Solve 3/5+2/3quadtext{(2)}31/4+22/7quadtext{(3)}1frac{1}{6}+frac{3}{10}

$$\frac { 3 } { 5 } + \frac { 2 } { 3 } \quad \text { (2) } 3 \frac { 1 } { 4 } + 2 \frac { 2 } { 7 } \quad \text { (3) } 1 \frac { 1 } { 6 } + \frac { 3 } { 10 }$$

Evaluate

$$\frac{397}{30}\approx 13.233333333$$

Solution Steps

Express \frac{2}{3}\times 2 as a single fraction.

$$\frac{3}{5}+\frac{2\times 2}{3}\times \frac{3\times 4+1}{4}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Multiply 2 and 2 to get 4.

$$\frac{3}{5}+\frac{4}{3}\times \frac{3\times 4+1}{4}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Multiply 3 and 4 to get 12.

$$\frac{3}{5}+\frac{4}{3}\times \frac{12+1}{4}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Add 12 and 1 to get 13.

$$\frac{3}{5}+\frac{4}{3}\times \frac{13}{4}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Multiply \frac{4}{3} times \frac{13}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{5}+\frac{4\times 13}{3\times 4}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Cancel out 4 in both numerator and denominator.

$$\frac{3}{5}+\frac{13}{3}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Least common multiple of 5 and 3 is 15. Convert \frac{3}{5} and \frac{13}{3} to fractions with denominator 15.

$$\frac{9}{15}+\frac{65}{15}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Since \frac{9}{15} and \frac{65}{15} have the same denominator, add them by adding their numerators.

$$\frac{9+65}{15}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Add 9 and 65 to get 74.

$$\frac{74}{15}+\frac{2\times 7+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Multiply 2 and 7 to get 14.

$$\frac{74}{15}+\frac{14+2}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Add 14 and 2 to get 16.

$$\frac{74}{15}+\frac{16}{7}\times 3\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Express \frac{16}{7}\times 3 as a single fraction.

$$\frac{74}{15}+\frac{16\times 3}{7}\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Multiply 16 and 3 to get 48.

$$\frac{74}{15}+\frac{48}{7}\times \frac{1\times 6+1}{6}+\frac{3}{10}$$

Multiply 1 and 6 to get 6.

$$\frac{74}{15}+\frac{48}{7}\times \frac{6+1}{6}+\frac{3}{10}$$

Add 6 and 1 to get 7.

$$\frac{74}{15}+\frac{48}{7}\times \frac{7}{6}+\frac{3}{10}$$

Multiply \frac{48}{7} times \frac{7}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{74}{15}+\frac{48\times 7}{7\times 6}+\frac{3}{10}$$

Cancel out 7 in both numerator and denominator.

$$\frac{74}{15}+\frac{48}{6}+\frac{3}{10}$$

Divide 48 by 6 to get 8.

$$\frac{74}{15}+8+\frac{3}{10}$$

Convert 8 to fraction \frac{120}{15}.

$$\frac{74}{15}+\frac{120}{15}+\frac{3}{10}$$

Since \frac{74}{15} and \frac{120}{15} have the same denominator, add them by adding their numerators.

$$\frac{74+120}{15}+\frac{3}{10}$$

Add 74 and 120 to get 194.

$$\frac{194}{15}+\frac{3}{10}$$

Least common multiple of 15 and 10 is 30. Convert \frac{194}{15} and \frac{3}{10} to fractions with denominator 30.

$$\frac{388}{30}+\frac{9}{30}$$

Since \frac{388}{30} and \frac{9}{30} have the same denominator, add them by adding their numerators.

$$\frac{388+9}{30}$$

Add 388 and 9 to get 397.

$$\frac{397}{30}$$

Factor

$$\frac{397}{2 \cdot 3 \cdot 5} = 13\frac{7}{30} = 13.233333333333333$$