Which simplified fraction is equal to 0.17 repeating
The Correct Answer and Explanation is :
To express the repeating decimal (0.17\overline{17}) (which means that the digits “17” repeat indefinitely) as a simplified fraction, we can use algebraic methods.
Step-by-Step Explanation
- Define the Decimal:
Let ( x = 0.171717…) - Multiply to Eliminate the Repeating Part:
Since the repeating part has two digits, we can multiply both sides of the equation by 100 to shift the decimal point two places to the right:
[
100x = 17.171717…
] - Set Up the Equation:
Now, we have two equations:
- ( x = 0.171717…) (1)
- ( 100x = 17.171717…) (2) We can subtract equation (1) from equation (2) to eliminate the repeating decimal:
[
100x – x = 17.171717… – 0.171717…
]
Simplifying both sides gives:
[
99x = 17
]
- Solve for (x):
To isolate (x), divide both sides by 99:
[
x = \frac{17}{99}
] - Check for Simplification:
Next, we need to check if the fraction (\frac{17}{99}) can be simplified. The numerator (17) is a prime number, and (99) can be factored into (3^2 \times 11). Since (17) does not share any common factors with (99), the fraction is already in its simplest form.
Conclusion
Thus, the simplified fraction equivalent to the repeating decimal (0.17\overline{17}) is:
[
\frac{17}{99}
]
This fraction represents the same value as the original repeating decimal, and through the algebraic method outlined above, we’ve successfully converted it into a simplified fraction. Repeating decimals can often be transformed into fractions using this method, which is particularly useful for understanding their precise values.