Split a sum with a linear condition: Divide 54 into two parts such that 10 times the first plus 22 times the second equals 780. What is the larger part?

Introduction / Context:
Partition problems with a linear condition are routine applications of two equations in two unknowns. Here, the total is fixed at 54 and a weighted sum equals 780. We will solve for the parts and identify the larger one.

Given Data / Assumptions:

• Let the first part be x and the second part be y.
• x + y = 54.
• 10x + 22y = 780.

Concept / Approach:
Use substitution from x + y = 54 ⇒ y = 54 − x, then plug into the second equation and solve for x. This produces an exact integer solution, making identification of the larger part direct.

Step-by-Step Solution:

Verification / Alternative check:
Compute 10*34 + 22*20 = 340 + 440 = 780, matching perfectly, while x + y = 54 holds.

Why Other Options Are Wrong:

• 24, 30, 32, 20: None satisfy both equations simultaneously as the larger part.

Common Pitfalls:
Mixing up which variable you substituted, or solving 10x + 22y = 780 without using x + y = 54. Always leverage both equations.

Final Answer:
34