Convert each of the following repeating decimals to $\frac{a}{b}$ form, where $a$ and $b$ are integers and $b \neq 0$.
a. $0 . \overline{7}$
b. $0 . \overline{46}$?
Answer (2)
Let x = 0. 7 , multiply by 10, subtract x , and solve for x = 9 7 .
Let y = 0. 46 , multiply by 100, subtract y , and solve for y = 99 46 .
The fraction form of 0. 7 is 9 7 .
The fraction form of 0. 46 is 99 46 . 9 7 , 99 46
Explanation
Problem Analysis We are asked to convert repeating decimals into fractions. Let's tackle each one separately.
Converting 0.7 (repeating) Let x = 0. 7 . This means x = 0.7777... . To eliminate the repeating part, we multiply x by 10, so 10 x = 7.7777... . Now, subtract x from 10 x : 10 x − x = 7.7777... − 0.7777... This simplifies to 9 x = 7 . Dividing both sides by 9, we get x = 9 7 .
Converting 0.46 (repeating) Now, let y = 0. 46 . This means y = 0.464646... . To eliminate the repeating part, we multiply y by 100, so 100 y = 46.464646... . Now, subtract y from 100 y : 100 y − y = 46.464646... − 0.464646... This simplifies to 99 y = 46 . Dividing both sides by 99, we get y = 99 46 .
Final Answer Therefore, the repeating decimals 0. 7 and 0. 46 can be expressed as the fractions 9 7 and 99 46 , respectively.
Examples
Repeating decimals are useful in various real-life scenarios, such as when dealing with measurements that are infinitely precise or when converting between different units of measurement. For example, when dividing 10 by 3, you get a repeating decimal 3.333.... Converting this to a fraction, 10/3, allows for exact calculations. Understanding how to convert repeating decimals to fractions ensures accuracy in calculations where approximations might lead to significant errors.
The repeating decimal 0.ar{7} converts to the fraction 9 7 , and the repeating decimal 0.ar{46} converts to the fraction 99 46 .
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