How do you write $0.8 \overline{3}$ as a fraction?
A. $\frac{10}{11}$
B. $\frac{5}{6}$
C. $\frac{7}{10}$
D. $\frac{2}{3}$
Answer (2)
Let x = 0.8 3 .
Multiply by 10 and 100 to get 10 x = 8. 3 and 100 x = 83. 3 .
Subtract the two equations: 100 x − 10 x = 83. 3 − 8. 3 , which simplifies to 90 x = 75 .
Solve for x and simplify the fraction: x = 90 75 = 6 5 .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0.8 3 into a fraction. This means 0.83333... where the 3 repeats infinitely.
Setting up the Equation Let x = 0.8 3 . To eliminate the repeating decimal, we can multiply by powers of 10. First, multiply by 10: 10 x = 8. 3 = 8.3333...
Multiplying by 100 Next, multiply by 100: 100 x = 83. 3 = 83.3333...
Eliminating the Repeating Decimal Now, subtract the equation 10 x = 8. 3 from the equation 100 x = 83. 3 to eliminate the repeating part:
100 x − 10 x = 83. 3 − 8. 3
This simplifies to 90 x = 75 .
Solving for x Solve for x by dividing both sides by 90: x = 90 75
Simplifying the Fraction Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15: x = 90 ÷ 15 75 ÷ 15 = 6 5
Final Answer Therefore, 0.8 3 as a fraction is 6 5 .
Examples
Repeating decimals can be useful in real-life situations such as calculating precise measurements or dividing resources equally. For instance, if you need to divide 5 pizzas equally among 6 people, each person would get 6 5 of a pizza. Converting this fraction to a decimal gives you 0.8 3 , which helps you understand that each person gets a little more than half a pizza, specifically 0.8333... of a whole pizza. This ensures fair distribution and accurate portioning.
The repeating decimal 0.8 3 can be expressed as the fraction 6 5 by setting up an equation, manipulating it, and simplifying the resulting fraction. The subtraction of two equations allows us to eliminate the repeating decimal part effectively. Therefore, the answer is 6 5 .
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