Number Operations and Properties

$8 : x :: 16 : 35$

Given, the first three terms of a proportion are $3,5$ and $21$ respectively.

Let the fourth number be $'x'$,

So, Numbers proportion are $3, 5, 21, x$

According to proportionality,

Product of means $=$ Product of extremes

$3 : 5 :: 21 : x$

$\dfrac{3}{5} = \dfrac{21}{x}$

Cross multiply,

$3 \times x = 21 \times 5$

$3x = 105$

$x = 35$

Therefore, The required fourth number is $35$

In $ a:b::c:d;   d $ is the fourth proportional.

In, $ 5: \sqrt {75} :: \sqrt {48} :d $, product of extremes $ = $ product of means

$  5 \times d = \sqrt {75} \times  \sqrt {48} $
$ d = \dfrac{\sqrt {75} \times  \sqrt {48}}{5} = \dfrac{5\sqrt {3} \times 4\sqrt {3}}{5} = 4 \times 3 = 12 $

In $ a:b::c:d;   d $ is the fourth proportional.
For, $ 3:42 :: 7 :d $, product of extremes $ = $ product of means
$ 3 \times d = 42 \times 7 $
$ d = 98 $

If a : b :: b : c, then we say that a, b, c are in continued

proportion, and c is the third proportional of a and b.





Here, $ {b}^{2} = ac $ or $ b = \sqrt {ac} $





So, for $ 12, 192 $, the third proportional is $ b = \sqrt {12 \times 192} = \sqrt {4

\times 3 \times 64 \times 3} = 2 \times 3 \times 8 = 48 $

$2:3 : :6:x$
$\Longrightarrow \dfrac{2}{3}=\dfrac{6}{x}$

Let the fourth proportional to $5,\,8,\,15$ be x
$5:8::15:x$

Multiples of 72 = 2,4,6,8,9,12,18,24,36,72
But in a year we have only 12 months and in a month we have only 30 days
So We have
$24.3.72 = 24\times 3 = 72$
$18.4.72 = 18\times 4 = 72$
$12.6.72= 12\times6 = 72$
$9.8.72 = 9\times8 = 72$
$8.9.72 = 8\times 9 = 72$
$6.12.72 = 6\times 12 = 72$

$14.79\times 1000 = 14790$

It can be solved as $27\times 1.\overline {2} \times 5.526\overline {2} \times 0.\overline {6}$
$= 27\times \left (1 + \dfrac {2}{9}\right )\times \dfrac {6}{9} \times (5.526\overline {2})$
$= 27\times \dfrac {11}{9}\times \dfrac {2}{3}\times (5.526\overline {2})$
$= 22\times (5.526\overline {2})$
$= 22\times (5.5 + 0.026\overline {2})$
$= 121 + (22\times 0.026\overline {2})$
$= 121 + 0.\overline {57}$
$= 121.\overline {57}$

$5 \times 10 \times 25 \times 40 \times 50 \times 55 \times 65 \times 125 \times 80 $
$= 5 \times 2 \times 5 \times 5^2 \times 2^3 \times 5 \times 2 \times 5^2 \times 11 \times 5 \times 13 \times 5 \times 5^3 \times 2^4 \times 5$
$= 2^9 \times 5^{13}\times 11 \times 13 = (2 \times 5)^9 \times 5^4 \times 11 \times 13$