Mark's geometric mean

Algebra Level 2

If two numbers have arithmetic mean 2700 and harmonic mean 75, then find their geometric mean.


Note :

  • The arithmetic mean of two numbers a a and b b is a + b 2 \frac{a+b}2 .
  • The harmonic mean of two numbers a a and b b is 2 1 a + 1 b \frac2{\frac1{a} + \frac1{b}} .


The answer is 450.

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15 solutions

Bryan Lee
May 20, 2014

The definitions of algebraic mean, geometric mean and harmonic mean of two numbers a , b a, b are as such:

A . M . = ( a + b ) / 2 A.M. = (a+b)/2

G . M . = a b G.M. = \sqrt{ab}

H . M . = 2 1 a + 1 b H.M. = \frac{2}{\frac{1}{a} + \frac{1}{b}}

Manipulating the formula for H.M, we get 2 a b a + b = 75 \frac{2ab}{a+b} = 75 .

Substituting the formula for A.M. into this equation, we have now

a b = 75 a + b 2 ab = 75 * \frac{a+b}{2} a b = 75 2700 ab = 75 * 2700 Thus G.M. = 75 2700 = 450 \sqrt{75*2700} = 450

Mohit Gupta
May 20, 2014

Let the numbers be A and B.

Then arithmetic mean = [ A + B ] 2 = 2700 =\frac{[A+B]}{2}=2700 ............................ (given).(equation 1)

Harmonic mean = 2 × A B A + B = 75 =\frac{2\times AB}{A+B}=75 ......................... (equation 2).

Now in equation 2 the value of 2 [ A + B ] \frac{2}{[A+B]} is 1/2700 {by equ. 1} now A B = 75 × 2700 AB =75\times2700 .

We know that geometric mean of two numbers is A B \sqrt{AB} . so finally we get 75 × 2700 = 450 \sqrt{75\times2700}=450 .

The harmonic mean m between two numbers a and b is defined by the equation 2 m = 1 a + 1 b \frac{2}{m} = \frac{1}{a} + \frac {1}{b} , thus

m = 2 1 a + 1 b m = \frac{2}{ \frac{1}{a} + \frac{1}{b} }

m = 2 1 a + 1 b a b a b m = \frac{2}{ \frac{1}{a} + \frac{1}{b} } \cdot \frac{ab}{ab}

m = 2 a b b + a m = \frac{2ab}{b+a}

m = = a b b + a 2 m = = \frac{ab}{ \frac{ b+a}{2}}

The denominator in the above expression is the arithmetic mean of a and b, so substituing numerical values, we have:

75 = a b 2700 75 = \frac{ab}{2700}

The geometric mean of two numbers, a and b, is defined as g = a b g= \sqrt{ab} , so from the above equation we get, g = 450

By the definition of arithmethic mean and the statement: x + y 2 = 2700 \frac {x + y}{2} = 2700 (I)

By the definition of harmonic mean: 2 1 x + 1 y = 75 \frac {2}{\frac {1}{x} + \frac{1}{y}} = 75 (II)

We can rearranje (II) to get:

x y = 75 ( x + y ) 2 x * y = \frac {75 * (x + y)}{2}

Using (I) we get:

x y = 75 ( 5400 ) 2 x * y = \frac {75 * (5400)}{2}

Supposing the numbers have same sign (otherwise there would be no definition of geometric mean for a pair of numbers), taking the square root of both sides we get:

x y = 450 \sqrt{x * y} = 450

Steve Gregg
May 20, 2014

Let the two numbers be a and b.

The arithmetic mean of the two numbers is 2700, so (a+b)/2 = 2700, so a+b=5400.

The harmonic mean of the two numbers is 75, so 2 / ( (1/a) + (1/b) ) = 75, which simplifies to 2ab / (a+b) = 75.

Using the first equation, substitute 5400 for a + b in this equation to get 2ab / 5400 = 75, so ab = 202500, so sqrt(ab) = 450. This is the definition of the geometric mean of a and b, so the answer is 450.

We can use the formula:

( G M ) 2 = ( A M ) ( H M ) \large \color{#D61F06}\boxed{(GM)^2=(AM)(HM)}

( G M ) 2 = ( 2700 ) ( 75 ) (GM)^2=(2700)(75)

( G M ) 2 = 202500 (GM)^2=202500

G M = 202500 = GM=\sqrt{202500}= 450 \boxed{450}

Zair Henrique
May 20, 2014

Arithmetic Mean = (a + b) / 2 = A Harmonic Mean = (2ab)/(a+b) = H So, 2A = a+b.

2ab = H * 2A ab = H*A

Geometric Mean = sqrt of ab = sqrt H*A = sqrt 2700 * 75 = 450.

Rik Tomalin
May 20, 2014

let: A = arithmetic mean H = harmonic mean G = geometric mean x,y = the two numbers

we know that: A = 0.5 (x+y) ..>> x+y = 5400 H = 1/((1/2) ((1/x)+(1/y))) which when simplified leads to x y = 75 2700 finally, G = sqrt(x*y) = 450

Bostang Palaguna
Jul 10, 2020

first thing to remember is A M G M H M AM \geqslant GM \geqslant HM , this will help you to make sure that your answer is true.

from the first information: A M = a + b 2 = 2700 a + b = 5400... ( ) AM = \frac{a+b}{2} = 2700 \implies a + b = 5400 ... (*)

from the second information: H M = 2 1 a + 1 b a b a + b = 75 2 . . . ( ) HM = \frac{2}{\frac{1}{a} + \frac{1}{b}} \implies \frac{ab}{a+b}=\frac{75}{2} ... (**)

by isolating a b ab , take the square root on both side, and substituting ( ) (*) to ( ) (**) , you get:

G M = a b = 81 25 100 = 450 GM = \sqrt {ab} = \sqrt{81 * 25 * 100} = \boxed{450}

Krish Shah
Apr 15, 2020

By definition, A M = a + b 2 AM = \frac{a+b}{2} and H M = 2 a b a + b HM = \frac{2ab}{a+b} (after simplication). Notice that if multiplied together, the two will yield a b ab which is simply G M 2 GM^{2} . So, the geometric mean is simply 2700 × 75 = 450 \sqrt{2700 \times 75} = 450 .

Jesse Nieminen
Jun 30, 2017

Let's denote the two numbers by a a and b b .

Now, by the definitions of arithmetic, geometric and harmonic means ( AM \text{AM} , GM \text{GM} and HM \text{HM} ), we have:

AM = a + b 2 = 2700 GM = a b 0 HM = 2 1 a + 1 b = 75 \begin{aligned} \\ \text{AM} &= \dfrac{a+b}2 = 2700 \\\\ \text{GM} &= \sqrt{ab} \geq 0 \\\\ \text{HM} &= \dfrac2{\dfrac1a+\dfrac1b} = 75 \end{aligned}

But since, a + b = a b ( 1 a + 1 b ) a + b = ab\left(\dfrac1a+\dfrac1b\right) ,

45 0 2 = 2700 75 = A M H M = a + b 2 2 1 a + 1 b = a b = GM 2 \begin{aligned} 450^2 &= 2700 \cdot 75 \\ &= AM \cdot HM \\\\ &= \dfrac{a+b}2 \cdot \dfrac2{\dfrac1a+\dfrac1b} \\\\ &= ab \\ &= \text{GM}^2 \end{aligned}

Since GM \text{GM} is non-negative, GM = 450 \text{GM} = \boxed{450} .

( G . M . ) 2 (G.M.)^2 = = ( A . M . ) ( H . M . ) (A.M.)(H.M.)

( G . M . ) 2 (G.M.)^2 = = ( 2700 ) ( 75 ) (2700)(75)

( G . M . ) 2 (G.M.)^2 = = 202500 202500

G . M . G.M. = = 450 450

Vishwash Kumar
Sep 28, 2016

Prasit Sarapee
Dec 11, 2015

Youssef Hassan F
Dec 11, 2015

w= 2/ harmonic mean x= arithmetic mean y= arithmetic mean 2 z= harmonic mean l= geometric mean (a+b)/2=x x 2=y 2/((1/a)+(1/b))=z 2/ z=w sqrt(y/w)=l then x=2700 y=5400 z=75 w= 0.0267 l= sqrt(202500) = 450

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