Question

2 If A.M: G.M = 13:12 for the two
positive numbers, then the ratio of
the numbers is *
04:1
0 4:9
O 3:4
02:3​

Answer 1

GIVEN :–

$\\ \implies \rm A.M.:G.M. = 13:12$

TO FIND :–

• Ratio of the numbers = ?

SOLUTION :–

• Let two numbers are a & b.

• We know that –

$\\ \implies \rm A.M.= \dfrac{a + b}{2}$

• And –

$\\ \implies \rm G.M.= \sqrt{ab}$

• So that –

$\\ \implies \rm \dfrac{ \dfrac{a + b}{2}}{ \sqrt{ab} } = \dfrac{13}{12}$

$\\ \implies \rm \dfrac{a + b}{ \cancel2\sqrt{ab} } = \dfrac{13}{ \cancel{12}}$

$\\ \implies \rm \dfrac{a + b}{\sqrt{ab} } = \dfrac{13}{6}$

$\\ \implies \rm \dfrac{a}{\sqrt{ab}} + \dfrac{b}{ \sqrt{ab} } = \dfrac{13}{6}$

$\\ \implies \rm \sqrt \dfrac{a}{b} +\sqrt \dfrac{b}{a}= \dfrac{13}{6}$

$\\ \implies \rm \sqrt \dfrac{a}{b} +\dfrac{1}{ \sqrt\dfrac{a}{b}}= \dfrac{13}{6}$

• Let's put $\: \: \rm \sqrt \dfrac{a}{b} = x \: \:$ –

$\\ \implies \rm x +\dfrac{1}{x}= \dfrac{13}{6}$

$\\ \implies \rm\dfrac{ {x}^{2} + 1}{x}= \dfrac{13}{6}$

$\\ \implies \rm6({x}^{2} + 1) = 13x$

$\\ \implies \rm6{x}^{2} +6= 13x$

$\\ \implies \rm6{x}^{2} - 13x+6=0$

$\\ \implies \rm6{x}^{2} -9x - 4x+6=0$

$\\ \implies \rm3x(2x - 3) -2(2x - 3)=0$

$\\ \implies \rm(3x - 2)(2x - 3)=0$

$\\ \large\implies{ \boxed{ \rm x = \dfrac{2}{3}, \dfrac{3}{2}}}$

• So that –

$\\ \implies\rm \sqrt \dfrac{a}{b}= \dfrac{2}{3} \: \: and \: \: \sqrt\dfrac{a}{b} = \dfrac{3}{2}$

• Square on both sides –

$\\ \implies \large { \boxed{\rm \dfrac{a}{b}= \dfrac{4}{9} \: \: and \: \:\dfrac{a}{b} = \dfrac{9}{4}}}$

▪︎ Hence , Second option is correct.

Answer 2

SOLUTION

GIVEN

A.M: G.M = 13:12 for the two positive numbers,

TO CHOOSE THE CORRECT OPTION

The ratio of the numbers is

(a) 4:1

(b) 4:9

(c) 3:4

(d) 2:3

EVALUATION

Let the two numbers are a and b

Then Arithmetic mean (AM)

$= \displaystyle \sf{ \frac{a + b}{2} \: }$

Then Geometric mean (GM)

$\sf{ = + \sqrt{ab} }$

( + sign is taken as the two numbers are positive)

Now by the given condition

AM : GM = 13 : 12

$\implies \displaystyle \sf{ \frac{a + b}{2} : \sqrt{ab} = 13 : 12}$

$\implies \displaystyle \sf{ \frac{a + b}{2\sqrt{ab} } = \frac{13}{12} }$

$\implies \displaystyle \sf{ \frac{{(a + b)}^{2} }{4{ab} } = \frac{169}{144} } \: (squaring \: both \: sides)$

$\implies \displaystyle \sf{ \frac{4ab}{ {(a + b)}^{2} } = \frac{144}{169} }$

$\implies \displaystyle \sf{ 1 - \frac{4ab}{ {(a + b)}^{2} } = 1 - \frac{144}{169} }$

$\implies \displaystyle \sf{ \frac{{(a + b)}^{2} - 4ab}{ {(a + b)}^{2} } = \frac{169 - 144}{169} }$

$\implies \displaystyle \sf{ \frac{{(a - b)}^{2} }{ {(a + b)}^{2} } = \frac{25}{169} }$

$\implies \displaystyle \sf{ \frac{{(a - b)} }{ {(a + b)} } = \frac{5}{13} }$ ( Taking positive sign on Square root of both sides)

$\implies \displaystyle \sf{ \frac{{(a + b)} }{ {(a - b)} } = \frac{13}{5} }$

Now use Componendo Dividendo Rule

$\implies \displaystyle \sf{ \frac{{(a + b )+ (a - b)} }{ {(a + b )- (a - b)} } = \frac{13 + 5}{13 - 5} }$

$\implies \displaystyle \sf{ \frac{{2a} }{ {2b} } = \frac{18}{8} }$

$\implies \displaystyle \sf{ \frac{{a} }{ {b} } = \frac{9}{4} }$

$\implies \displaystyle \sf{ \frac{{b} }{ {a} } = \frac{4}{9} }$

Hence the ratio of the numbers is 4 : 9

FINAL ANSWER

If A.M: G.M = 13:12 for the two positive numbers, then the ratio of the numbers is

(b) 4 : 9

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