Evaluate the expression (3.39)^2 − (2.69)^2 and choose the nearest integer value of the result.

Introduction / Context:
This question involves squaring decimal numbers and then subtracting those squares. It checks your comfort with decimal arithmetic and also allows the use of an algebraic identity for a faster solution. Such problems are common in aptitude exams where accuracy with decimals is crucial.

Given Data / Assumptions:

• We need to evaluate (3.39)^2 − (2.69)^2.
• We are asked to select the nearest integer value from the given options.
• All calculations use standard decimal arithmetic.

Concept / Approach:
Instead of squaring both numbers directly, we can use the algebraic identity for difference of squares. The identity is a^2 − b^2 = (a − b) * (a + b). Applying this identity with a = 3.39 and b = 2.69 avoids long multiplication of decimals and greatly simplifies the work.

Step-by-Step Solution:
Step 1: Identify a and b. Here a = 3.39 and b = 2.69. Step 2: Use the identity a^2 − b^2 = (a − b) * (a + b). Step 3: Compute a − b = 3.39 − 2.69 = 0.70. Step 4: Compute a + b = 3.39 + 2.69 = 6.08. Step 5: Now compute the product: (a − b) * (a + b) = 0.70 * 6.08. Step 6: 0.70 * 6.08 = 4.256. Step 7: The exact result is 4.256, which is closest to the integer 4 among the given options.

Verification / Alternative check:
As a verification, we can roughly estimate. 3.39 is just under 3.4 and 2.69 is just under 2.7. Using approximate values, (3.4)^2 = 11.56 and (2.7)^2 = 7.29, so the difference is about 4.27, which is also closest to 4. This agrees with the more accurate calculation of 4.256. Therefore option 4 is confirmed as the correct nearest integer.

Why Other Options Are Wrong:
3 is farther from 4.256 than 4 is, so it is not the nearest integer.
6 is significantly larger than 4.256 and does not match the computed value.
1 is far too small when compared with the exact result.

Common Pitfalls:
Many learners try to square the decimals directly and may make mistakes with decimal places. Others forget the difference of squares identity and therefore perform longer calculations than necessary. Some might also misinterpret the question and subtract the numbers first and then square, which would give (3.39 − 2.69)^2 instead of a^2 − b^2. Always read the expression carefully and apply the correct algebraic identity.

Final Answer:
The value of (3.39)^2 − (2.69)^2 is approximately 4.256, so the nearest integer is 4.