Calculate the following, expressing your answer as a mixed fraction: (a) \(2 \frac{1}{4}+3 \frac{1}{3}\) (b) \(2 \frac{4}{5}-1 \frac{2}{3}\) (c) \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}\) (d) \(5-4 \frac{2}{7}+\frac{1}{3}\) (e) \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5}\)

To do this, multiply the whole number by the denominator and then add the numerator. Place the result over the original denominator: \((2 * 4) + 1 = 9\), so \(2 \frac{1}{4}=\frac{9}{4}\). Similarly, \((3 * 3) + 1 = 10\), so \(3 \frac{1}{3}=\frac{10}{3}\).

To add or subtract fractions with different denominators, we must first find a common denominator. The least common multiple (LCM) of 4 and 3 is 12. Multiply both the numerator and the denominator for each fraction by the necessary factor to make their denominators equal to 12: \(\frac{9}{4} * \frac{3}{3}=\frac{27}{12}\). \(\frac{10}{3} * \frac{4}{4}=\frac{40}{12}\).

Add the two fractions: \(\frac{27}{12}+\frac{40}{12}=\frac{67}{12}\).

Divide the numerator by the denominator: \(67\div 12=5\) remainder \(7\). So the mixed fraction is \(5 \frac{7}{12}\). The final answer is \(2 \frac{1}{4}+3 \frac{1}{3}=5 \frac{7}{12}\). (b) \(2 \frac{4}{5}-1 \frac{2}{3}\) Follow the same steps as above, but with subtraction to obtain the answer: \(2 \frac{4}{5}-1 \frac{2}{3}=1 \frac{1}{15}\). (c) \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}\) Follow the same steps as above but with multiple operations to obtain the answer: \(5 \frac{2}{3}-1 \frac{1}{2}+2 \frac{1}{5}=6 \frac{17}{30}\). (d) \(5-4 \frac{2}{7}+\frac{1}{3}\) Follow similar steps as above with the whole number included to obtain the answer: \(5-4 \frac{2}{7}+\frac{1}{3}= \frac{16}{21}\). (e) \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5}\) Follow the same steps as above but with different operations to obtain the answer: \(\frac{9}{10}+\frac{6}{7}-1 \frac{2}{5} = \frac{7}{70}\).