When the recurring decimal 0.151515... is converted into a fraction in lowest terms, what is the resulting fraction?

Introduction / Context:
This question requires converting another recurring decimal into a fraction. The decimal 0.151515... has the digit pair 15 repeating indefinitely. The goal is to express this repeating decimal as a rational number in its simplest form. Understanding this process is useful for many algebra and number system problems.

Given Data / Assumptions:

• Recurring decimal: 0.151515... .
• The repeating block is 15.
• The sequence continues indefinitely.
• We must find an equivalent fraction in lowest terms.

Concept / Approach:
We use the standard algebraic method for repeating decimals:

• Let x equal the repeating decimal.
• Multiply x by a power of 10 such that the decimal point shifts beyond one full repeating block.
• Subtract the original x from this new multiple to eliminate the repeating part.
• Solve the resulting equation for x as a fraction and simplify.

Step-by-Step Solution:
Step 1: Let x = 0.151515... . Step 2: The repeating block (15) has two digits, so multiply both sides by 100: 100x = 15.151515... . Step 3: Note that 100x and x have the same repeating decimal part .151515..., which will cancel after subtraction. Step 4: Subtract the original equation from this new one: 100x - x = 15.151515... - 0.151515... . Step 5: This gives 99x = 15. Step 6: Solve for x: x = 15 / 99. Step 7: Simplify 15/99 by dividing numerator and denominator by their greatest common divisor 3: 15/99 = 5/33.

Verification / Alternative Check:
Check by converting 5/33 back to a decimal. Dividing 5 by 33 produces approximately 0.151515..., with 15 repeating indefinitely. This confirms that 5/33 is the correct fractional representation of the given decimal.

Why Other Options Are Wrong:
17/33 corresponds to a larger fraction and gives approximately 0.515151... , not 0.151515....
9/11 is about 0.818181..., with 81 repeating instead of 15.
7/33 produces approximately 0.212121..., again not matching the repeating block 15. Therefore, these fractions are not equivalent to the given decimal.

Common Pitfalls:
One common mistake is to treat 0.151515... as 15/100 plus a small remainder, rather than using the proper algebraic method. Another error is simplifying 15/99 incorrectly, for example stopping at 15/99 or dividing only by 9. Always divide by the greatest common divisor to reach lowest terms.

Final Answer:
The recurring decimal 0.151515... is equal to the fraction 5/33 in simplest form.