If the notation 6.46 represents the recurring decimal 6.464646..., what is the correct fractional form of this recurring decimal?

Introduction:
This question deals with converting a recurring decimal to a fraction. The notation 6.46 here is intended to represent the repeating decimal 6.464646..., where the block "46" repeats indefinitely. Converting recurring decimals to fractions is an important algebraic skill often tested in aptitude exams.

Given Data / Assumptions:

• Recurring decimal: 6.464646..., where 46 repeats forever.
• We must express this number as a fraction.
• The answer should be in simplest fractional form.

Concept / Approach:
The standard method is to use algebra. Let x be the recurring decimal. Because two digits repeat, we multiply x by 100 to shift the decimal point two places to the right, creating a new equation that allows the repeating part to cancel when we subtract x from 100x.

Step-by-Step Solution:
Step 1: Let x = 6.464646...Step 2: Multiply both sides by 100 to move the decimal point two places: 100x = 646.464646...Step 3: Now subtract the original equation from this new equation: 100x − x = 646.464646... − 6.464646...Step 4: On the left side, 100x − x = 99x.Step 5: On the right side, the repeating decimals cancel, leaving 646 − 6 = 640.Step 6: So 99x = 640.Step 7: Solve for x: x = 640 / 99.Step 8: Therefore, the recurring decimal 6.464646... is equal to 640/99.

Verification / Alternative check:
To verify, divide 640 by 99. The result is approximately 6.46464646..., confirming that the fractional form matches the recurring decimal. No further simplification is possible because 640 and 99 have no common factor greater than 1.

Why Other Options Are Wrong:
640/90 simplifies to 64/9, which is approximately 7.11, not 6.46 recurring. 64000/99 is far too large and corresponds to about 646.4646.... 640/9 is around 71.11. 646/99 is about 6.525..., not equal to 6.4646.... Only 640/99 reproduces the exact recurring decimal 6.464646....

Common Pitfalls:
Some learners mistakenly treat 6.46 as a terminating decimal 6.46 exactly, not recognizing the repeating pattern intended. Others may forget to subtract x from 100x correctly or mis-handle the decimal alignment. Carefully setting up the algebraic equations and subtracting to eliminate the repeating part is the key technique.

Final Answer:
The correct fractional representation of the recurring decimal 6.464646... is 640/99.