Express the repeating decimal as a fraction in lowest terms. 83 83 83 0.83 = + + 100 10,000 1,000,000 0.83= (Type an integer or a simplified fraction.)
The Correct Answer and Explanation is :
To convert the repeating decimal (0.\overline{83}) into a fraction, we follow the steps outlined below.
Step 1: Let (x = 0.\overline{83})
This means that (x = 0.838383…), where “83” repeats indefinitely.
Step 2: Eliminate the repeating decimal
To eliminate the repeating part, multiply both sides of the equation by 100. This shifts the decimal point two places to the right:
[
100x = 83.838383…
]
Now we have:
- (x = 0.838383…)
- (100x = 83.838383…)
Step 3: Subtract the two equations
Subtract the first equation ((x = 0.838383…)) from the second equation ((100x = 83.838383…)):
[
100x – x = 83.838383… – 0.838383…
]
Simplifying:
[
99x = 83
]
Step 4: Solve for (x)
Now, solve for (x) by dividing both sides of the equation by 99:
[
x = \frac{83}{99}
]
Step 5: Simplify the fraction
The fraction (\frac{83}{99}) is in its lowest terms because 83 is a prime number, and it does not divide 99. Therefore, (\frac{83}{99}) is the simplest form of the fraction.
Final Answer:
[
0.\overline{83} = \frac{83}{99}
]
Explanation:
To summarize, by using the method of subtracting equations and multiplying by a power of 10, we were able to express the repeating decimal (0.\overline{83}) as a fraction. This process is a common approach for converting repeating decimals into fractions. We ended up with the fraction (\frac{83}{99}), which is in its lowest terms.