what is 6.3 repeating as a fraction
The Correct Answer and Explanation is:
To express the repeating decimal 6.3 repeating as a fraction, we need to use algebra to solve for the value.
Step 1: Represent the decimal as a variable
Let xxx represent the repeating decimal:x=6.33333…x = 6.33333\ldotsx=6.33333…
Here, the digit “3” repeats indefinitely.
Step 2: Eliminate the repeating decimal
To isolate the repeating part, multiply both sides of the equation by 10, since the repeating digit “3” starts immediately after the decimal point:10x=63.33333…10x = 63.33333\ldots10x=63.33333…
Now, we have two equations:
- x=6.33333…x = 6.33333\ldotsx=6.33333…
- 10x=63.33333…10x = 63.33333\ldots10x=63.33333…
Step 3: Subtract the first equation from the second
To eliminate the repeating decimal, subtract equation (1) from equation (2):10x−x=63.33333…−6.33333…10x – x = 63.33333\ldots – 6.33333\ldots10x−x=63.33333…−6.33333…
This simplifies to:9x=579x = 579x=57
Step 4: Solve for xxx
Now, divide both sides of the equation by 9:x=579x = \frac{57}{9}x=957
Step 5: Simplify the fraction
The fraction 579\frac{57}{9}957 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:x=57÷39÷3=193x = \frac{57 \div 3}{9 \div 3} = \frac{19}{3}x=9÷357÷3=319
Final Answer
Therefore, the repeating decimal 6.3 repeating can be expressed as the fraction 193\frac{19}{3}319.
Explanation:
The method we used to convert the repeating decimal into a fraction is based on algebraic manipulation. By multiplying the equation by 10, we shift the decimal point, and subtracting allows us to isolate the repeating portion. Once the equation is simplified, we solve for the fraction and simplify it to its lowest terms. This method is applicable to any repeating decimal, whether it’s a single digit or multiple digits repeating.
