Question
A repeating decimal, like 0.327327327..., can be written as a fraction.(1 point) true or false explain
Answers
GPT-4o mini
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.
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.
Is a repeating decimal rational or irrational