What is 0.16666666666 as a fraction?
The Correct Answer and Explanation is :
The decimal number (0.16666666666) can be expressed as a fraction. This decimal is a repeating decimal, specifically (0.1\overline{6}), which means it has a repeating pattern of 6 after the decimal point. To convert it to a fraction, we can follow a systematic approach.
Step 1: Define the Decimal
Let (x = 0.1\overline{6}).
Step 2: Eliminate the Repeating Part
To eliminate the repeating decimal, we can multiply (x) by 10 to shift the decimal point:
[
10x = 1.66666666666\ldots
]
Step 3: Set Up an Equation
Now we have two equations:
- (x = 0.16666666666\ldots)
- (10x = 1.66666666666\ldots)
Step 4: Subtract the First Equation from the Second
Subtract the first equation from the second:
[
10x – x = 1.66666666666\ldots – 0.16666666666\ldots
]
[
9x = 1.5
]
Step 5: Solve for (x)
Now, we solve for (x):
[
x = \frac{1.5}{9}
]
Step 6: Simplify the Fraction
We can simplify (\frac{1.5}{9}). To make it easier, we can express (1.5) as (\frac{15}{10}):
[
x = \frac{15}{10 \times 9} = \frac{15}{90}
]
Now we can simplify (\frac{15}{90}) by dividing both the numerator and denominator by 15:
[
x = \frac{1}{6}
]
Conclusion
Thus, the decimal (0.16666666666) is equivalent to the fraction (\frac{1}{6}). This method illustrates the process of converting a repeating decimal into a fraction, demonstrating the utility of algebraic manipulation to solve problems involving decimal representations. The result shows that fractions can provide a precise representation of decimal values, particularly when dealing with repeating patterns.