$\displaystyle a + b + c$

$\displaystyle = \frac{3}{5}+ \frac{2}{3}+ \frac{4}{7}$

$\displaystyle = \frac{3 \times 21}{5 \times 21}+ \frac{2\times 35}{3 \times 35}+ \frac{4\times 15}{7 \times 15}$

$\displaystyle = \frac{63}{105}+ \frac{70}{105}+ \frac{60}{105} = \frac{193}{105}$

$\displaystyle = 193 \div 105 \approx 1.84$

The correct choice is 1.8.

$\displaystyle a - b = 3\frac{3}{5} - 1\frac{5}{6}$

$\displaystyle = \frac{18}{5} - \frac{11}{6}$

$\displaystyle = \frac{18\times 6 }{5 \times 6} - \frac{11\times 5}{6\times 5}$

$\displaystyle = \frac{108}{30} - \frac{55}{30}$

$\displaystyle = \frac{53}{30}$

$\displaystyle = 53 \div 30 \approx 1.77$

The correct choice is 1.8.

$\displaystyle a\blacklozenge b = a^{2} - b^{2}$

$\displaystyle \frac{1}{2} \blacklozenge \frac{1}{3} =\left ( \frac{1}{2} \right ) ^{2} - \left ( \frac{1}{3} \right )^{2}= \frac{1}{4} - \frac{1}{9}= \frac{9}{36} - \frac{4}{36}= \frac{5}{36}$

$\displaystyle = 5 \div 36 = 0.13888...$

The correct choice is 0.15.

$\displaystyle a bc=1 \frac{4}{9} \times 1\frac{3}{8} \times \frac{1}{2}= \frac{13}{9} \times \frac{11}{8} \times \frac{1}{2}= \frac{143}{144}= 143 \div 144 \approx 0.99$

The correct response is 1.

$\displaystyle \frac{4\frac{2}{3}}{5\frac{3}{5}} = 4\frac{2}{3} \div 5\frac{3}{5} = \frac{14}{3} \div \frac{28}{5} = \frac{14}{3} \times \frac{5} {28} = \frac{1}{3} \times \frac{5} {2} = \frac{5}{6}$

$\displaystyle \frac{5}{6}= 5 \div 6 = 0.8333...$, 

so the correct response is $\displaystyle 0.8\overline{3}$.

$\displaystyle a\blacklozenge b = \frac{ab}{a+b}$

$\displaystyle \frac{1}{2} \blacklozenge \frac{1}{3}=\frac{\frac{1}{2} \times \frac{1}{3}}{\frac{1}{2} +\frac{1}{3}}=\frac{\frac{1}{6} }{\frac{5}{6} } = \frac{1}{6} \div \frac{5}{6} = \frac{1}{6} \times \frac{6} {5} = \frac{1}{1} \times \frac{1} {5} = \frac{1}{5}$

$\displaystyle \frac{1}{5} = 1 \div 5 = 0.2$,

is the correct response.

Divide 7 by 13. The result is as follows:

The correct choice is that $\displaystyle 0.53 < \frac{7}{13} < 0.54$.

$\displaystyle \frac{2}{3} + \frac{4}{5} = \frac{2 \cdot 5}{3\cdot 5} + \frac{3\cdot 4}{3\cdot 5} = \frac{10}{15} + \frac{12}{15} =\frac{22}{15}$

Divide 22 by 15:

Of the five choices, the closest is 1.45.

Divide 19 by 27:

$\displaystyle 0.70 < \frac{19}{27}< 0.71$, 

so

$\displaystyle -0.71< -\frac{19}{27}< -0.70$.

Since all of the fractions have 77 as a denominator, one way to solve this problem is to let $\displaystyle N$ be the numerator of the correct fraction. Therefore, the equivalent problem is to find integer $\displaystyle N$ such that

$\displaystyle 0.32 < \frac{N}{77} < 0.33$

Multiply each expression by $\displaystyle 77$:

$\displaystyle 0.32 \cdot 77 < \frac{N}{77} \cdot 77 < 0.33 \cdot 77$

$\displaystyle 24.64 < N < 25.41$

Therefore, $\displaystyle N = 25$, making $\displaystyle \frac{25}{77}$ the correct choice.