How do you write $0.91 \overline{6}$ as a fraction?
$\frac{17}{18}$
$\frac{17}{20}$
$\frac{11}{12}$
$\frac{2}{3}$
Answer (2)
Let x = 0.91 6 .
Multiply by powers of 10 to get 100 x = 91. 6 and 1000 x = 916. 6 .
Subtract to eliminate the repeating part: 1000 x − 100 x = 900 x = 825 .
Solve for x and simplify: x = 900 825 = 12 11 .
Explanation
Understanding the Problem We are given the repeating decimal 0.91 6 and asked to express it as a fraction. Let's break down the process step by step. The key idea is to eliminate the repeating part through subtraction.
Setting up the Equation Let x = 0.91 6 . Our goal is to find a fraction that is equal to x . To eliminate the repeating decimal, we'll multiply x by powers of 10.
Multiplying by Powers of 10 First, multiply x by 100: 100 x = 91. 6 . Then, multiply x by 1000: 1000 x = 916. 6 .
Eliminating the Repeating Part Now, subtract 100 x from 1000 x to eliminate the repeating part:
1000 x − 100 x = 916. 6 − 91. 6
Simplifying the Equation Simplify the equation:
900 x = 825
Solving for x Solve for x :
x = 900 825
Simplifying the Fraction Now, we simplify the fraction 900 825 by finding the greatest common divisor (GCD) of 825 and 900. The GCD of 825 and 900 is 75. Divide both the numerator and the denominator by 75:
x = 900 ÷ 75 825 ÷ 75 = 12 11
Final Answer Therefore, the repeating decimal 0.91 6 can be written as the fraction 12 11 .
Examples
Repeating decimals and their fractional equivalents are useful in various real-world scenarios. For instance, when dealing with precise measurements in engineering or finance, converting a repeating decimal to a fraction allows for exact calculations. Imagine you're calculating the dimensions of a component that needs to be 12 11 of an inch. Using the fractional representation ensures accuracy, whereas using the decimal approximation 0.91 6 might introduce small errors that accumulate over multiple calculations. This conversion is also crucial in financial calculations where even tiny discrepancies can lead to significant differences in outcomes.
The repeating decimal 0.91 6 can be written as the fraction 12 11 . This is achieved by defining the decimal as a variable and manipulating it through multiplication and subtraction. Ultimately, the simplified fraction is 12 11 .
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