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

### Concept & Logic This is a classic linear equation problem based on dividing a known total into two unknown parts. By expressing one part as $x$, the other part is naturally forced to be $(\text{Total} - x)$. This allows us to translate the given conditions into a single variable linear equation. ### Step-by-Step Solution * **Given:** * Total sum = 54 * Condition: $10 \times (\text{First Part}) + 22 \times (\text{Second Part}) = 780$ * **Calculation:** 1. Let the first part be $x$. 2. Since the two parts add up to 54, the second part must be $(54 - x)$. 3. Translate the condition into an algebraic equation: $$ 10x + 22(54 - x) = 780 $$ 4. Distribute the 22 into the parenthesis: $$ 10x + (22 \times 54) - 22x = 780 $$ 5. Calculate $22 \times 54$. (Hint: $22 \times 50 = 1100$, and $22 \times 4 = 88$. So, $1188$). $$ 10x + 1188 - 22x = 780 $$ 6. Combine the $x$ terms: $$ -12x + 1188 = 780 $$ 7. Isolate the $x$ term by moving constants to one side: $$ 1188 - 780 = 12x $$ $$ 408 = 12x $$ 8. Solve for $x$: $$ x = \frac{408}{12} $$ $$ x = 34 $$ 9. The first part is 34. The second part is $54 - 34 = 20$. 10. The question asks for the bigger part, which is 34. ### Exam Strategy & Shortcut **Mixture & Alligation Method (Fastest Method):** Treat this as finding the weighted average that blends two groups. Imagine if the whole 54 was just the first part. The sum would be $54 \times 10 = 540$. Imagine if the whole 54 was just the second part. The sum would be $54 \times 22 = 1188$. The actual sum is 780. Using Alligation layout: Part 1 (10x) : 540 Part 2 (22x) : 1188 Middle (Target) : 780 Diagonal Differences: Ratio of Part 1 to Part 2 = $(1188 - 780) : (780 - 540)$ $= 408 : 240$ Divide by 24 to simplify: $= 17 : 10$ The parts are split in the ratio $17 : 10$. Total parts = $17 + 10 = 27$. Value of 1 ratio unit = $\frac{54}{27} = 2$. Bigger part (17 units) = $17 \times 2 = 34$. ### Common Pitfall A common error is solving for $x$ and assuming it is automatically the final answer. If you set $x$ as the second part in your initial assumption, you would find $x = 20$, and you might incorrectly select an option if 20 was listed. Always verify which part is the "bigger part" before answering. ### Final Answer **Therefore, the correct answer is 34.**