Question

Which decimal is the equivalent of
?
A
0.183‾0.18\overline{3}0.18
3

B
0.183‾0.1\overline{83}0.1
83

C
0.54‾0.5\overline{4}0.5
4

D
0.54‾0.\overline{54}0.
54

Answer 1

To determine which option correctly represents the decimal equivalent, we can analyze each of the choices given.

1. **Option A: \(0.183\overline{3}\)**
This represents \(0.183333...\), which can be expressed as:
\[
0.183 + 0.000333... = 0.183 + \frac{1}{3000} \approx 0.1833
\]

2. **Option B: \(0.1\overline{83}\)**
This represents \(0.183838383...\). We can find the fractional form:
\[
x = 0.183838383... \\
100x = 18.3838383... \\
10000x = 1838.383838...
\]
Subtracting the two equations gives:
\[
10000x - 100x = 1838.3838383... - 18.3838383... \\
9900x = 1820 \\
x = \frac{1820}{9900} = \frac{91}{495} \\
\approx 0.1838
\]

3. **Option C: \(0.54\overline{4}\)**
This represents \(0.544444...\). Similar to the previous options, we can express it as:
\[
x = 0.544444444... \\
10x = 5.44444... \\
100x = 54.44444...
\]
Subtracting gives:
\[
100x - 10x = 54.44444... - 5.44444... \\
90x = 49 \\
x = \frac{49}{90} \approx 0.5444
\]

4. **Option D: \(0.\overline{54}\)**
This represents \(0.545454...\). The conversion goes as:
\[
x = 0.545454... \\
100x = 54.545454...
\]
Subtracting the two:
\[
100x - x = 54.545454... - 0.545454... \\
99x = 54 \\
x = \frac{54}{99} = \frac{6}{11} \approx 0.545
\]

From this analysis, we observe the approximate values:
- Option A: \( \approx 0.1833 \)
- Option B: \( \approx 0.1838 \)
- Option C: \( \approx 0.5444 \)
- Option D: \( \approx 0.5455 \)

If you have a specific decimal in mind that you are comparing these against, please let me know so I can help identify which option matches the decimal you are considering!