If a = 1.2, b = 0.8 and c = 0.7, then what is the value of (a^3 + b^3 + c^3 - 2.016) ÷ [1.35((1.2)^2 + (0.8)^2 + (0.7)^2 - 0.96 - 0.84 - 0.56)]?

Introduction / Context:
This question is based on the algebraic identity for the sum of cubes of three numbers. The expression looks complicated because of decimals and many terms, but it is cleverly structured so that well known identities simplify it dramatically.

Given Data / Assumptions:
We are told:

• a = 1.2, b = 0.8, c = 0.7
• Numerator: a^3 + b^3 + c^3 - 2.016
• Denominator: 1.35[(1.2)^2 + (0.8)^2 + (0.7)^2 - 0.96 - 0.84 - 0.56]
• Note that 2.016 = 3 * 1.2 * 0.8 * 0.7 and that 0.96, 0.84 and 0.56 are pairwise products of a, b and c.

Concept / Approach:
The key algebraic identities are: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) and the structure in the denominator resembles (a + b + c) or a multiple of it times (a^2 + b^2 + c^2 - ab - bc - ca). Recognising that 2.016 = 3abc and that 1.35 equals (a + b + c) / 2 allows us to reduce the entire expression to a simple constant.

Step-by-Step Solution:
First compute a + b + c = 1.2 + 0.8 + 0.7 = 2.7. Compute abc = 1.2 * 0.8 * 0.7 = 0.672. Then 3abc = 3 * 0.672 = 2.016, which matches the term in the numerator. So numerator = a^3 + b^3 + c^3 - 3abc. Using the identity, numerator = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). Compute a^2 + b^2 + c^2: 1.2^2 = 1.44, 0.8^2 = 0.64, 0.7^2 = 0.49, total = 2.57. Compute ab + bc + ca: ab = 0.96, bc = 0.56, ca = 0.84, total = 2.36. Hence a^2 + b^2 + c^2 - ab - bc - ca = 2.57 - 2.36 = 0.21. So numerator = (2.7) * (0.21) = 0.567. Next, observe that 1.35 = 2.7 / 2 = (a + b + c) / 2. Denominator = 1.35 * (a^2 + b^2 + c^2 - ab - bc - ca) = (2.7 / 2) * 0.21 = 0.2835. Therefore the required value is 0.567 / 0.2835 = 2.

Verification / Alternative check:
Direct calculator based computation also yields about 2.000, confirming the algebraic simplification. The identity based approach is more instructive and avoids round off errors in competitive exams.

Why Other Options Are Wrong:
1/4, 1/2 and 1 are much smaller than 2 and would arise only if a student forgot the factor 1.35 in the denominator or misapplied the identity. 3/4 is also inconsistent with the ratio 0.567 to 0.2835, which is almost exactly 2.

Common Pitfalls:
A frequent mistake is to cube decimals incorrectly or to ignore that 2.016 equals 3abc. Another common error is to assume that 1.35 is arbitrary and not related to a + b + c. Recognising patterns in expressions and linking them to standard identities is crucial for solving such problems quickly and accurately.

Final Answer:
The value of the given expression is 2.