Question

Add the following rational numbers:
(i) $\frac{3}{4}\mathrm{and}\frac{-3}{5}$
(ii) $\frac{5}{8}\mathrm{and}\frac{-7}{12}$
(iii) $\frac{-8}{9}\mathrm{and}\frac{11}{6}$
(iv) $\frac{-5}{16}\mathrm{and}\frac{7}{24}$
(v) $\frac{7}{-18}and\frac{8}{27}$
(vi) $\frac{1}{-12}\mathrm{and}\frac{2}{-15}$
(vii) $-1\mathrm{and}\frac{3}{4}$
(viii) $2\mathrm{and}\frac{-5}{4}$
(ix) $0\mathrm{and}\frac{-2}{5}$

Answer 1

1. The denominators of the given rational numbers are 4 and 5.

LCM of 4 and 5 is 20.

Now,

$\frac{3}{4}=\frac{3\times 5}{4\times 5}=\frac{15}{20}$ and $\frac{-3}{5}=\frac{-3\times 4}{5\times 4}=\frac{-12}{20}$

∴ $\frac{3}{4}+\frac{-3}{5}=\frac{15}{20}+\frac{-12}{20}=\frac{15+(-12)}{20}=\frac{15-12}{20}=\frac{3}{20}$

2.​ The denominators of the given rational numbers are 8 and 12.

LCM of 8 and 12 is 24.

Now,

$\frac{5}{8}=\frac{5\times 3}{8\times 3}=\frac{15}{24}$ and $\frac{-7}{12}=\frac{-7\times 2}{12\times 2}=\frac{-14}{24}$

∴​ $\frac{5}{8}+\frac{-7}{12}=\frac{15}{24}+\frac{-14}{24}=\frac{15+(-14)}{24}=\frac{15-14}{24}=\frac{1}{24}$

3. ​The denominators of the given rational numbers are 9 and 6.

LCM of 9 and 6 is 18.

Now,

$\frac{-8}{9}=\frac{-8\times 2}{9\times 2}=\frac{-16}{18}$ and $\frac{11}{6}=\frac{11\times 3}{6\times 3}=\frac{33}{18}$

∴​ $\frac{-8}{9}+\frac{11}{6}=\frac{-16}{18}+\frac{33}{18}=\frac{-16+33}{18}=\frac{-16+33}{18}=\frac{17}{18}$

4.​ The denominators of the given rational numbers are 16 and 24.

LCM of 16 and 24 is 48.

Now,

$\frac{-5}{16}=\frac{-5\times 3}{16\times 3}=\frac{-15}{48}$ and $\frac{7}{24}=\frac{7\times 2}{24\times 2}=\frac{14}{48}$

∴​ $\frac{-5}{16}+\frac{7}{24}=\frac{-15}{48}+\frac{14}{48}=\frac{-15+14}{48}=\frac{-1}{48}$

5. We will first write each of the given numbers with positive denominators.

$\frac{7}{-18}=\frac{7\times (-1)}{-18\times (-1)}=\frac{-7}{18}$

​The denominators of the given rational numbers are 18 and 27.

LCM of 18 and 27 is 54.

Now,

$\frac{-7}{18}=\frac{-7\times 3}{18\times 3}=\frac{-21}{54}$ and $\frac{8}{27}=\frac{8\times 2}{27\times 2}=\frac{16}{54}$

∴ $\frac{7}{-18}+\frac{8}{27}=\frac{-21}{54}+\frac{16}{54}=\frac{-21+16}{54}=\frac{-5}{54}$

6. ​We will first write each of the given numbers with positive denominators.

$\frac{1}{-12}=\frac{1\times (-1)}{-12\times (-1)}=\frac{-1}{12}$ and $\frac{2}{-15}=\frac{2\times (-1)}{-15\times (-1)}=\frac{-2}{15}$

​The denominators of the given rational numbers are 12 and 15.

LCM of 12 and 15 is 60.

Now,

$\frac{-1}{12}=\frac{-1\times 5}{12\times 5}=\frac{-5}{60}$ and $\frac{-2}{15}=\frac{-2\times 4}{15\times 4}=\frac{-8}{60}$

∴ $\frac{1}{-12}+\frac{2}{-15}=\frac{-5}{60}+\frac{-8}{60}=\frac{-5+(-8)}{60}=\frac{-5-8}{60}=\frac{-13}{60}$

7. We can write -1 as$\frac{-1}{1}$.

The denominators of the given rational numbers are 1 and 4.

LCM of 1 and 4 is 4.

Now,

$\frac{-1}{1}=\frac{-1\times 4}{1\times 4}=\frac{-4}{4}$ and $\frac{3}{4}=\frac{3\times 1}{4\times 1}=\frac{3}{4}$

∴ $-1+\frac{3}{4}=\frac{-4}{4}+\frac{3}{4}=\frac{-4+3}{4}=\frac{-1}{4}$

8. ​We can write 2 as$\frac{2}{1}$.

The denominators of the given rational numbers are 1 and 4.

LCM of 1 and 4 is 4.

Now,

$\frac{2}{1}=\frac{2\times 4}{1\times 4}=\frac{8}{4}$ and $\frac{-5}{4}=\frac{-5\times 1}{4\times 1}=\frac{-5}{4}$

∴ $2+\frac{(-5)}{4}=\frac{8}{4}+\frac{(-5)}{4}=\frac{8+(-5)}{4}=\frac{8-5}{4}=\frac{3}{4}$

9. ​We can write 0 as$\frac{0}{1}$.

The denominators of the given rational numbers are 1 and 5.

LCM of 1 and 5 is 5, that is, (1 $\times$ 5).

Now,

$\frac{0}{1}=\frac{0\times 5}{1\times 5}=\frac{0}{5}=0$ and $\frac{-2}{5}=\frac{-2\times 1}{5\times 1}=\frac{-2}{5}$

∴ $0+\frac{(-2)}{5}=\frac{0}{5}+\frac{(-2)}{5}=\frac{0+(-2)}{5}=\frac{0-2}{5}=\frac{-2}{5}$