How do you write $3.5 \overline{3}$ as a fraction?
Answer (1)
Let x = 3.5\[\]\overline{3} . - Multiply by 10 and 100 to get 10 x = 35. 3 and 100 x = 353. 3 . - Subtract the equations: 100 x โ 10 x = 353. 3 โ 35. 3 , which simplifies to 90 x = 318 . - Solve for x and simplify: x = 90 318 โ = 15 53 โ = 3 15 8 โ . ### Explanation 1. Understanding the Problem We are asked to express the repeating decimal 3.5\[\]\overline{3} as a fraction. This means we need to convert 3.53333... into a fraction in the form q p โ , where p and q are integers and q ๎ = 0 .
Setting up Equations Let x = 3.53333... . To eliminate the repeating decimal, we can multiply x by powers of 10. First, multiply by 10: 10 x = 35.3333... . Next, multiply by 100: 100 x = 353.3333... .
Eliminating the Repeating Decimal Now, subtract the first equation from the second equation to eliminate the repeating decimal part: 100 x โ 10 x = 353.3333... โ 35.3333... . This simplifies to 90 x = 318 .
Solving for x and Simplifying Solve for x by dividing both sides by 90: x = 90 318 โ . Now, we simplify the fraction by finding the greatest common divisor (GCD) of 318 and 90. The GCD of 318 and 90 is 6. Divide both the numerator and the denominator by 6: x = 90 รท 6 318 รท 6 โ = 15 53 โ .
Converting to a Mixed Number Finally, convert the improper fraction 15 53 โ to a mixed number. Divide 53 by 15: 53 รท 15 = 3 with a remainder of 8. So, 15 53 โ = 3 15 8 โ .
Final Answer Therefore, 3.5 3 can be written as the fraction 3 15 8 โ .
Examples
Understanding how to convert repeating decimals to fractions is useful in various real-life situations, such as when dealing with precise measurements or financial calculations. For example, if you are calculating the area of a garden bed and one of the dimensions is a repeating decimal, converting it to a fraction allows for more accurate calculations. Similarly, in finance, converting interest rates or currency exchange rates that are expressed as repeating decimals into fractions can help in precise financial planning and analysis. This skill ensures accuracy and avoids rounding errors in important calculations.
