If a sum of Rs 782 is divided into three parts in the ratio 12 : 23 : 34, what is the value of the first part?

Introduction / Context:
This aptitude question tests the basic concept of dividing a total amount into proportional parts. Such questions are very common in bank exams, SSC exams and campus placement tests where ratio and proportion is an essential topic.

Given Data / Assumptions:

• Total amount = Rs 782
• The amount is divided into three parts in the ratio 12 : 23 : 34
• We are required to find the value of the first part corresponding to ratio term 12

Concept / Approach:
When a sum is divided in a given ratio a : b : c, the individual shares are found by first computing the sum of ratio terms and then multiplying the total amount by each term divided by that sum. That is, share for a = Total * a / (a + b + c). This is a direct application of the idea of proportional division.

Step-by-Step Solution:
Step 1: Compute the sum of the ratio terms: 12 + 23 + 34 = 69. Step 2: The first part corresponds to 12 out of the 69 total parts. Step 3: First part = 782 * 12 / 69. Step 4: Simplify 782 / 69 = 782 ÷ 69 = 11.333... = 34 / 3 exactly. Step 5: So first part = (34 / 3) * 12 = 34 * 4 = 136. Step 6: Therefore, the value of the first part is Rs 136.

Verification / Alternative check:
We can find all three parts to verify. First part = 136. Remaining amount = 782 - 136 = 646. The remaining ratio terms are 23 and 34, whose sum is 57. One part size here is 646 / 57 = 11.333..., again equal to 34 / 3. Second part = 23 * 34 / 3 = 782 / 3. Third part = 34 * 34 / 3. Adding all three parts gives back 782, which confirms that 136 is correct for the first part.

Why Other Options Are Wrong:
190, 192, 182 and 204 all correspond to different ratios when plugged back with the other parts and do not maintain the exact 12 : 23 : 34 proportion with a total of 782. None of them satisfies the proportional division formula Total * 12 / 69 exactly.

Common Pitfalls:
A common mistake is to divide 782 directly by 3 or by one of the ratio numbers instead of by the sum 69. Another error is to round intermediate values too early instead of keeping the exact fraction. Forgetting that every ratio problem must keep the same scale factor for all terms also leads to wrong answers.

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