The number 54 is to be divided into two parts such that the sum of 10 times the first part and 22 times the second part is 780. What is the value of the bigger part?

Introduction:
This is a standard linear equation word problem involving two parts of a given total. The problem asks you to split 54 into two numbers so that a weighted sum (10 times one part plus 22 times the other) equals 780. Such questions test your ability to convert a word statement into a pair of algebraic equations and solve them consistently.

Given Data / Assumptions:

• The total of the two parts is 54.
• Let the first part be x and the second part be y.
• x + y = 54.
• 10x + 22y = 780.
• We must find the bigger of x and y.

Concept / Approach:
Use simultaneous equations. From the total equation, express one variable in terms of the other (for example, y = 54 - x). Substitute this into the second equation to obtain a single equation in x. Solve for x, compute y, and determine which is larger. This is a direct application of solving two linear equations in two unknowns.

Step-by-Step Solution:
Let first part = x and second part = y.Total: x + y = 54 => y = 54 - x.Condition: 10x + 22y = 780.Substitute y: 10x + 22(54 - x) = 780.10x + 1188 - 22x = 780.-12x + 1188 = 780.-12x = 780 - 1188 = -408.x = (-408)/(-12) = 34.Then y = 54 - 34 = 20.So the bigger part is 34.

Verification / Alternative check:
Check the condition: 10 * 34 + 22 * 20 = 340 + 440 = 780, which matches exactly. The total 34 + 20 = 54 is also satisfied. Thus the split (34, 20) is correct and 34 is indeed the larger part.

Why Other Options Are Wrong:
24, 44, 30, 20: none of these values, when paired with 54 minus that value, satisfy the equation 10x + 22y = 780. A quick substitution shows the weighted sum is different from 780.

Common Pitfalls:
Confusing which coefficient (10 or 22) multiplies which part when writing the equation.Arithmetic errors while simplifying 10x + 22(54 - x).Forgetting to verify both the total 54 and the weighted sum 780 after solving.

Final Answer:
34