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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].