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Which is greater in each of the following:
$(i)$. $\frac{2}{3},\ \frac{5}{2}$
$(ii)$. $-\frac{5}{6},\ -\frac{4}{3}$
$(iii)$. $-\frac{3}{4},\ \frac{2}{-3}$
$(iv)$. $-\frac{1}{4},\ \frac{1}{4}$
$(v)$. $-3\frac{2}{7\ },\ -3\frac{4}{5}$
Given:
$(i)$. $\frac{2}{3},\ \frac{5}{2}$
$(ii)$. $-\frac{5}{6},\ -\frac{4}{3}$
$(iii)$. $-\frac{3}{4},\ \frac{2}{-3}$
$(iv)$. $-\frac{1}{4},\ \frac{1}{4}$
$(v)$. $-3\frac{2}{7\ },\ -3\frac{4}{5}$
To do: To find the greater rational number in each of the given pairs.
Solution:
$(i)$. $\frac{2}{3},\ \frac{5}{2}$
Taking the LCM of the denominators $3$ and $2$ of both the rational numbers, we get $6$.
So, $\frac{2}{3}=\frac{2}{3}\times\frac{2}{2}$
$=\frac{4}{6}$
And $\frac{5}{2}=\frac{5}{2}\times\frac{3}{3}$
$=\frac{15}{6}$
On comparing both the rational numbers we have:
$\frac{4}{6}<\frac{15}{6}$
Therefore, $\frac{2}{3}$< $\frac{5}{2}$
$(ii)$. $-\frac{5}{6},\ -\frac{4}{3}$
Taking the LCM of the denominators $6$ and $3$ of both the rational numbers, we get $6$.
So, $-\frac{5}{6}=-\frac{5}{6}\times\frac{1}{1}$
$=-\frac{5}{6}$
And $-\frac{4}{3}=\frac{-4}{3}\times\frac{2}{2}$
$=\frac{-8}{6}$
On comparing both the fractions we have: $-\frac{5}{6}$> $-\frac{8}{6}$
Therefore, $-\frac{5}{6}$>$-\frac{4}{3}$
$(iii)$. $-\frac{3}{4},\ \frac{2}{-3}$
Taking the LCM of the denominators $4$ and $3$ of both the rational numbers, we get $12$.
So, $-\frac{3}{4}=-\frac{3}{4}\times\frac{3}{3}$
$=-\frac{9}{12}$
And $\frac{2}{-3}=\frac{2}{-3}\times\frac{4}{4}$
$= \frac{8}{-12}$
On comparing both fractions we have,
$-\frac{9}{12}$<$\frac{8}{-12}$
Or $-\frac{3}{4}$ < $\frac{2}{-3}$
$(iv)$. $-\frac{1}{4},\ \frac{1}{4}$
It is known that all the negative integers are less than $0$ and all the positive integers are greater than $0$.
Here, $\frac{-1}{4}$<$0$
And $\frac{1}{4}$>$0$
Therefore, $\frac{-1}{4}$ < $\frac{1}{4} $
$(v)$. $-3\frac{2}{7},\ -3\frac{4}{5}$
$-3\frac{2}{7}=-\frac{21+2}{7}$
$=-\frac{23}{7}$
And $-3\frac{4}{5}=-\frac{15+4}{5}$
$=-\frac{19}{5}$
Taking the LCM of the denominators $7$ and $5$ of both the rational numbers, we get $35$.
$-\frac{23}{7}=-\frac{23}{7}\times\frac{5}{5}$
$=\frac{-115}{35}$
And $-\frac{19}{5}=-\frac{19}{5}\times\frac{7}{7}$
$=\frac{-133}{35}$
On comparing both the fractions we have:
$-\frac{115}{35}$>$-\frac{133}{35}$
Or $-3\frac{2}{7}\ >\ -3\frac{4}{5}$
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