A fraction multiplied by its reciprocal will equal 1. To find the reciprocal of a fraction, switch the denominator and numerator. The reciprocal of 18/27 is 27/18.

We need to find a common denominator to add and subtract these fractions. Let's do the addition first. The lowest common denominator of 5 and 7 is 5 * 7 = 35, so 3/5 + 4/7 = 21/35 + 20/35 = 41/35. 

Now to the subtraction. The lowest common denominator of 35 and 3 is 35 * 3 = 105, so altogether, 3/5 + 4/7 – 1/3 = 41/35 – 1/3 = 123/105 – 35/105 = 88/105. This does not simplify and is therefore the correct answer.

We need to find a common denominator to add and subtract these fractions. Let's do the addition first. The lowest common denominator of 5 and 7 is 5 * 7 = 35, so 3/5 + 4/7 = 21/35 + 20/35 = 41/35. 

Now to the subtraction. The lowest common denominator of 35 and 3 is 35 * 3 = 105, so altogether, 3/5 + 4/7 – 1/3 = 41/35 – 1/3 = 123/105 – 35/105 = 88/105. This does not simplify and is therefore the correct answer.

$\displaystyle \textup{Find LCD: }5\times7=35\: \: \: \: \: \: \frac{2(7)}{5(7)}+\frac{3(5)}{7(5)}=\frac{14}{35}+\frac{15}{35}=\frac{29}{35}$

$\displaystyle \frac{35}{35}-\frac{29}{35}=\frac{6}{35}$

First, find the common denominator of the fractions in order to add them together. Looking at the first two fractions, we can see that the common denominator is 15 because 3 times five is 15. We can now change both fractions to have a denominator of 15 so that we can add them together:

$\displaystyle \frac{7}{15}+\frac{2\cdot 5}{3\cdot 5}+\frac{13}{20}$

$\displaystyle \frac{7}{15}+\frac{10}{15}+\frac{13}{20}$

$\displaystyle \frac{17}{15}+\frac{13}{20}$

Now, find a common denominator for the remaining fractions. This can be done by listing multiples of 15 and 20 and finding the lowest common multiple:

15: 15, 30, 45, 60, 75

20: 20, 40, 60, 80

60 is the lowest common multiple, so it is our least common denominator.

$\displaystyle \frac{17\cdot 4}{15\cdot 4}+\frac{13\cdot 3}{20\cdot 3}$

$\displaystyle \frac{68}{60}+\frac{39}{60}$

$\displaystyle \frac{107}{60}$

$\displaystyle \frac{56}{1-\frac{1}{49}}$

$\displaystyle = 56 \div \left ( 1-\frac{1}{49} \right )$

$\displaystyle = 56 \div \left (\frac{49}{49} -\frac{1}{49} \right )$

$\displaystyle = \frac{56 }{1}\div \frac{48}{49}$

$\displaystyle = \frac{56 }{1}\cdot \frac{49}{48}$

$\displaystyle = \frac{7}{1}\cdot \frac{49}{6}$

$\displaystyle = \frac{343}{6}$

$\displaystyle = 57 \frac{1}{6}$

$\displaystyle \frac{\frac{3}{4}-\frac{2}{5}}{2- \frac{9}{10}}$

$\displaystyle = \left ( \frac{3}{4}-\frac{2}{5} \right ) \div \left ( 2- \frac{9}{10} \right )$

$\displaystyle = \left ( \frac{15}{20}-\frac{8}{20} \right ) \div \left ( \frac{20}{10}- \frac{9}{10} \right )$

$\displaystyle = \frac{7}{20} \div \frac{11}{10}$

$\displaystyle = \frac{7}{20} \cdot \frac{10}{11}$

$\displaystyle = \frac{7}{2} \cdot \frac{1}{11}$

$\displaystyle = \frac{7}{22}$

$\displaystyle \frac{1-\frac{1}{4}}{1-\frac{1}{8}}$

$\displaystyle = \left (1 -\frac{1}{4} \right ) \div \left (1-\frac{1}{8} \right )$

$\displaystyle = \left (\frac{4}{4}-\frac{1}{4} \right ) \div \left (\frac{8}{8}-\frac{1}{8} \right )$

$\displaystyle = \frac{3}{4} \div \frac{7}{8}$

$\displaystyle = \frac{3}{4} \cdot \frac{8}{7}$

$\displaystyle = \frac{3}{1} \cdot \frac{2}{7}$

$\displaystyle = \frac{6}{7}$

$\displaystyle \frac{2-\frac{1}{4}}{1+\frac{1}{7}}$

$\displaystyle = \left ( 2-\frac{1}{4} \right ) \div \left ( 1+\frac{1}{7} \right )$

$\displaystyle = \left (\frac{8}{4}-\frac{1}{4} \right ) \div \left ( \frac{7}{7}+\frac{1}{7} \right )$

$\displaystyle = \frac{7}{4} \div \frac{8}{7}$

$\displaystyle = \frac{7}{4} \cdot \frac{7}{8}$

$\displaystyle = \frac{49}{32}$

$\displaystyle = 1 \frac{17}{32}$

$\displaystyle \frac{\frac{1}{3}+\frac{4}{5} }{\frac{2}{3}+\frac{4}{5} }$

$\displaystyle =\left ( \frac{1}{3}+\frac{4}{5} \right ) \div\left ( \frac{2}{3}+\frac{4}{5} \right )$

$\displaystyle =\left ( \frac{5}{15}+\frac{12}{15} \right ) \div\left ( \frac{10}{15}+\frac{12}{15} \right )$

$\displaystyle = \frac{17}{15} \div \frac{22}{15}$

$\displaystyle = \frac{17}{15} \cdot \frac{15}{22}$

$\displaystyle = \frac{17}{1} \cdot \frac{1}{22}$

$\displaystyle = \frac{17} {22}$