If A = (0.216 + 0.008) / (0.36 + 0.04 - 0.12) and B = (0.729 - 0.027) / (0.81 + 0.09 + 0.27), then what is the value of A^2 + B^2?

Introduction / Context:
This question checks your ability to recognise algebraic patterns inside decimal expressions. The decimals used resemble perfect cubes and squares, and the fractions are constructed so that algebraic identities can simplify them nicely. After simplifying A and B separately, you are asked to compute A^2 + B^2. Problems like this often appear in aptitude tests to see whether you can spot structure rather than simply perform long decimal arithmetic.

Given Data / Assumptions:

• A = (0.216 + 0.008) / (0.36 + 0.04 - 0.12).
• B = (0.729 - 0.027) / (0.81 + 0.09 + 0.27).
• We must compute A^2 + B^2.
• All decimals are exact and represent fractions derived from simple numbers.

Concept / Approach:
Notice that 0.216 is 0.6^3 and 0.008 is 0.2^3. Similarly, 0.36 is 0.6^2, 0.04 is 0.2^2 and 0.12 equals 0.6 * 0.2. These fit the identity (a^3 + b^3) / (a^2 - ab + b^2) = a + b. For B, 0.729 is 0.9^3, 0.027 is 0.3^3, 0.81 is 0.9^2, 0.09 is 0.3^2 and 0.27 equals 0.9 * 0.3, matching (a^3 - b^3) / (a^2 + ab + b^2) = a - b. Recognising these identities allows us to replace A and B with simple decimal values, and then we can easily compute A^2 + B^2.

Step-by-Step Solution:
Step 1: Let a = 0.6 and b = 0.2. Then a^3 = 0.216, b^3 = 0.008, a^2 = 0.36, b^2 = 0.04 and ab = 0.12. Step 2: A becomes (a^3 + b^3) / (a^2 - ab + b^2). By the identity, this equals a + b = 0.6 + 0.2 = 0.8. Step 3: Now let c = 0.9 and d = 0.3. Then c^3 = 0.729, d^3 = 0.027, c^2 = 0.81, d^2 = 0.09 and cd = 0.27. Step 4: B becomes (c^3 - d^3) / (c^2 + cd + d^2). By the identity, this equals c - d = 0.9 - 0.3 = 0.6. Step 5: Calculate A^2 + B^2 = (0.8)^2 + (0.6)^2 = 0.64 + 0.36 = 1.0.

Verification / Alternative check:
As an alternative, you can directly compute the decimals to confirm. The numerator of A is 0.224, the denominator is 0.28, so A = 0.224 / 0.28 = 0.8. For B, the numerator is 0.702 and the denominator is 1.17, which simplifies to 0.702 / 1.17 = 0.6. Squaring and adding gives 0.64 + 0.36 = 1. These direct calculations agree with the identity based approach and confirm that A^2 + B^2 is exactly 1.

Why Other Options Are Wrong:

• Option a (0.8) corresponds to A, not to the sum A^2 + B^2.
• Option c (1.4) might arise from adding A and B directly (0.8 + 0.6) instead of their squares.
• Option d (2.2) is too large and could come from incorrectly doubling 1.1 or miscomputing squares.
• Option e (1.0) repeats the correct value but as a different formatting of option b; the intended correct choice is the one labelled as 1.

Common Pitfalls:
A frequent mistake is to perform all operations using approximate decimals without noticing the underlying algebraic structure, which can lead to rounding errors. Some students might also confuse A^2 + B^2 with A + B or A^2 + B. Recognising that the decimals come from perfect cubes and squares, and applying identities, makes the computation more elegant and exact. Always read the expression carefully to see whether squares, sums, or differences are required.

Final Answer:
The value of A^2 + B^2 is 1.