If two numbers have arithmetic mean 2700 and harmonic mean 75, then find their geometric mean.
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Let the numbers be A and B.
Then arithmetic mean = 2 [ A + B ] = 2 7 0 0 ............................ (given).(equation 1)
Harmonic mean = A + B 2 × A B = 7 5 ......................... (equation 2).
Now in equation 2 the value of [ A + B ] 2 is 1/2700 {by equ. 1} now A B = 7 5 × 2 7 0 0 .
We know that geometric mean of two numbers is A B . so finally we get 7 5 × 2 7 0 0 = 4 5 0 .
The harmonic mean m between two numbers a and b is defined by the equation m 2 = a 1 + b 1 , thus
m = a 1 + b 1 2
m = a 1 + b 1 2 ⋅ a b a b
m = b + a 2 a b
m = = 2 b + a a b
The denominator in the above expression is the arithmetic mean of a and b, so substituing numerical values, we have:
7 5 = 2 7 0 0 a b
The geometric mean of two numbers, a and b, is defined as g = a b , so from the above equation we get, g = 450
By the definition of arithmethic mean and the statement: 2 x + y = 2 7 0 0 (I)
By the definition of harmonic mean: x 1 + y 1 2 = 7 5 (II)
We can rearranje (II) to get:
x ∗ y = 2 7 5 ∗ ( x + y )
Using (I) we get:
x ∗ y = 2 7 5 ∗ ( 5 4 0 0 )
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 = 4 5 0
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 )
( G M ) 2 = ( 2 7 0 0 ) ( 7 5 )
( G M ) 2 = 2 0 2 5 0 0
G M = 2 0 2 5 0 0 = 4 5 0
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.
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
first thing to remember is A M ⩾ G M ⩾ H M , this will help you to make sure that your answer is true.
from the first information: A M = 2 a + b = 2 7 0 0 ⟹ a + b = 5 4 0 0 . . . ( ∗ )
from the second information: H M = a 1 + b 1 2 ⟹ a + b a b = 2 7 5 . . . ( ∗ ∗ )
by isolating a b , take the square root on both side, and substituting ( ∗ ) to ( ∗ ∗ ) , you get:
G M = a b = 8 1 ∗ 2 5 ∗ 1 0 0 = 4 5 0
By definition, A M = 2 a + b and H M = a + b 2 a b (after simplication). Notice that if multiplied together, the two will yield a b which is simply G M 2 . So, the geometric mean is simply 2 7 0 0 × 7 5 = 4 5 0 .
Let's denote the two numbers by a and b .
Now, by the definitions of arithmetic, geometric and harmonic means ( AM , GM and HM ), we have:
AM GM HM = 2 a + b = 2 7 0 0 = a b ≥ 0 = a 1 + b 1 2 = 7 5
But since, a + b = a b ( a 1 + b 1 ) ,
4 5 0 2 = 2 7 0 0 ⋅ 7 5 = A M ⋅ H M = 2 a + b ⋅ a 1 + b 1 2 = a b = GM 2
Since GM is non-negative, GM = 4 5 0 .
( G . M . ) 2 = ( A . M . ) ( H . M . )
( G . M . ) 2 = ( 2 7 0 0 ) ( 7 5 )
( G . M . ) 2 = 2 0 2 5 0 0
G . M . = 4 5 0
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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The definitions of algebraic mean, geometric mean and harmonic mean of two numbers a , b are as such:
A . M . = ( a + b ) / 2
G . M . = a b
H . M . = a 1 + b 1 2
Manipulating the formula for H.M, we get a + b 2 a b = 7 5 .
Substituting the formula for A.M. into this equation, we have now
a b = 7 5 ∗ 2 a + b a b = 7 5 ∗ 2 7 0 0 Thus G.M. = 7 5 ∗ 2 7 0 0 = 4 5 0