / 69.Calculate the numerator: 12 * 782 = 9384.Now divide 9384 by 69 to get 136.Therefore, the first part is Rs. 136.

Verification / Alternative check:
Compute all three parts to verify the total.First part: 12 * (782 / 69) = 136.Second part: 23 * (782 / 69) = 23 * 136 / 12 ≈ 260.67, but more precisely 260 is near when rounded; in exam contexts the first part is usually the focus.Third part: 34 * (782 / 69) gives the remaining share so that the total is 782.Adding the three exact parts computed using the ratio method gives back the original total 782, so the division is consistent with the ratio 12 : 23 : 34.

Why Other Options Are Wrong:
Values like Rs. 182, Rs. 190, Rs. 196 or Rs. 204 do not fit the ratio correctly. For instance, if the first part were Rs. 182, then the value per ratio unit would be 182 / 12, and the implied total amount would not equal Rs. 782 when the three parts are recomputed. Only Rs. 136 leads to a consistent division of the total in the ratio 12 : 23 : 34.

Common Pitfalls:
Many students mistakenly divide 782 directly by 3 or confuse the ratio itself with the actual amounts. Another error is to miscalculate the sum of the ratio terms or to perform division incorrectly. Always remember the formula: first part = (first ratio term / sum of all ratio terms) * total amount. Careful arithmetic avoids unnecessary mistakes.

Final Answer:
The amount of the first part is Rs. 136.