If the following fractions were converted to decimals, which one would result in a repeating decimal?
A) $5 / 11$
B) $1 / 9$
C) $3 / 7$
D) $3 / 4$
Answer (1)
Determine if a fraction results in a repeating decimal by checking if its denominator, in simplest form, contains prime factors other than 2 and 5.
Analyze each fraction: 5/11 , 1/9 , 3/7 , and 3/4 .
Identify that 5/11 , 1/9 , and 3/7 have denominators with prime factors other than 2 and 5, thus they are repeating decimals.
Conclude that the answer is 5/11 .
Explanation
Problem Analysis We are given four fractions: 5/11 , 1/9 , 3/7 , and 3/4 . Our goal is to determine which of these fractions, when converted to a decimal, results in a repeating decimal.
Terminating vs. Repeating Decimals A fraction will result in a terminating decimal if and only if its denominator, when written in simplest form, only contains the prime factors 2 and 5. If the denominator has any other prime factors, the decimal representation will repeat. Let's examine each fraction:
Analyzing 5/11 A) 5/11 : The denominator is 11, which is a prime number other than 2 or 5. Therefore, this fraction will result in a repeating decimal. When we divide 5 by 11, we get 0.454545...
Analyzing 1/9 B) 1/9 : The denominator is 9, which can be written as 3 2 . Since the prime factor is 3 (not 2 or 5), this fraction will result in a repeating decimal. When we divide 1 by 9, we get 0.111111...
Analyzing 3/7 C) 3/7 : The denominator is 7, which is a prime number other than 2 or 5. Therefore, this fraction will result in a repeating decimal. When we divide 3 by 7, we get 0.428571428571...
Analyzing 3/4 D) 3/4 : The denominator is 4, which can be written as 2 2 . Since the only prime factor is 2, this fraction will result in a terminating decimal. When we divide 3 by 4, we get 0.75 .
Conclusion Since the question asks for only one fraction that results in a repeating decimal, and options A, B, and C all result in repeating decimals, there might be an issue with the question. However, based on the options, we can say that 5/11 , 1/9 , and 3/7 result in repeating decimals, while 3/4 results in a terminating decimal. The question asks which ONE of the fractions results in a repeating decimal. Since A, B, and C all repeat, we need to determine which one is the correct answer based on the question. The question asks which ONE would result in a repeating decimal. Since all of A, B, and C result in repeating decimals, there may be an error in the question or the options. However, based on the options given, we can determine which one is the most likely answer. Since the question is a multiple-choice question, we can assume that only one of the options is correct. Therefore, we need to choose the option that is most likely to be correct. In this case, the most likely answer is A) 5/11 , since it is the first option that results in a repeating decimal.
Examples
Understanding repeating decimals is useful in various real-life scenarios, such as financial calculations or measurements. For instance, when dividing a bill of $5 between 11 people, the amount each person owes is a repeating decimal ($0.454545...). In manufacturing, if you need to cut a piece of material into 9 equal parts, the length of each part might be a repeating decimal. Knowing how to identify and work with repeating decimals ensures accuracy and fairness in these situations.
