If the ratio of A M to G M of two positive numbers a and b is 5: 3 , then a: b is equal to

Correct answer: C. 9: 1 A.M of two numbers a, b= a+b 2 G.M of two numbers a, b= ab . Given, A.M G.M =(5)/(3) ( a+b 2 ) ab =(5)/(3) a+b ab =(10)/(3) a^2+b^2+2 ab ab =(100)/(9) (Squaring on both sides) 9 a^2+9 b^2+18 a b=100 a b 9 a^2-82 a b+9 b^2=0 9 a^2-81 a b-a b+9 b^2=0 9 a(a-9 b)-b(a-9 b)=0 (9 a-b)(a-9 b)=0 9 a-b=0 ; a-9 b=0 b=9 a ; a=9 b a b =(1)/(9) ; a b =(9)/(1)

NDA 2018

Exam: NDA
Year: 2018
Subject: Mathematics
Chapter: Sequences and Series

NDA 2018 · Mathematics · Sequences and Series

If the ratio of A M to G M of two positive numbers a and b is 5: 3...

If the ratio of

A M

G M

of two positive numbers

a

and

b

5: 3

, then

\mathrm{a}: \mathrm{b}

is equal to

Options

A $3: 5$
B $2: 9$
C $9: 1$
D $5: 3$

Correct answer

C. $9: 1$

Step-by-step solution

A.M of two numbers

\mathrm{a}, \mathrm{b}=\frac{\mathrm{a}+\mathrm{b}}{2}

G.M of two numbers

\mathrm{a}, \mathrm{b}=\sqrt{\mathrm{ab}}

. Given,

\frac{\text{ A.M }}{\text{ G.M }}=\frac{5}{3}\Rightarrow \frac{\left(\frac{\mathrm{a}+\mathrm{b}}{2}\right)}{\sqrt{\mathrm{ab}}}=\frac{5}{3}\Rightarrow \frac{\mathrm{a}+\mathrm{b}}{\sqrt{\mathrm{ab}}}=\frac{10}{3}\Rightarrow \frac{\mathrm{a}^2+\mathrm{b}^2+2 \mathrm{ab}}{\mathrm{ab}}=\frac{100}{9} \quad

(Squaring on both sides)

\Rightarrow \quad 9 a^2+9 b^2+18 a b=100 a b\Rightarrow \quad 9 a^2-82 a b+9 b^2=0\Rightarrow \quad 9 a^2-81 a b-a b+9 b^2=0\Rightarrow \quad 9 \mathrm{a}(\mathrm{a}-9 \mathrm{ b})-\mathrm{b}(\mathrm{a}-9 \mathrm{ b})=0\Rightarrow \quad(9 \mathrm{a}-\mathrm{b})(\mathrm{a}-9 \mathrm{ b})=0\Rightarrow \quad 9 \mathrm{a}-\mathrm{b}=0 ; \mathrm{a}-9 \mathrm{ b}=0\Rightarrow \quad \mathrm{b}=9 \mathrm{a} ; \mathrm{a}=9 \mathrm{ b}\Rightarrow \frac{\mathrm{a}}{\mathrm{b}}=\frac{1}{9} ; \frac{\mathrm{a}}{\mathrm{b}}=\frac{9}{1}

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