Simplify the expression (1.5^3 + 4.7^3 + 3.8^3 - 3 * 1.5 * 4.7 * 3.8) divided by (1.5^2 + 4.7^2 + 3.8^2 - 1.5 * 4.7 - 4.7 * 3.8 - 1.5 * 3.8) by using an algebraic identity and find its numerical value.

Introduction / Context:
This is an algebraic simplification problem that checks whether you recognise and correctly apply a standard identity for the sum of cubes. Instead of expanding all powers and computing a very long numerical expression, the problem becomes easy if you see the pattern. This type of question is common in aptitude and algebra sections because it rewards conceptual understanding of identities rather than brute force calculation.

Given Data / Assumptions:

• The expression in the numerator is 1.5^3 + 4.7^3 + 3.8^3 - 3 * 1.5 * 4.7 * 3.8.
• The expression in the denominator is 1.5^2 + 4.7^2 + 3.8^2 - 1.5 * 4.7 - 4.7 * 3.8 - 1.5 * 3.8.
• All numbers are real and standard arithmetic rules apply.
• We assume the denominator is not zero so the division is valid.

Concept / Approach:
The numerator and denominator match the well known algebraic identity involving three variables a, b, and c. The identity is: a^3 + b^3 + c^3 - 3abc = (a + b + c) * (a^2 + b^2 + c^2 - ab - bc - ca). This means that the given numerator is exactly equal to (a + b + c) multiplied by the denominator. Therefore, when we divide the numerator by the denominator, the complicated factor cancels out and the result is simply a + b + c. We only need to identify a, b, and c and then compute their sum.

Step-by-Step Solution:
Compare the numerator with a^3 + b^3 + c^3 - 3abc. We see that a = 1.5, b = 4.7, and c = 3.8. The denominator matches a^2 + b^2 + c^2 - ab - bc - ca. By the identity, a^3 + b^3 + c^3 - 3abc = (a + b + c) * (a^2 + b^2 + c^2 - ab - bc - ca). Therefore, the given fraction is: [(a^3 + b^3 + c^3 - 3abc)] / [a^2 + b^2 + c^2 - ab - bc - ca] = a + b + c. Compute the sum a + b + c = 1.5 + 4.7 + 3.8. First, 1.5 + 4.7 = 6.2. Then, 6.2 + 3.8 = 10. So, the value of the expression is 10.

Verification / Alternative check:
A direct verification method is to approximate both numerator and denominator numerically and then divide. Although it is tedious to compute each cube and each product exactly by hand, a calculator would confirm that the numerator is equal to 10 times the denominator. This agrees perfectly with the identity based solution. The identity method is preferred in exams, because it is much faster and avoids arithmetic errors with decimals.

Why Other Options Are Wrong:
6 and 8: These values are smaller than the true sum a + b + c and would correspond to an incorrect or partial application of the identity, or to miscalculation in adding the decimal numbers.
12 and 14: These are larger than the correct sum and might be guessed by students who think that the identity produces something like 2(a + b + c) or a more complicated factor. Correct use of the identity gives exactly a + b + c, which is 10.

Common Pitfalls:
A very common mistake is to not recognise the identity and start expanding each cube and product manually. This is time consuming and increases the chance of making arithmetic errors, especially with decimal numbers. Another pitfall is to misremember the identity, for example writing the denominator incorrectly. Also, some students add 1.5, 4.7, and 3.8 incorrectly by misplacing the decimal points. Careful use of the identity and correct addition of decimals avoids these issues.

Final Answer:
The simplified value of the given expression is 10.