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Question

To convert 0.82¯¯¯¯¯ to a rational number, you would set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by what number?

Answer 1

To convert 0.82¯¯¯¯¯ to a rational number, we can set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by a number that eliminates the repeating decimal.

To do this, we can multiply both sides by a power of 10 equal to the number of digits in the repeating pattern. In this case, the repeating pattern is 82, which has 2 digits. Therefore, we multiply both sides by 100 (10 raised to the power of 2).

0.82¯¯¯¯¯ * 100 = x * 100

82.¯¯¯¯¯ = 100x

Now, we can simplify the left side by subtracting the original equation from it:

82 - 0.82¯¯¯¯¯ = 100x - 0.82¯¯¯¯¯

81.18 = 99.18x

Finally, dividing both sides by 99.18:

\(\frac{{81.18}}{{99.18}} = \frac{{99.18x}}{{99.18}}\)

The simplified fraction is the rational representation of 0.82¯¯¯¯¯.