To determine the correct statement about the number [tex]\(6.\overline{24}\)[/tex], we first need to identify its nature and its expression as a fraction.
1. Understanding the Notation:
- The notation [tex]\(6.\overline{24}\)[/tex] indicates that [tex]\(24\)[/tex] is the repeating part of the decimal. This means the number can be written as [tex]\(6.24242424\ldots\)[/tex].
2. Converting the Repeating Decimal to a Fraction:
- Let's denote the repeating decimal as [tex]\(x\)[/tex]. So, [tex]\(x = 6.\overline{24}\)[/tex].
- To eliminate the repeating part, we can multiply [tex]\(x\)[/tex] by 100 (since the repeating block has 2 digits).
Hence, [tex]\(100x = 624.242424\ldots\)[/tex].
3. Creating an Equation:
- Now, subtract the original [tex]\(x\)[/tex] from this equation to eliminate the repeating part:
[tex]\[
100x = 624.242424\ldots \][/tex]
[tex]\[
- \ \ \ x = \ \ \ 6.242424\ldots \][/tex]
[tex]\[
\ \ \ \ \ \ \ \ 99x = 618.
\][/tex]
4. Solving for [tex]\(x\)[/tex]:
- Solving the equation [tex]\(99x = 618\)[/tex]:
[tex]\[
x = \frac{618}{99}.
\][/tex]
5. Simplifying the Fraction:
- We simplify [tex]\(\frac{618}{99}\)[/tex] by finding the greatest common divisor (GCD) of 618 and 99.
- Both numbers are divisible by 3:
[tex]\[
\frac{618 \div 3}{99 \div 3} = \frac{206}{33}.
\][/tex]
Thus, the fraction representing [tex]\(6.\overline{24}\)[/tex] is [tex]\(\frac{206}{33}\)[/tex], confirming that [tex]\(6.\overline{24}\)[/tex] is a rational number because it can be expressed as a fraction.
Now, examining the given statements:
- Statement A: It's rational because [tex]\(6.\overline{24} = \frac{68}{33}\)[/tex].
- This is incorrect because the correct fraction is [tex]\(\frac{206}{33}\)[/tex], not [tex]\(\frac{68}{33}\)[/tex].
- Statement B: It's rational because [tex]\(6.\overline{24} = \frac{6\%}{25}\)[/tex].
- This is incorrect as [tex]\(6.\overline{24}\)[/tex] does not equal [tex]\(\frac{6\%}{25}\)[/tex].
- Statement C: It's irrational because [tex]\(6.\overline{24} = \frac{68}{33}\)[/tex].
- This is incorrect on two counts: first, the fraction given is incorrect; secondly, [tex]\(6.\overline{24}\)[/tex] is actually rational.
- Statement D: It's irrational because [tex]\(6.\overline{24} = \frac{6\%}{25}\)[/tex].
- This is incorrect because [tex]\(6.\overline{24}\)[/tex] is rational and also [tex]\(\frac{6\%}{25}\)[/tex] is not equal to [tex]\(6.\overline{24}\)[/tex].
Therefore, none of the given statements are true.
However, if we are only to use the provided math code result, we could argue that the function correctly computed the numbers and the actual correct fraction derived from [tex]\(6.\overline{24}\)[/tex] is [tex]\(\frac{206}{33}\)[/tex], fitting the rationale in the context of a math class. Nevertheless, as per our provided choices, none accurately reflect the correct working process and valid output.
