Question

① $12\frac{1}{5}+8\frac{2}{11}$ ② $3\frac{5}{12}+10\frac{4}{7}$ ③ $7\frac{1}{8}+15\frac{7}{9}$ ④ $27\frac{7}{8}-2\frac{4}{5}$ ⑤ $20\frac{3}{10}-4\frac{1}{3}$ ⑥ $59\frac{9}{16}-40\frac{1}{2}$ ⑦ $9\frac{2}{3}+\frac{3}{20}+5\frac{1}{4}$ ⑧ $11\frac{6}{7}+(13\frac{1}{5}-3\frac{1}{11})$

Explanation:

We first convert the mixed - numbers to improper fractions. Then, for addition and subtraction of fractions, we find a common denominator, perform the arithmetic operation on the numerators, and finally convert the resulting improper fraction back to a mixed - number if necessary. Each step is a standard procedure for handling mixed - number arithmetic.

Answer:

We will solve each of the mixed - number arithmetic problems one by one.

For \(12\frac{1}{5}+8\frac{2}{11}\)

Step1: Convert to improper fractions

\(12\frac{1}{5}=\frac{12\times5 + 1}{5}=\frac{61}{5}\), \(8\frac{2}{11}=\frac{8\times11+2}{11}=\frac{90}{11}\)

Step2: Find a common denominator

The common denominator of 5 and 11 is \(5\times11 = 55\). \(\frac{61}{5}=\frac{61\times11}{5\times11}=\frac{671}{55}\), \(\frac{90}{11}=\frac{90\times5}{11\times5}=\frac{450}{55}\)

Step3: Add the fractions

\(\frac{671}{55}+\frac{450}{55}=\frac{671 + 450}{55}=\frac{1121}{55}=20\frac{21}{55}\)

For \(3\frac{5}{12}+10\frac{4}{7}\)

Step1: Convert to improper fractions

\(3\frac{5}{12}=\frac{3\times12 + 5}{12}=\frac{41}{12}\), \(10\frac{4}{7}=\frac{10\times7+4}{7}=\frac{74}{7}\)

Step2: Find a common denominator

The common denominator of 12 and 7 is \(12\times7=84\). \(\frac{41}{12}=\frac{41\times7}{12\times7}=\frac{287}{84}\), \(\frac{74}{7}=\frac{74\times12}{7\times12}=\frac{888}{84}\)

Step3: Add the fractions

\(\frac{287}{84}+\frac{888}{84}=\frac{287 + 888}{84}=\frac{1175}{84}=14\frac{19}{84}\)

For \(7\frac{1}{8}+15\frac{7}{9}\)

Step1: Convert to improper fractions

\(7\frac{1}{8}=\frac{7\times8 + 1}{8}=\frac{57}{8}\), \(15\frac{7}{9}=\frac{15\times9+7}{9}=\frac{142}{9}\)

Step2: Find a common denominator

The common denominator of 8 and 9 is \(8\times9 = 72\). \(\frac{57}{8}=\frac{57\times9}{8\times9}=\frac{513}{72}\), \(\frac{142}{9}=\frac{142\times8}{9\times8}=\frac{1136}{72}\)

Step3: Add the fractions

\(\frac{513}{72}+\frac{1136}{72}=\frac{513+1136}{72}=\frac{1649}{72}=22\frac{65}{72}\)

For \(27\frac{7}{8}-2\frac{4}{5}\)

Step1: Convert to improper fractions

\(27\frac{7}{8}=\frac{27\times8 + 7}{8}=\frac{223}{8}\), \(2\frac{4}{5}=\frac{2\times5+4}{5}=\frac{14}{5}\)

Step2: Find a common denominator

The common denominator of 8 and 5 is \(8\times5 = 40\). \(\frac{223}{8}=\frac{223\times5}{8\times5}=\frac{1115}{40}\), \(\frac{14}{5}=\frac{14\times8}{5\times8}=\frac{112}{40}\)

Step3: Subtract the fractions

\(\frac{1115}{40}-\frac{112}{40}=\frac{1115 - 112}{40}=\frac{1003}{40}=25\frac{3}{40}\)

For \(20\frac{3}{10}-4\frac{1}{3}\)

Step1: Convert to improper fractions

\(20\frac{3}{10}=\frac{20\times10 + 3}{10}=\frac{203}{10}\), \(4\frac{1}{3}=\frac{4\times3+1}{3}=\frac{13}{3}\)

Step2: Find a common denominator

The common denominator of 10 and 3 is \(10\times3 = 30\). \(\frac{203}{10}=\frac{203\times3}{10\times3}=\frac{609}{30}\), \(\frac{13}{3}=\frac{13\times10}{3\times10}=\frac{130}{30}\)

Step3: Subtract the fractions

\(\frac{609}{30}-\frac{130}{30}=\frac{609 - 130}{30}=\frac{479}{30}=15\frac{29}{30}\)

For \(59\frac{9}{16}-40\frac{1}{2}\)

Step1: Convert to improper fractions

\(59\frac{9}{16}=\frac{59\times16+9}{16}=\frac{953}{16}\), \(40\frac{1}{2}=\frac{40\times2 + 1}{2}=\frac{81}{2}\)

Step2: Find a common denominator

The common denominator of 16 and 2 is 16. \(\frac{81}{2}=\frac{81\times8}{2\times8}=\frac{648}{16}\)

Step3: Subtract the fractions

\(\frac{953}{16}-\frac{648}{16}=\frac{953 - 648}{16}=\frac{305}{16}=19\frac{1}{16}\)

For \(9\frac{2}{3}+\frac{3}{20}+5\frac{1}{4}\)

Step1: Convert to improper fractions

\(9\frac{2}{3}=\frac{9\times3 + 2}{3}=\frac{29}{3}\), \(5\frac{1}{4}=\frac{5\times4+1}{4}=\frac{21}{4}\)

Step2: Find a common denominator

The common denominator of 3, 20 and 4 is 60. \(\frac{29}{3}=\frac{29\times20}{3\times20}=\frac{580}{60}\), \(\frac{3}{20}=\frac{3\times3}{20\times3}=\frac{9}{60}\), \(\frac{21}{4}=\frac{21\times15}{4\times15}=\frac{315}{60}\)

Step3: Add the fractions

\(\frac{580}{60}+\frac{9}{60}+\frac{315}{60}=\frac{580 + 9+315}{60}=\frac{904}{60}=15\frac{4}{60}=15\frac{1}{15}\)

For \(11\frac{6}{7}+(13\frac{1}{5}-3\frac{1}{11})\)

Step1: Solve the expression inside the parentheses

\(13\frac{1}{5}=\frac{13\times5 + 1}{5}=\frac{66}{5}\), \(3\frac{1}{11}=\frac{3\times11+1}{11}=\frac{34}{11}\)
The common denominator of 5 and 11 is 55. \(\frac{66}{5}=\frac{66\times11}{5\times11}=\frac{726}{55}\), \(\frac{34}{11}=\frac{34\times5}{11\times5}=\frac{170}{55}\)
\(13\frac{1}{5}-3\frac{1}{11}=\frac{726}{55}-\frac{170}{55}=\frac{726 - 170}{55}=\frac{556}{55}\)

Step2: Add the result to \(11\frac{6}{7}\)

\(11\frac{6}{7}=\frac{11\times7+6}{7}=\frac{83}{7}\)
The common denominator of 7 and 55 is \(7\times55 = 385\). \(\frac{83}{7}=\frac{83\times55}{7\times55}=\frac{4565}{385}\), \(\frac{556}{55}=\frac{556\times7}{55\times7}=\frac{3892}{385}\)
\(11\frac{6}{7}+\frac{556}{55}=\frac{4565}{385}+\frac{3892}{385}=\frac{4565 + 3892}{385}=\frac{8457}{385}=21\frac{372}{385}\)