Find the fraction, in lowest terms, represented by the repeating decimal 0.15
Answer (2)
Let x = 0.15 621 .
Calculate 100 x = 15. 621 and 100000 x = 15621. 621 .
Subtract to eliminate the repeating part: 99900 x = 15606 .
Solve for x and simplify the fraction: x = 99900 15606 = 1850 289 .
The fraction in lowest terms is 1850 289 .
Explanation
Understanding the Problem We are given the repeating decimal 0.15 621 and we want to express it as a fraction in lowest terms.
Moving the Decimal Let x = 0.15 621 . To eliminate the repeating part, we first multiply x by 100 to move the decimal point two places to the right: 100 x = 15. 621 .
Shifting the Repeating Block Next, we multiply 100 x by 1000 to shift the repeating block to the left of the decimal point: 1000 ( 100 x ) = 100000 x = 15621. 621 .
Eliminating the Repeating Part Now, we subtract 100 x from 100000 x to eliminate the repeating decimal part: 100000 x − 100 x = 15621. 621 − 15. 621 99900 x = 15606
Solving for x Solving for x , we get: x = 99900 15606
Simplifying the Fraction To simplify the fraction, we find the greatest common divisor (GCD) of 15606 and 99900. Using the Euclidean algorithm or a calculator, we find that GCD(15606, 99900) = 54. Dividing both the numerator and the denominator by their GCD, we get: 99900 ÷ 54 15606 ÷ 54 = 1850 289
Checking for Further Simplification Now we check if 289 and 1850 have any common factors. The prime factorization of 289 is 1 7 2 , and the prime factorization of 1850 is 2 × 5 2 × 37 . Since they have no common factors, the fraction is in its simplest form.
Final Answer Therefore, the fraction in lowest terms represented by the repeating decimal 0.15 621 is 1850 289 .
Examples
Repeating decimals can be used to represent quantities that don't have a finite decimal representation, such as when dividing certain fractions. For example, if you are splitting a bill of $156.06 equally among 100 people, each person would owe $1.5606. If you continue the division, you'll find that the '621' repeats indefinitely. Converting this repeating decimal to a fraction helps in understanding the exact, irreducible share each person has to pay, ensuring accuracy in financial calculations.
The repeating decimal 0.15 621 is expressed as the fraction 1850 289 in lowest terms. This conversion involves shifting the decimal point and eliminating the repeating portion through subtraction. After careful simplification, we arrive at the final fraction.
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