If 6.46 represents the recurring decimal 6.464646..., which of the following fractions is exactly equal to this recurring decimal?

Introduction / Context:
This question involves converting a recurring decimal into an exact fraction. Recurring decimals are common in division results and understanding how to express them as fractions is a key concept in number systems and aptitude exams.

Given Data / Assumptions:

• Recurring decimal: 6.464646..., where the block 46 repeats.
• We must write this recurring decimal as an exact fraction.
• The repeating block has two digits, namely 46.

Concept / Approach:
For a recurring decimal with repeating block of length two, we can use an algebraic method. Let x be the recurring decimal. By multiplying x by 100, we shift the decimal point two places to the right, aligning the repeating parts, and then subtract the original x. This eliminates the repeating part and leaves a simple equation in x that we can solve. Finally, we simplify the resulting fraction.

Step-by-Step Solution:
Step 1: Let x = 6.464646... Step 2: Since the repeating block has two digits, multiply x by 100: 100x = 646.464646... Step 3: Subtract the original x from this equation: 100x − x = 646.464646... − 6.464646... Step 4: The repeating parts cancel, leaving 99x = 640. Step 5: Solve for x: x = 640 / 99. Step 6: Therefore the recurring decimal 6.464646... equals the fraction 640/99.

Verification / Alternative check:
We can perform a rough decimal check by dividing 640 by 99. The result is a value a little above 6.46, and the digits 46 repeat, which matches the given repeating decimal. Since the algebraic method is standard and the arithmetic is straightforward, 640/99 is confirmed as the correct fraction.

Why Other Options Are Wrong:
640/90 simplifies to 64/9, which is about 7.11, not 6.46 recurring.
64000/99 is roughly 646.46, which is much larger than 6.46.
640/9 equals about 71.11, which again does not match the given decimal.

Common Pitfalls:
A common mistake is to treat 6.46 as a terminating decimal instead of a recurring one and write it as 646/100. Another error occurs if you multiply by 10 rather than 100 for a two digit repeating block. Always match the length of the repeating block with the power of 10 used in the multiplication step.

Final Answer:
The recurring decimal 6.464646... is exactly equal to 640/99.