Convert the following decimal numbers into the form of $\frac{p}{q}$.

(A) $12.6$

(B) $0.0120$

(C) $3.0\overline{52}$

(D) $1.2\overline{35}$

(E) $0.\overline{23}$

(F) $2.0\overline{5}$

(G) $2.12\overline{5}$

(H) $3.12\overline{5}$

(I) $2.\overline{1625}$

(A) $12.6 = \frac{126}{10} = \frac{63}{5}$

(B) $0.0120 = \frac{120}{10000} = \frac{3}{250}$

(C) $3.0\overline{52}$: Let $x = 3.0525252...$

$$10x = 30.52525... \quad ...(i)$$

$$1000x = 3052.52525... \quad ...(ii)$$

Subtract eq (i) from (ii):

$$1000x - 10x = 3052.5252... - 30.5252...$$

$$= 3022$$

$$990x = 3022$$

$$\Rightarrow x = \frac{3022}{990} = \frac{1511}{495}$$

(D) $1.2\overline{35}$: Let $x = 1.23535353...$

$$10x = 12.35353... \quad ...(i)$$

$$1000x = 1235.3535... \quad ...(ii)$$

Subtract eq (i) from (ii):

$$1000x - 10x = 1235.3535 - 12.3535$$

$$= 1223$$

$$990x = 1223$$

$$\Rightarrow x = \frac{1223}{990}$$

$$(E) \ 0.\overline{23}: \text{Let} \quad x = 0.232323... \quad ...(i)$$

$$100x = 23.2323... \quad ...(ii)$$

Subtract eq(i) from (ii):

$$99x = 23$$

$$\Rightarrow x = \frac{23}{99}$$

$$(F) \ 2.0\overline{5}: \text{Let } x = 2.0555...$$

$$10x = 20.555... \quad ...(i)$$

$$100x = 205.555... \quad ...(ii)$$

Subtract eq(i) from (ii):

$$100x - 10x = 185$$

$$\Rightarrow 90x = 185$$

$$\Rightarrow x = \frac{185}{90} = \frac{37}{18}$$

$$(G) \ 2.12\overline{5}: \text{Let} \quad x = 2.12555...$$

$$100x = 212.555...;$$

$$1000x = 2125.555...$$

$$1000x - 100x = 2125.555 - 212.555$$

$$= 1913$$

$$900x = 1913$$

$$\Rightarrow x = \frac{1913}{900}$$

$$(H) \ 3.12\overline{5}: \text{Let} \quad x = 3.12555...$$

$$100x = 312.555... \quad ...(i)$$

$$1000x = 3125.555... \quad ...(ii)$$

Subtract eq(i) from (ii):

$$1000x - 100x = 2813$$

$$900x = 2813$$

$$\Rightarrow x = \frac{2813}{900}$$

$$(I) \ 2.\overline{1625}:$$

$$\text{Let} \quad x = 2.1625162516251625... \quad ...(i)$$

$$10000x = 21625.\overline{1625} \quad ...(ii)$$

Subtract eq(i) from (ii):

$$10000x - x = 21625.1625... - 2.1625...$$

$$= 21623$$

$$9999x = 21623$$

$\Rightarrow x = \frac{21623}{9999}$

Concept Applied
To convert a repeating decimal to $\frac{p}{q}$ form, let $x$ equal the decimal, multiply by a suitable power of $10$ to align the repeating block, then subtract to eliminate it.