Decimal Fraction

$$ = \frac{{3.6 \times 0.48 \times 2.50}}{{0.12 \times 0.09 \times 0.5}}$$
$$ = \frac{{36 \times 48 \times 250}}{{12 \times 9 \times 5}}$$
$$ = 800$$

Converting each of the given fractions in to decimal form, we get
$$\frac{5}{9}$$ = 0.55
$$\frac{7}{11}$$ = 0.63
$$\frac{8}{15}$$ = 0.533
$$\frac{11}{17}$$ = 0.647
Clearly, 0.647 > 0.63 > 0.55 > 0.533
So, $$\frac{11}{17}$$ > $$\frac{7}{11}$$ > $$\frac{5}{9}$$ > $$\frac{8}{15}$$

Given expression :
$$ = 3927 + \frac{{5526}}{{12.5}}$$
$$ = 3927 + \frac{{55260}}{{125}}$$
= 3927 + 442.08
= 4369.08

$$\frac{4}{5}$$ = 0.8
$$\frac{7}{13}$$ = 0.53
$$\frac{1}{2}$$ = 0.5
$$\frac{2}{3}$$ = 0.66
$$\frac{3}{4}$$ = 0.75
$$\frac{5}{7}$$ = 0.714
Clearly, 0.5 does not lie between 0.53 and 0.8
∴ $$\frac{1}{2}$$ does not lie between $$\frac{4}{5}$$ and $$\frac{7}{13}$$

Let the missing number be x
$$\frac{{21.5}}{5} + \frac{{21}}{6} - \frac{{13.5}}{{15}} = \frac{{{{\left(×\right)}^{\frac{1}{3}}}}}{4} + \frac{{17}}{{30}}$$
$$\frac{{21.5}}{5} + \frac{{21}}{6} - \frac{{13.5}}{{15}} - \frac{{17}}{{30}} = \frac{{{{\left(×\right)}^{\frac{1}{3}}}}}{4}$$
L.C.M of 5, 6, 15 and 30 is 30
$$\frac{{129 + 105 - 27 - 17}}{{30}} = \frac{{{{\left(×\right)}^{\frac{1}{3}}}}}{4}$$
$$\root 3 \of x = \frac{{190 \times 4}}{{30}}$$
$$\root 3 \of x = 25.33 \approx 25 \cr &×= {25^3} \cr &×= 15625$$
Hence, the numbers 15625

$${\text{Given,}}$$
$$\frac{{52416}}{{312}} = 168$$
$$ ⇒ \frac{{52416}}{{168}} = 312$$
$${\text{Now,}}$$
$$\frac{{52.416}}{{0.0168}}$$
$$ = \frac{{524160}}{{168}}$$
$$ = \frac{{52416}}{{168}} \times 10$$
$$ = 312 \times 10$$
$$ = 3120$$

Converting each of the given fractions into decimal form, we get :
$$\frac{2}{3}$$ = 0.66
$$\frac{3}{4}$$ = 0.75
$$\frac{4}{5}$$ = 0.8
$$\frac{5}{6}$$ = 0.833
Since 0.833 > 0.8 > 0.75 > 0.66
So, $$\frac{5}{6}$$ > $$\frac{4}{5}$$ > $$\frac{3}{4}$$ > $$\frac{2}{3}$$
∴ Required difference = $$\left( {\frac{5}{6} - \frac{2}{3}} \right) = \frac{1}{6}$$

$$0.121212......$$
$$ = 0.\overline {12} $$
$$ = \frac{{12}}{{99}}$$
$$ = \frac{4}{{33}}$$

Given expression :
$$\left[ {1 + \frac{1}{{1 \times 2}} + \frac{1}{{1 \times 2 \times 4}} + \frac{1}{{1 \times 2 \times 4 \times 8}} + \frac{1}{{1 \times 2 \times 4 \times 8 \times 16}}} \right]$$
$$ = \frac{{2 \times 4 \times 8 \times 16 + 4 \times 8 \times 16 + 8 \times 16 + 16 + 1}}{{2 \times 4 \times 8 \times 16}}$$
$$ = \frac{{1024 + 512 + 128 + 16 + 1}}{{1024}}$$
$$ = \frac{{1681}}{{1024}}$$
$$ = 1.6416$$

Given expression :
$$\frac{{{{\left( {0.013} \right)}^3} + 0.000000343}}{{{{\left( {0.013} \right)}^2} - 0.000091 + 0.000049}}$$
$$a = 0.013 \text{ and } b = 0.007 $$
$$ = \left( {\frac{{{a^3} + {b^3}}}{{{a^2} - ab + {b^2}}}} \right)$$
$$ = a + b$$
$$ = 0.013 + 0.007$$
$$ = 0.020$$