Express the repeating decimal as the ratio of two integers. \overline{0.15} = 0.151515… The ratio of two integers is (Type an integer or a simplified fraction.
The Correct Answer and Explanation is:
Let x=0.15‾x = 0.\overline{15}x=0.15, which represents the repeating decimal 0.151515…0.151515\ldots0.151515….
Step 1: Set up the equation
Let x=0.15‾x = 0.\overline{15}x=0.15.
Step 2: Multiply both sides by a power of 10 to shift the decimal
Since the decimal repeats every two digits, multiply both sides of the equation by 100 to shift the decimal point two places to the right:100x=15.15‾100x = 15.\overline{15}100x=15.15
Step 3: Subtract the original equation from the new equation
Now, subtract the original equation x=0.15‾x = 0.\overline{15}x=0.15 from the new equation 100x=15.15‾100x = 15.\overline{15}100x=15.15:100x−x=15.15‾−0.15‾100x – x = 15.\overline{15} – 0.\overline{15}100x−x=15.15−0.1599x=1599x = 1599x=15
Step 4: Solve for xxx
Now solve for xxx by dividing both sides by 99:x=1599x = \frac{15}{99}x=9915
Step 5: Simplify the fraction
To simplify the fraction 1599\frac{15}{99}9915, divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3:x=15÷399÷3=533x = \frac{15 \div 3}{99 \div 3} = \frac{5}{33}x=99÷315÷3=335
Thus, the repeating decimal 0.15‾0.\overline{15}0.15 can be expressed as the ratio of two integers:x=533x = \frac{5}{33}x=335
Conclusion:
The repeating decimal 0.15‾0.\overline{15}0.15 is equivalent to the fraction 533\frac{5}{33}335.
