Question

express the following rational number as a decimal.
\\(\frac{1}{7}\\)
choose the correct answer below.
a. 0.142857
b. 0.5714286
c. \\(0.\overline{142857}\\)
d. \\(0.\overline{5714286}\\)

Explanation:

Step1: Recall the decimal expansion of $\frac{1}{7}$

We know that when we divide 1 by 7, we get a repeating decimal. The decimal expansion of $\frac{1}{7}$ is $0.\overline{142857}$, where the bar indicates that the sequence 142857 repeats indefinitely.

Step2: Compare with the options

• Option A: 0.142857 (non - repeating, but $\frac{1}{7}$ is repeating, so A is incorrect)
• Option B: 0.5714286 (this is not the decimal expansion of $\frac{1}{7}$, $\frac{4}{7}=0.571428\overline{571428}$ approximately, so B is incorrect)
• Option C: $0.\overline{142857}$ which matches the decimal expansion of $\frac{1}{7}$
• Option D: $0.\overline{5714286}$ (incorrect as it does not match the expansion of $\frac{1}{7}$)

Answer:

C. $0.\overline{142857}$