Question

A repeating decimal, like 0.327327327..., can be written as a fraction.(1 point) true or false explain

Answer 1

Is a repeating decimal rational or irrational

Answer 2

True. A repeating decimal can be expressed as a fraction.

To understand why, consider the repeating decimal \( x = 0.327327327... \). We can manipulate the equation to eliminate the repeating part:

1. Let \( x = 0.327327327... \).
2. Since the repeating portion "327" has 3 digits, multiply both sides of the equation by 1000 (which shifts the decimal point three places to the right):
\[
1000x = 327.327327327...
\]
3. Now, we have two equations:
\[
x = 0.327327327...
\]
\[
1000x = 327.327327327...
\]
4. Subtract the first equation from the second:
\[
1000x - x = 327.327327327... - 0.327327327...
\]
This simplifies to:
\[
999x = 327
\]
5. Solve for \( x \):
\[
x = \frac{327}{999}
\]

Now, \( \frac{327}{999} \) can be simplified (if possible) to reach the final fraction representing the repeating decimal. This shows that repeating decimals can indeed be represented as fractions.