Express the repeating decimal as the ratio of two integers.
The ratio of two integers is
. (Type an integer or a simplified fraction.
The Correct Answer and Explanation is:
To express a repeating decimal as the ratio of two integers, we can follow a systematic approach using algebra.
Let’s use the example of the repeating decimal 0.ab‾0.\overline{ab}0.ab, where “ab” represents the repeating part (two digits). For the sake of explanation, let’s consider a concrete example: 0.15‾0.\overline{15}0.15, which means 0.151515…0.151515…0.151515….
Step 1: Set up the equation
Let x=0.15‾x = 0.\overline{15}x=0.15. This means x=0.151515…x = 0.151515…x=0.151515….
Step 2: Eliminate the repeating decimal
To eliminate the repeating part, multiply both sides of the equation by 100 (because there are two digits in the repeating part). This gives:100x=15.151515…100x = 15.151515…100x=15.151515…
Now you have the original repeating decimal part (0.151515…) on both sides of the equation.
Step 3: Subtract the original equation from the new equation
Subtract the original equation x=0.151515…x = 0.151515…x=0.151515… from 100x=15.151515…100x = 15.151515…100x=15.151515…. This gives:100x−x=15.151515…−0.151515…100x – x = 15.151515… – 0.151515…100x−x=15.151515…−0.151515…
Simplifying both sides:99x=1599x = 1599x=15
Step 4: Solve for xxx
Now, solve for xxx:x=1599x = \frac{15}{99}x=9915
Step 5: Simplify the fraction
Simplify the fraction 1599\frac{15}{99}9915. The greatest common divisor (GCD) of 15 and 99 is 3, so divide both the numerator and the denominator by 3:x=15÷399÷3=533x = \frac{15 \div 3}{99 \div 3} = \frac{5}{33}x=99÷315÷3=335
Final Answer
Thus, the repeating decimal 0.15‾0.\overline{15}0.15 can be expressed as the fraction 533\frac{5}{33}335.
This method works for any repeating decimal. The key is to multiply both sides of the equation by a power of 10 that matches the number of repeating digits, subtract the two equations, and then simplify the resulting fraction.
