If 3a = 4b = 6c and a + b + c = 279, then what is the value of a^2 + b^2 + c^2?

Introduction / Context:
This question involves proportional relationships between three variables and asks for a sum of squares. It is typical of algebra and aptitude tests where candidates must convert a relation like 3a = 4b = 6c into useful expressions for a, b, and c, and then compute required symmetric combinations.

Given Data / Assumptions:

• 3a = 4b = 6c = k for some constant k.
• a + b + c = 279.
• We need to find a^2 + b^2 + c^2.
• All variables are assumed to be real numbers.

Concept / Approach:
When an expression like 3a = 4b = 6c is given, we usually set each equal to a common constant k. Then we express a, b, and c in terms of k. Using the sum a + b + c, we find k. Once we know a, b, and c explicitly, we can compute their squares and sum them. This method avoids solving simultaneous equations separately.

Step-by-Step Solution:
Step 1: Let 3a = 4b = 6c = k.Step 2: Express the variables: a = k / 3, b = k / 4, c = k / 6.Step 3: Use the given sum a + b + c = 279.Step 4: Substitute in terms of k: k/3 + k/4 + k/6 = 279.Step 5: Find a common denominator, which is 12. Then k/3 = 4k/12, k/4 = 3k/12, k/6 = 2k/12.Step 6: Add: (4k + 3k + 2k)/12 = 9k/12 = 3k/4.Step 7: So 3k/4 = 279, giving k = 279 * 4 / 3.Step 8: Compute k = (279 / 3) * 4 = 93 * 4 = 372.Step 9: Now a = 372 / 3 = 124, b = 372 / 4 = 93, c = 372 / 6 = 62.Step 10: Compute squares: a^2 = 124^2 = 15376, b^2 = 93^2 = 8649, c^2 = 62^2 = 3844.Step 11: Sum them: a^2 + b^2 + c^2 = 15376 + 8649 + 3844.Step 12: 15376 + 8649 = 24025, and 24025 + 3844 = 27869.

Verification / Alternative check:
We can verify by checking that our values of a, b, and c satisfy the original relationship 3a = 4b = 6c. Compute 3*124 = 372, 4*93 = 372, and 6*62 = 372. All are equal to 372, so the relationship holds, confirming that the values are correct and the sum of squares 27869 is reliable.

Why Other Options Are Wrong:

• 27969, 27789, 27896: These numbers are close to 27869 but arise from slight arithmetic errors such as mis adding the squares or mis squaring one of the numbers. Only 27869 exactly matches 124^2 + 93^2 + 62^2.

Common Pitfalls:
A common source of error is misinterpreting 3a = 4b = 6c as three separate equations rather than introducing a single constant k. Another frequent mistake is arithmetic error in squaring large numbers or in adding them. Breaking the calculation into clear steps reduces such mistakes.

Final Answer:
The value of a^2 + b^2 + c^2 is 27869.