How do you write $0.08 \overline{3}$ as a fraction?
Answer (2)
Let x = 0.08\[3\overline{\]} .
Multiply by 100: 100x = 8.3\[3\overline{\]} .
Multiply by 1000: 1000x = 83.3\[3\overline{\]} .
Subtract: 900 x = 75 , so x = 900 75 = 12 1 .
The fraction is 12 1 .
Explanation
Understanding the Problem We are given the repeating decimal 0.08\[3\overline{\]} and asked to express it as a fraction. To do this, we will use algebraic manipulation to eliminate the repeating part of the decimal.
Assigning a Variable Let x = 0.08\[3\overline{\]} . Our goal is to express x as a fraction.
Multiplying by 100 Multiply x by 100 to move the decimal point two places to the right: 100x = 8.3\[3\overline{\]} .
Multiplying by 1000 Now, multiply 100 x by 10 to move the decimal point one more place to the right: 1000x = 83.3\[3\overline{\]} .
Subtracting to Eliminate the Repeating Decimal Subtract 100 x from 1000 x to eliminate the repeating decimal part:
1000x - 100x = 83.3\[3\overline{\]} - 8.3\[3\overline{\]}
This simplifies to:
900 x = 75
Solving for x Now, solve for x by dividing both sides of the equation by 900:
x = 900 75
Simplifying the Fraction Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 75:
x = 900 ÷ 75 75 ÷ 75 = 12 1
Final Answer Therefore, the repeating decimal 0.08\[3\overline{\]} can be written as the fraction 12 1 .
Examples
Understanding how to convert repeating decimals to fractions is useful in various real-life scenarios, such as financial calculations or measurements. For instance, if you're calculating the exact amount of a payment that involves a repeating decimal, converting it to a fraction ensures accuracy. This skill is also fundamental in understanding mathematical concepts related to rational numbers and their representations.
The repeating decimal 0.08 3 is converted into the fraction 12 1 by using algebraic manipulation to eliminate the repeating part. This involves multiplying the decimal by powers of 10, setting up an equation, and solving for the variable. The fraction is then simplified to its lowest terms.
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