Question

Express 1.32̅ + 0.3̅5̅ as a fraction in its simplest form

Answer 1

Answer :

We can express 1.3$\overline{2}$ + 0. $\overline{35}$ as a fraction As :

Firsts we convert 1.3$\overline{2}$ into fraction form , As :

1.3$\overline{2}$ = 1.32222222.....

1.322222222... = 1.3 + 0.02222222.....
Now we multiply and divide by 10 , in number 1.3 and get

1.3222222222... = $\frac{13}{10}$ + 0.02222222..... --------------- ( 1 )

Now Let

x = 0.02222222...... ------------- ( 2 )

Now we multiply by 100 on both hand side and get

100x = 2.2222222...... ------------ ( 3 )

Now we subtract equation 2 from equation 3 and get

99x = 2.2

Now we multiply by 10 on both hand side , we get

990x = 22

Now we divide by 22 on both hand side , we get

45x = 1

So,

x = $\frac{1}{45}$ , So

0.02222222..... = $\frac{1}{45}$

Substitute that value in equation and get

1.3222222222... = $\frac{13}{10}$ + $\frac{1}{45}$ Taking LCM we get

1.3222222222... = $\frac{13\times 9+2\times 1}{90}=\frac{117+\hspace{0.17em}2}{90}$ = $\frac{119}{90}$

Now we convert 0. $\overline{35}$ into fraction form , As :

0. $\overline{35}$ = 0.353535353535...

Now Let

y = 0.353535353535... ------------- ( A )

Now we multiply by 100 on both hand side and get

100y = 35.353535353535... ------------ ( B )

Now we subtract equation 2 from equation 3 and get

99 y = 35

y = $\frac{35}{99}$
So,
0. $\overline{35}$ = $\frac{35}{99}$

Now we get

1.3$\overline{2}$ + 0. $\overline{35}$ = $\frac{119}{90}$ + $\frac{35}{99}$ taking LCM we get

1.3$\overline{2}$ + 0. $\overline{35}$ = $\frac{119\times 11+35\times 10}{990}$

1.3$\overline{2}$ + 0. $\overline{35}$ = $\frac{1309+350}{990}$

1.3$\overline{2}$ + 0. $\overline{35}$ = $\frac{1659}{990}$ ( Ans )