Decimal Fraction

$$6.\overline {46} $$
$$ = 6 + 0.\overline {46} $$
$$ = 6 + \frac{{46}}{{99}}$$
$$ = \frac{{594 + 46}}{{99}}$$
$$ = \frac{{640}}{{99}}$$

$$101\frac{{27}}{{100000}}$$
$$ = 101 + \frac{{27}}{{100000}}$$
$$ = 101 + .00027$$
$$ = 101.00027$$

$$\frac{{0.0203 \times 2.92}}{{0.0073 \times 14.5 \times 0.7}}$$
$$ = \frac{{203 \times 292}}{{73 \times 145 \times 7}}$$
$$ = \frac{4}{5}$$
$$ = 0.8$$

$$\frac{{4.036}}{{0.04}}$$
$$ = \frac{{403.6}}{4}$$
$$ = 100.9$$

$$3.\overline {87} - 2.\overline {59} $$
$$ = \left( {3 + 0.\overline {87} } \right) - \left( {2 + 0.\overline {59} } \right)$$
$$ = \left( {3 + \frac{{87}}{{99}}} \right) - \left( {2 + \frac{{59}}{{99}}} \right)$$
$$ = 1 + \left( {\frac{{87}}{{99}} - \frac{{59}}{{99}}} \right)$$
$$ = 1 + \frac{{28}}{{99}}$$
$$ = 1.\overline {28} $$

$$0.125125....$$
$$ = 0.\overline {125} $$
$$ = \frac{{125}}{{999}}$$

$$\frac{2}{3} = 0.666$$
$$\frac{3}{5} = 0.6$$
$$\frac{2}{5} = 0.4$$
$$\frac{1}{3} = 0.333$$
$$\frac{1}{{15}} = 0.066$$
$$\frac{{31}}{{50}} = 0.62$$
Clearly, 0.62 lies between 0.6 and 0.666
So, $$\frac{31}{50}$$ lies between $$\frac{2}{3}$$ and $$\frac{3}{5}$$

Let the third number be 2960
∵ Second number = $$\frac{1}{4}$$ of the third number = $$\frac{1}{4}$$ × 2960 = 740
$$\frac{5}{9}$$ of first number = 25% of second number
$$\frac{5}{9}$$ first number = $$\frac{25 × 740}{100}$$  = 185
⇒ First number = $$\frac{185 × 9}{5}$$  = 333
∴ 30% of 333 = $$\frac{30}{100}$$ × 333 = 99.9

Given expression
= $$\frac{35 × 1.4}{7}$$
= 5 × 1.4
= 7

$$\therefore \frac{{1999}}{{2111}} = 0.946$$