How do you write $0.3 \overline{8}$ as a fraction?
$\frac{2}{9}$
$\frac{2}{3}$
$\frac{7}{18}$
$\frac{11}{20}$
Answer (2)
Let x = 0.3 8 .
Multiply by 10 and 100 to get 10 x = 3. 8 and 100 x = 38. 8 .
Subtract the equations: 100 x โ 10 x = 38. 8 โ 3. 8 , which simplifies to 90 x = 35 .
Solve for x and simplify the fraction: x = 90 35 โ = 18 7 โ . The final answer is 18 7 โ โ .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0.3\[\]\overline{8} into a fraction. This means we need to express the number 0.388888... in the form q p โ , where p and q are integers and q ๎ = 0 .
Setting up Equations Let x = 0.3 8 . To eliminate the repeating part, we'll multiply x by powers of 10. First, multiply by 10: 10 x = 3. 8 = 3.8888... Next, multiply by 100: 100 x = 38. 8 = 38.8888...
Eliminating the Repeating Decimal Now, subtract the first equation from the second equation to eliminate the repeating decimal: 100 x โ 10 x = 38. 8 โ 3. 8 This simplifies to: 90 x = 35
Solving for x Solve for x by dividing both sides by 90: x = 90 35 โ
Simplifying the Fraction Now, we simplify the fraction by finding the greatest common divisor (GCD) of 35 and 90. The GCD of 35 and 90 is 5. Divide both the numerator and the denominator by 5: x = 90 รท 5 35 รท 5 โ = 18 7 โ
Final Answer Therefore, the repeating decimal 0.3 8 can be written as the fraction 18 7 โ .
Examples
Converting repeating decimals to fractions is useful in various real-life scenarios, such as calculating precise measurements in engineering or finance. For example, if a manufacturing process requires a component to be exactly 0.3 8 inches long, converting this to 18 7 โ inches allows for accurate machining and reduces errors. This conversion ensures that the component meets the required specifications, preventing potential issues in the final product.
The repeating decimal 0.3 8 can be converted into the fraction 18 7 โ by using algebraic manipulation. This involves setting the decimal as a variable, multiplying to shift the decimal, and solving a resulting equation. The final answer is 18 7 โ .
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