IDNLearn.com makes it easy to find the right answers to your questions. Get comprehensive answers to all your questions from our network of experienced experts.

Write the following infinite decimal expansion as a fraction:

[tex]\[ 0.601601601... \][/tex]


Sagot :

To convert the repeating decimal [tex]\(0.\overline{601}\)[/tex] into a fraction, we can follow these steps:

1. Assign a variable to the repeating decimal:
Let [tex]\( x = 0.601601601\ldots \)[/tex]

2. Multiply [tex]\( x \)[/tex] by a power of 10 that shifts the repeating part to the left of the decimal point:
Since the repeating part "601" is three digits long, we multiply [tex]\( x \)[/tex] by [tex]\( 1000 \)[/tex]:
[tex]\[ 1000x = 601.601601601\ldots \][/tex]

3. Set up the equations:
[tex]\[ x = 0.601601601\ldots \quad \text{(1)} \][/tex]
[tex]\[ 1000x = 601.601601601\ldots \quad \text{(2)} \][/tex]

4. Subtract the first equation from the second to eliminate the repeating part:
[tex]\[ 1000x - x = 601.601601601\ldots - 0.601601601\ldots \][/tex]
[tex]\[ 999x = 601 \][/tex]

5. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{601}{999} \][/tex]

Thus, the infinite decimal [tex]\(0.\overline{601}\)[/tex] can be expressed as the fraction [tex]\(\frac{601}{999}\)[/tex].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.