Decimal Fraction

$$ = 7\frac{1}{2} - $$  $$\left[ {2\frac{1}{4} \div \left\{ {1\frac{1}{4} - \frac{1}{2}\left( {1\frac{1}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right]$$
$$ = \frac{{15}}{2} - $$   $$\left[ {\frac{9}{4} \div \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{3}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right]$$
$$ = \frac{{15}}{2} - $$   $$\left[ {\frac{9}{4} \div \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{{9 - 2 - 1}}{6}} \right)} \right\}} \right]$$
$$ = \frac{{15}}{2} - \left[ {\frac{9}{4} \div \left\{ {\frac{5}{4} - \frac{1}{2}} \right\}} \right]$$
$$ = \frac{{15}}{2} - \left[ {\frac{9}{4} \div \left\{ {\frac{{5 - 2}}{4}} \right\}} \right]$$
$$ = \frac{{15}}{2} - \left[ {\frac{9}{4} \div \frac{3}{4}} \right]$$
$$ = \frac{{15}}{2} - \left[ {\frac{9}{4} \times \frac{4}{3}} \right]$$
$$ = \frac{{15}}{2} - 3$$
$$ = \frac{{15 - 6}}{2}$$
$$ = \frac{9}{2}$$
$$ = 4\frac{1}{2}$$

Clearly,
$$\frac{{11}}{{14}} = 0.785$$
$$\frac{{16}}{{19}} = 0.842$$
$$\frac{{19}}{{21}} = 0.904$$
Now, 0.785 < 0.842 < 0.904
So, $$\frac{11}{14}$$ < $$\frac{16}{19}$$ < $$\frac{19}{21}$$

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

9 is at thousand^ths place in 0.06945
So, place value of 9 = $$\frac{9}{1000}$$

$$\frac{1}{3} = 0.333$$
$$\frac{3}{4} = 0.75$$
$$\frac{{117}}{{300}} = 0.39$$
$$\frac{{287}}{{400}} = 0.7175$$
$$\frac{{95}}{{300}} = 0.316$$
$$\frac{{301}}{{400}} = 0.7525$$
$$\frac{{99}}{{300}} = 0.33$$
$$\frac{{97}}{{300}} = 0.323$$
$$\frac{{299}}{{500}} = 0.598$$
Clearly, each one of 0.39 and 0.7175 lies between 0.333 and 0.75
So, $$\frac{117}{300}$$ and $$\frac{287}{400}$$ lie between $$\frac{1}{3}$$ and $$\frac{3}{4}$$

4×162 = 648. Sum of decimal places = 6.
So,
0.04×0.0162
= 0.000648
= 6.48×10^-4

$$\text{Let } a = 4.2 \text{ and } b = 1.9 $$
$$ {\text{Given Expression}}$$
$$ = \frac{{ {{a^2} - {b^2}} }}{{\left( {a + b} \right)\left( {a - b} \right)}}$$
$$ = \frac{{ {{a^2} - {b^2}} }}{{ {{a^2} - {b^2}} }}$$
$$ = 1$$

$$\frac{{144}}{{0.144}} = \frac{{14.4}}{x}$$
$$ ⇒ \frac{{144 \times 1000}}{{144}} = \frac{{14.4}}{x}$$
$$ ⇒ x = \frac{{14.4}}{{1000}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 0.0144$$

Suppose commodity X will cost 40 paise more than Y after z years
Then,(4.20 + 0.40z) − (6.30 + 0.15z) = 0.40
⇒ 0.25z = 0.40 + 2.10
$$ ⇒ z = \frac{{2.50}}{{0.25}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{250}}{{25}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\, = 10$$
∴ X will cost 40 paise more than Y 10 years
After 2001 i.e., 2011

Converting each of the given fractions in to decimal form, we get
$$\frac{1}{3} = 0.33$$
$$\frac{2}{5} = 0.4$$
$$\frac{3}{7} = 0.42$$
$$\frac{4}{5} = 0.8$$
$$\frac{5}{6} = 0.83$$
$$\frac{6}{7} = 0.85$$
Clearly, 0.85 > 0.83 > 0.8 > 0.42 > 0.4 > 0.33
So, $$\frac{6}{7}$$ > $$\frac{5}{6}$$ > $$\frac{4}{5}$$ > $$\frac{3}{7}$$ > $$\frac{2}{5}$$ > $$\frac{1}{3}$$