Question

Which of the two rational numbers is greater in the given pair?
(i) $\frac{-4}{3}\mathrm{or}\frac{-8}{7}$
(ii) $\frac{7}{-9}\mathrm{or}\frac{-5}{8}$
(iii) $\frac{-1}{3}\mathrm{or}\frac{4}{-5}$
(iv) $\frac{9}{-13}\mathrm{or}\frac{7}{-12}$
(v) $\frac{4}{-5}\mathrm{or}\frac{-7}{10}$
(vi) $\frac{-12}{5}\mathrm{or}-3$

Answer 1

1. The two rational numbers are $\frac{-4}{3}\mathrm{and}\frac{-8}{7}$.

The LCM of the denominators 3 and 7 is 21.

Now,

$\frac{-4}{3}=\frac{-4\times 7}{3\times 7}=\frac{-28}{21}$

Also,

$\frac{-8}{7}=\frac{-8\times 3}{7\times 3}=\frac{-24}{21}$

Further,

$\frac{-28}{21}<\frac{-24}{21}$

∴ $\frac{-4}{3}<\frac{-8}{7}$

2. ​The two rational numbers are $\frac{7}{-9}\mathrm{and}\frac{-5}{8}$.

The first fraction can be expressed as $\frac{7}{-9}=\frac{7\times -1}{-9\times -1}=\frac{-7}{9}$.

The LCM of the denominators 9 and 8 is 72.

Now,

$\frac{-7}{9}=\frac{-7\times 8}{9\times 8}=\frac{-56}{72}$

Also,

$\frac{-5}{8}=\frac{-5\times 9}{8\times 9}=\frac{-45}{72}$

Further,

$\frac{-56}{72}<\frac{-45}{72}$

∴​ $\frac{7}{-9}<\frac{-5}{8}$

3. ​The two rational numbers are $\frac{-1}{3}\mathrm{and}\frac{4}{-5}$.

$\frac{4}{-5}=\frac{4\times -1}{-5\times -1}=\frac{-4}{5}$

The LCM of the denominators 3 and 5 is 15.

Now,

$\frac{-1}{3}=\frac{-1\times 5}{3\times 5}=\frac{-5}{15}$

Also,

$\frac{-4}{5}=\frac{-4\times 3}{5\times 3}=\frac{-12}{15}$

Further,

$\frac{-12}{15}<\frac{-5}{15}$

∴ $\frac{4}{-5}<\frac{-1}{3}$

4. The two rational numbers are $\frac{9}{-13}\mathrm{and}\frac{7}{-12}$.

$\mathrm{Now},\frac{9}{-13}=\frac{9\times -1}{-13\times -1}=\frac{-9}{13}\mathrm{and}\frac{7}{-12}=\frac{7\times -1}{-12\times -1}=\frac{-7}{12}$

The LCM of the denominators 13 and 12 is 156.

Now,

$\frac{-9}{13}=\frac{-9\times 12}{13\times 12}=\frac{-108}{156}$

Also,

$\frac{-7}{12}=\frac{-7\times 13}{12\times 13}=\frac{-91}{156}$

Further,

$\frac{-108}{156}<\frac{-91}{156}$

∴ $\frac{9}{-13}<\frac{7}{-12}$

5. The two rational numbers are $\frac{4}{-5}\mathrm{and}\frac{-7}{10}$.

∴​ $\frac{4}{-5}=\frac{4\times -1}{-5\times -1}=\frac{-4}{5}$

The LCM of the denominators 5 and 10 is 10.

Now,

$\frac{-4}{5}=\frac{-4\times 2}{5\times 2}=\frac{-8}{10}$

Also,

$\frac{-7}{10}=\frac{-7\times 1}{10\times 1}=\frac{-7}{10}$

Further,

$\frac{-8}{10}<\frac{-7}{10}$

∴ $\frac{-4}{5}<\frac{-7}{10},or,\frac{4}{-5}<\frac{-7}{10}$

6. The two rational numbers are $\frac{-12}{5}\mathrm{and}-3.$
$-3\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as}\frac{-3}{1}$.

The LCM of the denominators is 5.

Now,

$\frac{-3}{1}=\frac{-3\times 5}{1\times 5}=\frac{-15}{5}$

Because $\frac{-15}{5}<\frac{-12}{5}$, we can conclude that $-3<\frac{-12}{5}$.