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