If the sum of the arithmetic mean and the har | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the two positive numbers be $a$ and $b$.Arithmetic Mean $A = \frac{a+b}{2}$,Geometric Mean $G = \sqrt{ab} = 12$,Harmonic Mean $H = \frac{2ab}{

Vedclass · Accepted Answer

$32$

Vedclass · Answer

(C) Let the two positive numbers be $a$ and $b$.Arithmetic Mean $A = \frac{a+b}{2}$,Geometric Mean $G = \sqrt{ab} = 12$,Harmonic Mean $H = \frac{2ab}{a+b}$.Given $A + H = 25$ and $G = 12$.Since $G^2 = AH$,we have $AH = 12^2 = 144$.Substituting $H = \frac{144}{A}$ into $A + H = 25$:$A + \frac{144}{A} = 25 \implies A^2 - 25A + 144 = 0$.$(A - 16)(A - 9) = 0$,so $A = 16$ or $A = 9$.Since $A \geq G$,we must have $A = 16$.Thus,$\frac{a+b}{2} = 16 \implies a+b = 32$.

If the sum of the arithmetic mean and the harmonic mean between two positive numbers is $25$ and their geometric mean is $12$,then what is the sum of the numbers?

Similar Questions

If the ratio of $A.M.$ between two positive real numbers $a$ and $b$ to their $H.M.$ is $m:n$,then $a:b$ is

Let ${a_1, a_2, \dots, a_{10}}$ be in $A.P.$ and ${h_1, h_2, \dots, h_{10}}$ be in $H.P.$ If ${a_1 = h_1 = 2}$ and ${a_{10} = h_{10} = 3}$,then the value of ${a_4 h_7}$ is:

If $A, G,$ and $H$ are the arithmetic mean,geometric mean,and harmonic mean of two given numbers respectively,then which of the following is true?

The sum of three decreasing numbers in $A.P.$ is $27$. If $-1, -1, 3$ are added to them respectively,the resulting series is in $G.P.$ The numbers are

If ${G_1}$ and ${G_2}$ are two geometric means and $A$ is the arithmetic mean inserted between two numbers,then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module