Convert 3.5 (repeating) into a fraction
The Correct answer and Explanation is:
To convert the repeating decimal 3.55‾3.5\overline{5}3.55 into a fraction, follow these steps:
- Let xxx equal the repeating decimal:x=3.55555…x = 3.55555\ldotsx=3.55555…
- Multiply by a power of 10 to move the decimal point: Since the digit “5” repeats, we can multiply by 10:10x=35.55555…10x = 35.55555\ldots10x=35.55555…
- Set up an equation by subtracting the first equation from the second:10x−x=35.55555…−3.55555…10x – x = 35.55555\ldots – 3.55555\ldots10x−x=35.55555…−3.55555…This simplifies to:9x=329x = 329x=32
- Solve for xxx:x=329x = \frac{32}{9}x=932
Thus, the fraction representation of 3.55‾3.5\overline{5}3.55 is 329\frac{32}{9}932.
Explanation
Repeating decimals can be effectively converted into fractions using algebraic methods. The repeating decimal 3.55‾3.5\overline{5}3.55 is expressed as 3.55555…3.55555\ldots3.55555…, where the “5” is the digit that repeats indefinitely. To isolate the repeating part, we start by defining the decimal as xxx.
Next, we multiply xxx by 10 to shift the decimal point one place to the right. This gives us a new equation 10x=35.55555…10x = 35.55555\ldots10x=35.55555…. Subtracting the original x=3.55555…x = 3.55555\ldotsx=3.55555… from this equation helps eliminate the repeating part. By subtracting, we effectively eliminate the infinite “5s,” allowing us to isolate the non-repeating parts on one side.
The subtraction leads to a straightforward equation: 9x=329x = 329x=32. This step is crucial because it demonstrates how the infinite nature of the decimal can be handled algebraically. Solving for xxx yields x=329x = \frac{32}{9}x=932, a simple fraction representing the original repeating decimal.
This method of using algebra to convert repeating decimals into fractions is systematic and can be applied to any repeating decimal. Understanding this process is beneficial in mathematics, particularly in number theory and when performing operations with rational numbers.