Please help me to solve this sum.
Answers
The two observations are 20 and 5
Given :-
The Arithmetic Mean and Geometric Mean of two observations are 12.5 and 10.
To find :-
The two observations.
Solution :-
Let the two observations be X and Y
We know that
The Arithmetic Mean of X and Y = (X+Y)/2
According to the given problem
Arithmetic Mean of the two observations = 12.5
Therefore, (X+Y)/2 = 12.5
=> X+Y = 12.5×2
=> X+Y = 25 ------------(1)
We know that
The Geometric Mean of X and Y = √(XY)
According to the given problem
The Geometric Mean of the two observations
= 10
Therefore , √(XY) = 10
On squaring both sides then
=> (√XY)² = 10²
=> XY = 100 ----------(2)
We know that
(a-b)² = (a+b)² - 4ab
=> (X-Y)² = 25²-4(100)
=> (X-Y)² = 625-400
=> (X-Y)² = 225
=> X-Y = ±√225
=> X-Y = ±15
=> X-Y = 15 -----------(3)
Since, The AP and GP are positive.
On adding (1) and (3) then
X+Y = 25
X-Y = 15
(+)
________
2X+0 = 40
________
=> 2X = 40
=> X = 40/2
=> X = 20
Therefore, X = 20
On substituting the value of X in (1) then
20+Y = 25
=> Y = 25-20
=> Y = 5
Therefore, X = 20 and Y = 5
Answer :-
The two observations are 20 and 5
Used formulae:-
• The Arithmetic Mean of X and Y
= (X+Y)/2
• The Geometric Mean of X and Y
= √(XY)