Evaluate each of the following:
(i) $ \frac{2}{3}-\frac{3}{5} $
(ii) $ \frac{-4}{7}-\frac{2}{-3} $
(iii) $ \frac{4}{7}-\frac{-5}{-7} $
(iv) $ -2-\frac{5}{9} $
(v) $ \frac{-3}{-8}-\frac{-2}{7} $
(vi) $ \frac{-4}{13}-\frac{-5}{26} $
(vii) $ \frac{-5}{14}-\frac{-2}{7} $.
(viii) $ \frac{13}{15}-\frac{12}{25} $
(ix) $ \frac{-6}{13}-\frac{-7}{13} $
(x) $ \frac{7}{24}-\frac{19}{36} $
(xi) $ \frac{5}{63}-\frac{-8}{21} $

To do:

We have to evaluate the given expressions.

Solution:

We know that,

$(-)\times(-)=(+)$

Therefore,

(i) $\frac{2}{3}-\frac{3}{5}=\frac{2\times5-3\times3}{15}$           (LCM of 3 and 5 is 15)

$=\frac{10-9}{15}$

$=\frac{1}{15}$    

Therefore,

$\frac{2}{3}-\frac{3}{5}=\frac{1}{15}$ .

(ii) $\frac{-4}{7}-\frac{2}{-3}=\frac{-4}{7}+\frac{2}{3}$

$=\frac{-4\times3+7\times2}{21}$           (LCM of 7 and 3 is 21)

$=\frac{-12+14}{21}$

$=\frac{2}{21}$    

Therefore,

$\frac{-4}{7}-\frac{2}{-3}=\frac{2}{21}$ . 

(iii) $\frac{4}{7}-\frac{-5}{-7}=\frac{4}{7}-\frac{5}{7}$

$=\frac{4-5}{7}$

$=\frac{-1}{7}$    

Therefore,

$\frac{4}{7}-\frac{-5}{-7}=\frac{-1}{7}$ .  

(iv) $-2-\frac{5}{9}=\frac{-2}{1}-\frac{5}{9}$

$=\frac{-2\times9-5\times1}{9}$           (LCM of 1 and 9 is 9)

$=\frac{-18-5}{9}$

$=\frac{-23}{9}$    

Therefore,

$-2-\frac{5}{9}=\frac{-23}{9}$ .  

(v) $\frac{-3}{-8}-\frac{-2}{7}=\frac{3}{8}+\frac{2}{7}$

$=\frac{3\times7+8\times2}{56}$           (LCM of 8 and 7 is 56)

$=\frac{21+16}{56}$

$=\frac{37}{56}$    

Therefore,

$\frac{-3}{-8}-\frac{-2}{7}=\frac{37}{56}$ .   

(vi) $\frac{-4}{13}-\frac{-5}{26}=\frac{-4}{13}+\frac{5}{26}$

$=\frac{-4\times2+5\times1}{26}$           (LCM of 13 and 26 is 26)

$=\frac{-8+5}{26}$

$=\frac{-3}{26}$    

Therefore,

$\frac{-4}{13}-\frac{-5}{26}=\frac{-3}{26}$ .

(vii) $\frac{-5}{14}-\frac{-2}{7}=\frac{-5}{14}+\frac{2}{7}$

$=\frac{-5\times1+2\times2}{14}$           (LCM of 14 and 7 is 14)

$=\frac{-5+4}{14}$

$=\frac{-1}{14}$    

Therefore,

$\frac{-5}{14}-\frac{-2}{7}=\frac{-1}{14}$ . 

(viii) $\frac{13}{15}-\frac{12}{25}=\frac{13\times5-12\times3}{75}$           (LCM of 15 and 25 is 75)

$=\frac{65-36}{75}$

$=\frac{29}{75}$    

Therefore,

$\frac{13}{15}-\frac{12}{25}=\frac{29}{75}$ .  

(ix) $\frac{-6}{13}-\frac{-7}{13}=\frac{-6+7}{13}$

$=\frac{1}{13}$    

Therefore,

$\frac{-6}{13}-\frac{-7}{13}=\frac{1}{13}$ .   

(x) $\frac{7}{24}-\frac{19}{36}=\frac{7\times3-19\times2}{72}$           (LCM of 24 and 36 is 72)

$=\frac{21-38}{72}$    

$=\frac{-17}{72}$

Therefore,

$\frac{7}{24}-\frac{19}{36}=\frac{-17}{72}$ .    

(xi) $\frac{5}{63}-\frac{-8}{21}=\frac{5}{63}+\frac{8}{21}$

$=\frac{5\times1+8\times3}{63}$           (LCM of 63 and 21 is 63)

$=\frac{5+24}{63}$    

$=\frac{29}{63}$

Therefore,

$\frac{5}{63}-\frac{-8}{21}=\frac{29}{63}$ .