Two-part split — Split 24 into two parts so that 7 times the first plus 5 times the second equals 146. What is the first part?

Introduction / Context:
This is a standard two-variable linear system framed as a partition problem. You divide a fixed total into two parts subject to a weighted-sum condition. Solving such problems efficiently is basic but essential practice for algebra and quantitative aptitude tests.

Given Data / Assumptions:

• Let the first part be x and the second part be y.
• x + y = 24 (the split of 24).
• 7x + 5y = 146 (weighted sum condition).

Concept / Approach:
Use substitution from the sum equation into the weighted equation, then solve for x. This minimizes steps and avoids unnecessary simultaneous manipulations. After finding x, back-calculate y if needed (not required here).

Step-by-Step Solution:
From x + y = 24, express y = 24 - x.Substitute into 7x + 5y = 146 → 7x + 5(24 - x) = 146.Simplify: 7x + 120 - 5x = 146 → 2x = 26.Solve: x = 13. Thus, the first part is 13.

Verification / Alternative check:
Compute y = 24 - 13 = 11; then 7*13 + 5*11 = 91 + 55 = 146, confirming correctness.

Why Other Options Are Wrong:

• 11 / 16 / 17 / 19: Substituting any of these into the equations fails to satisfy 7x + 5(24 - x) = 146.

Common Pitfalls:
Arithmetic slips when distributing 5; mixing x and y; forgetting to use the total sum equation first to reduce variables.

Final Answer:
13