\(\displaystyle 0.3 = \frac{3}{10}\), so its reciprocal is \(\displaystyle \frac{10}{3}\).

\(\displaystyle 0.7= \frac{7}{10}\), so its reciprocal is \(\displaystyle \frac{10}{7}\).

Add them:

\(\displaystyle \frac{10}{3} + \frac{10}{7} = \frac{10 \cdot 7 }{3 \cdot 7} + \frac{10\cdot 3}{7\cdot 3} = \frac{70}{21}+\frac{30}{21} = \frac{100}{21}\)

\(\displaystyle 100 \div 21 = 4 \textup{ R }16\), so

\(\displaystyle \frac{100}{21}= 4\frac{16}{21}\)

\(\displaystyle 1.5 = \frac{15}{10} = \frac{3}{2}\), so its reciprocal is \(\displaystyle \frac{2}{3}\).

\(\displaystyle 0.125 = \frac{125}{1,000} = \frac{1}{8}\), so its reciprocal is 8.

The requested difference is \(\displaystyle 8-\frac{2}{3} = 7\frac{1}{3}\)

\(\displaystyle 0.22 = \frac{22}{100}\), so its reciprocal is \(\displaystyle N= \frac{100}{22}\).

Since \(\displaystyle 100 \div 22 = 4\textup{ R }12\),

\(\displaystyle N= \frac{100}{22} = 4 \frac{12}{22}=4\frac{6}{11}\)

Therefore, \(\displaystyle 4< N < 5\).

\(\displaystyle 1.77= \frac{177}{100}\), so its reciprocal is \(\displaystyle N= \frac{100}{177}\).

If 100 is divided by 177, the decimal equivalent is seen to be \(\displaystyle N = 0.564...\) so 

\(\displaystyle 0.56 < N < 0.57\).

The reciprocals of 11 and 13 are, respectively, \(\displaystyle \frac{1}{11}\) and \(\displaystyle \frac{1}{13}\). The requested difference is

\(\displaystyle \frac{1}{13}-\frac{1}{11} = \frac{1\cdot 11}{13 \cdot 11}-\frac{1\cdot 13}{11 \cdot 13} = \frac{1 1}{143 }-\frac{ 13}{143} =-\frac{2}{143}\)

This answer is not among the given choices.

The reciprocals of 4, 6, and 8 are, respectively, \(\displaystyle \frac{1}{4}, \frac{1}{6}, \frac{1}{8}\). Their sum is

\(\displaystyle \frac{1}{4}+\frac{1}{6}+ \frac{1}{8}\)

First we find the least common denominator between all three fractions.

\(\displaystyle = \frac{1}{4}\bigg(\frac{6}{6}\bigg) + \frac{1}{6}\bigg(\frac{4}{4}\bigg) +\frac{1}{8}\bigg(\frac{3}{3}\bigg)\)

\(\displaystyle =\frac{6}{24}+\frac{4}{24}+\frac{3}{24} = \frac{13}{24}\)

To find the least common denominator, list out the multiples for each denominator then choose the lowest one that appears in both sets.

\(\displaystyle 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80\)

\(\displaystyle 16: 16, 32, 48, 64, 80\)

To find the least common denominator, list out the multiples for each denominator then choose the lowest one that appears in both sets.

\(\displaystyle 14: 14, 28\)

\(\displaystyle 28:28, 56\)

To find the least common denominator, list out the multiples for each denominator then choose the lowest one that is in both sets.

\(\displaystyle 5: 5, 10, 15, 20, 25, 30\)

\(\displaystyle 6: 6, 12, 18, 24, 30\)

To find the least common denominator, list out the multiples for each denominator then choose the lowest one that appears in both sets.

\(\displaystyle 8: 8, 16, 24, 32, 48, 56, 64, 72\)

\(\displaystyle 9: 9, 18, 27, 36, 45, 54, 63, 72\)