Determine the nearest integer to
(a) $\frac{19}{15}+\frac{19}{3}$
(b) $\frac{85}{42}+\frac{43}{21}+\frac{29}{14}+\frac{15}{7}$
(c) $-\frac{11}{10}-\frac{1}{2}-\frac{7}{5}+\frac{2}{3}$
(a) We have:

$$
\frac{19}{15}+\frac{19}{3}=1+\frac{4}{15}+6+\frac{1}{3}=7+\frac{9}{15}=7+\frac{3}{5}
$$

Thus, the given sum is between 7 and 8. Since $\frac{3}{5}>\frac{1}{2}$, the nearest integer is 8.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-083.jpg?height=251&width=668&top_left_y=1859&top_left_x=797)

(b) We have:

$$
\begin{aligned}
\frac{85}{42}+\frac{43}{21} & +\frac{29}{14}+\frac{15}{7}=2+\frac{1}{42}+2+\frac{1}{21}+2+\frac{1}{14}+2+\frac{1}{7} \\
& =8+\frac{1}{42}+\frac{1}{21}+\frac{1}{14}+\frac{1}{7}=8+\frac{1}{7}\left(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+1\right)=8+\frac{2}{7}
\end{aligned}
$$

Thus, the given sum is between 8 and 9. Since $\frac{2}{7}<\frac{1}{2}$, the nearest integer is 8.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-084.jpg?height=234&width=648&top_left_y=497&top_left_x=858)