Simplify the fractional expression 3/4 + {3/4 + (3/4 ÷ (3/4 + 3/4))} using the correct order of operations and find its numerical value.

Introduction / Context:
This question tests simplification of a nested fractional expression using the correct order of operations. Many aptitude and competitive exams include such problems to check comfort with fractions, brackets, and division. The expression has the same fraction 3/4 repeated in different places, which can tempt a candidate to rush. A systematic approach based on BODMAS or PEMDAS avoids mistakes and gives the exact numerical value without needing a calculator.

Given Data / Assumptions:

• The given expression is 3/4 + {3/4 + (3/4 ÷ (3/4 + 3/4))}.

• All operations are ordinary real number operations on fractions.

• We must respect the standard order: brackets first, then division and multiplication, then addition and subtraction.

Concept / Approach:
The key idea is to simplify from the innermost brackets outward. First we add the fractions inside the innermost bracket, then perform the division 3/4 ÷ something, and finally add the outer 3/4 terms. Division of fractions is performed by multiplying by the reciprocal of the divisor. Keeping everything as exact fractions rather than decimals prevents rounding errors and makes checking easier.

Step-by-Step Solution:
Step 1: Compute the inner sum: 3/4 + 3/4 = 6/4 = 3/2. Step 2: Evaluate the division: 3/4 ÷ (3/2) = 3/4 × 2/3 = (3 × 2) / (4 × 3) = 6/12 = 1/2. Step 3: Now the bracket becomes 3/4 + 1/2. Convert 1/2 to a denominator of 4: 1/2 = 2/4. Step 4: Add these: 3/4 + 2/4 = 5/4. Step 5: Finally add the first 3/4: 3/4 + 5/4 = 8/4 = 2.

Verification / Alternative check:
We can convert each fraction to decimals as a rough check. We have 3/4 = 0.75. Inside the denominator, 3/4 + 3/4 = 1.5, so 0.75 ÷ 1.5 = 0.5. Then the whole expression is 0.75 + (0.75 + 0.5) = 0.75 + 1.25 = 2.00. This matches the exact fraction result of 2, so the calculation is consistent. Any other order that respects brackets will also lead to the same value.

Why Other Options Are Wrong:
Option 3/4 is obtained if someone mistakenly adds only the outer 3/4 terms and ignores the division part. Option 1 arises if 3/4 ÷ (3/4 + 3/4) is wrongly treated as 3/4 ÷ 3/4 = 1 without adding the second 3/4. Option 5/4 is the value of the expression before adding the very first 3/4, so it is an intermediate step, not the final answer. Option 0 would require negative or cancelling terms which do not appear here, so it is clearly incorrect.

Common Pitfalls:
A common mistake is to ignore the curly brackets and perform operations from left to right without respecting nesting. Another frequent error is to treat division of fractions as dividing numerators and denominators separately rather than multiplying by the reciprocal. Some learners also simplify 3/4 + 3/4 incorrectly as 3/8 instead of 6/4. Writing each step clearly and converting to a common denominator whenever fractions are added helps to avoid these issues in any similar aptitude question.

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