Express the following recurring decimals as vulgar fractions :

(i) $1.3\overline{45}$

(ii) $2.\overline{357}$

(i) Let x = $1.3\overline{45}$ = 1.3454545 … $\qquad \text{….(i)}$

So multiplying both sides of (i) by 10

we get,

10x = 13.4545…$\qquad \text{….(ii)}$

Again multiply by 100 on both sides ,

1000x =1345.4545…..$\qquad \text{….(iii)}$

Subtracting (ii) from (iii), we get

1000x - 10x = 1345.4545… - 13.4545…

990x = 1332

x = $\dfrac{1332}{990}$ = $\bold{\dfrac{74}{55}}$

which is in the form of $\dfrac{p}{q}$, q ≠ 0

(ii) Let x = $2.\overline{357}$ = 2.357357… $\qquad \text{….(i)}$

So multiplying both sides of (i) by 1000,

we get,

1000x = 2357.357357…$\qquad \text{….(ii)}$

Subtracting (i) from (ii), we get

1000x - x = 2357.357357… - 2.357357…

999x = 2355

x = $\bold{\dfrac{2355}{999}}$

which is in the form of $\dfrac{p}{q}$, q ≠ 0.