Question

(a) 7 is the mean proportion between two numbers x and y and 56 is third proportion to x and y. Find the numbers.
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(b) If $\frac{x}{1}=\frac{\sqrt{a+3b}+\sqrt{a-3b}}{\sqrt{a+3b}-\sqrt{a-3b}}$, then, show that $3b{x}^2+3b-2ax=0$.
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(c) There are 36 members in a student council in a school and the ratio of the number of boys to the number of girls is 3 : 1. How many more girls should be added to the council so that the ratio of number of boys to the number of girls may be 9 :5 ?
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Answer 1

(a) Since, 7 is mean proportional between x and y.

$\Rightarrow x:7=7:y\Rightarrow xy=49\cdots \cdots \left(1\right)$
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and 56 is third proportional to x and y

$\Rightarrow x:y=y:56\Rightarrow y^2=56x\cdots \cdots \left(2\right)$
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From eq (1)

$xy=49\Rightarrow x=\frac{49}{y}$
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Substituting $x=\frac{49}{y}$ in eq (2)

$y^2=56\times \frac{49}{y}\Rightarrow y^3=56\times 49$

$\Rightarrow y^3=(7^3\times 2^3)$
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$y=7\times 2=14$

$\therefore x=\frac{49}{14}=3.5$
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$\therefore$ The required numbers are 3.5 and 14.
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(b) $\frac{x}{1}=\frac{\sqrt{a+3b}+\sqrt{a-3b}}{\sqrt{a+3b}-\sqrt{a-3b}}$
Applying componendo and dividendo
$\frac{x+1}{x-1}=\frac{\sqrt{a+3b}+\sqrt{a-3b}+\sqrt{a+3b}-\sqrt{a-3b}}{\sqrt{a+3b}+\sqrt{a-3b}-\sqrt{a+3b}+\sqrt{a-3b}}$
= $\frac{2\sqrt{a+3b}}{2\sqrt{a-3b}}$
= $\frac{\sqrt{a+3b}}{\sqrt{a-3b}}$

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Squaring on both sides, we get:
$\frac{(x+1{)}^2}{(x-1{)}^2}=\frac{(\sqrt{a+3b}{)}^2}{(\sqrt{a-3b}{)}^2}$
= $\frac{(x+1{)}^2}{(x-1{)}^2}=\frac{a+3b}{a-3b}$

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Again, applying componendo and dividendo
$\frac{(x+1{)}^2+(x-1{)}^2}{(x+1{)}^2-(x-1{)}^2}=\frac{a+3b+a-3b}{a+3b-a+3b}$
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$\frac{2(x^2+1)}{4x}=\frac{2a}{6b}$
=$\frac{(x^2+1)}{2x}=\frac{a}{3b}$
=$3b{x}^2+3b=2ax$
= $3b{x}^2+3b-2ax=0$
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(c) Total numbers of boys in student council = $\frac{3}{3+1}\times 36$
= $\frac{3}{4}\times 36$
= 27 boys
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And

Total numbers of girls in student council = $\frac{1}{3+1}\times 36$
= $\frac{1}{4}\times 36$
= 9 Girls
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Let x girls be added to get the required ratio ( 9 : 5 ) ,
Than we have
$\frac{27}{9+x}=\frac{9}{5}$
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By invertendo we have
$\frac{9+x}{27}=\frac{5}{9}$
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$\frac{9+x}{3}=\frac{5}{1}$
or $9+x=3\times 5$
or $9+x=15$

or x= 15-9

$\therefore x=6$

So,

6 girls need to be added to get our ratio as 9 : 5.
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