what is 0.16, 6 repeating as a fraction?
The Correct Answer and Explanation is:
To convert the decimal 0.16 repeating (written as 0.166666… where the 6 repeats) into a fraction, we can follow a step-by-step process.
Step 1: Set up an equation
Let x=0.166666… x = 0.166666…x=0.166666…, where the 6 repeats indefinitely. Now, multiply both sides of the equation by 10 to shift the decimal point one place to the right:10x=1.666666…10x = 1.666666…10x=1.666666…
Step 2: Subtract the original equation from the new equation
Now subtract the original equation from this new one:10x−x=1.666666…−0.166666…10x – x = 1.666666… – 0.166666…10x−x=1.666666…−0.166666…
Simplifying both sides:9x=1.59x = 1.59x=1.5
Step 3: Solve for xxx
Now, divide both sides of the equation by 9:x=1.59x = \frac{1.5}{9}x=91.5
To eliminate the decimal, multiply both the numerator and denominator by 10:x=1590x = \frac{15}{90}x=9015
Step 4: Simplify the fraction
Now simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 15:x=15÷1590÷15=16x = \frac{15 \div 15}{90 \div 15} = \frac{1}{6}x=90÷1515÷15=61
Final Answer:
Thus, 0.166‾0.16 \overline{6}0.166 as a fraction is 16\frac{1}{6}61.
Explanation:
The repeating decimal 0.16 repeating is essentially equivalent to the fraction 16\frac{1}{6}61. This method works for any repeating decimal by setting up an equation and then using subtraction and simplification to convert it into a fraction. Repeating decimals are a common occurrence in real numbers, and they can often be converted into fractions for easier calculation or representation.
