How do you convert 3.2 (2 repeating) as a fraction?
The Correct Answer and Explanation is:
To convert 3.22‾3.2 \overline{2}3.22 (where the “2” repeats indefinitely) into a fraction, we can follow these steps:
Step 1: Represent the number as xxx
Let x=3.22‾x = 3.2 \overline{2}x=3.22, which means x=3.22222…x = 3.22222\ldotsx=3.22222….
Step 2: Eliminate the repeating decimal
To eliminate the repeating decimal, multiply both sides of the equation by 10. This will shift the decimal point one place to the right:10x=32.22222…10x = 32.22222\ldots10x=32.22222…
Notice that the decimal part is still the same as in xxx, so we can now set up another equation.
Step 3: Subtract the original equation from this new one
Now, subtract the original equation x=3.22222…x = 3.22222\ldotsx=3.22222… from the equation 10x=32.22222…10x = 32.22222\ldots10x=32.22222…:10x−x=32.22222…−3.22222…10x – x = 32.22222\ldots – 3.22222\ldots10x−x=32.22222…−3.22222…
This simplifies to:9x=299x = 299x=29
Step 4: Solve for xxx
Now, solve for xxx by dividing both sides by 9:x=299x = \frac{29}{9}x=929
Final Answer:
So, 3.22‾=2993.2 \overline{2} = \frac{29}{9}3.22=929.
Explanation:
The repeating decimal 3.22‾3.2 \overline{2}3.22 is a rational number, which means it can be expressed as a fraction. By setting up an equation and eliminating the repeating part through multiplication and subtraction, we found that the fraction representation of 3.22‾3.2 \overline{2}3.22 is 299\frac{29}{9}929. This method can be applied to any repeating decimal to convert it into a fraction.
