convert 3.5 in to a fraction with 5 repeating
The Correct Answer and Explanation is:
Correct Answer:
The decimal 3.5 with 5 repeating means:
3.55555… (i.e., 5 repeats infinitely).
This is a repeating decimal, and we can convert it to a fraction as follows:
Let:
x = 3.55555…
Step 1: Multiply both sides by 10 to shift the decimal point one place:
10x = 35.55555…
Step 2: Subtract the original equation from this new equation:
10x – x = 35.55555… – 3.55555…
9x = 32
Step 3: Solve for x:
x = 32 / 9
Final Answer:
3.5 with 5 repeating = 32/9
Explanation:
Repeating decimals can be converted to fractions using algebraic techniques. The repeating decimal in question is 3.55555…, where only the digit “5” repeats endlessly. To convert it into a fraction, we begin by setting the decimal equal to a variable, say xxx, so that x=3.5555…x = 3.5555\ldotsx=3.5555….
The repeating part is only one digit long, so we multiply both sides by 10 to shift the decimal one place to the right, which gives us 10x=35.5555…10x = 35.5555\ldots10x=35.5555…. The key idea here is to eliminate the repeating decimal part by subtraction. We subtract the original equation x=3.5555…x = 3.5555\ldotsx=3.5555… from the new one: 10x−x=35.5555…−3.5555…10x – x = 35.5555\ldots – 3.5555\ldots10x−x=35.5555…−3.5555… 9x=329x = 329x=32
When we subtract, the infinite decimals cancel out because they are the same after the decimal point. We’re left with a solvable linear equation: 9x=329x = 329x=32. Solving for xxx, we get x=329x = \frac{32}{9}x=932. This fraction is already in its simplest form because 32 and 9 have no common factors other than 1.
This method works reliably for any repeating decimal, and it shows the close relationship between repeating decimals and rational numbers. Every repeating decimal can be expressed as a fraction, which means it’s a rational number. So, 3.5‾=3293.\overline{5} = \frac{32}{9}3.5=932 is a rational expression of that repeating decimal.
