Find the fraction, in lowest terms, represented by the repeating decimal 0.15
Answer (2)
Let x = 0.1 5 .
Multiply by 10 and 100 to get 10 x = 1.5 5 and 100 x = 15.5 5 .
Subtract 10 x from 100 x to eliminate the repeating part: 90 x = 14 .
Solve for x and reduce the fraction: x = 90 14 = 45 7 . The final answer is 45 7 .
Explanation
Understanding the Problem We are given the repeating decimal 0.1 5 and asked to find the equivalent fraction in lowest terms.
Setting up the Equation Let x = 0.1 5 = 0.1555... . To convert this repeating decimal to a fraction, we can multiply by powers of 10 to shift the decimal point.
Multiply by 10 First, multiply by 10: 10 x = 1.555...
Multiply by 100 Next, multiply by 100: 100 x = 15.555...
Subtract to Eliminate Repeating Part Subtract the first equation from the second to eliminate the repeating part: 100 x − 10 x = 15.555... − 1.555... which simplifies to 90 x = 14 .
Solve for x Now, solve for x : x = 90 14 .
Reduce to Lowest Terms To reduce the fraction to lowest terms, we find the greatest common divisor (GCD) of 14 and 90. The GCD of 14 and 90 is 2. Divide both the numerator and the denominator by 2: 90 ÷ 2 14 ÷ 2 = 45 7 .
Final Answer Therefore, the fraction in lowest terms is 45 7 .
Examples
Repeating decimals appear in various real-world scenarios, such as when dividing quantities that don't result in whole numbers. For example, if you divide 7 cookies among 45 people, each person would get approximately 0.1555... of a cookie. Converting this repeating decimal to the fraction 45 7 allows for precise calculations and fair distribution.
The repeating decimal 0.1 5 can be converted to the fraction 45 7 by defining the decimal as x , multiplying to shift the decimal, and solving for x after eliminating the repeating part. Finally, the fraction is simplified to its lowest terms. Therefore, the answer is 45 7 .
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