Arithmetic, Geometric, and Harmonic Means
Kiara Crawford
Period: 3
Find the mean for a=25 and b=115
Solution
2ab
Step 1: Set up the equation
25 + 115
2
Step 2 : Solve
140/2 = 70
a+b
or
2
a+b
Find the mean when a= 49 and b= 16
1 1
Harmonic Means
+
a b
2
If the arithmetic mean of x and y is double their geometric mean, with x>y>0, then a possible value for the ratio x/y, to the nearest tenth, is:
Step 1 : Set up the equation
(49)(16)^1/2
Step 2: Solve
784^1/2=
28
a. 5
b.8
c.11
d.14
e. none of these
The inverse of the arithmetic mean's reciprocals of the observations in the set.
Solution
Remember that arithmetic is x+y/2 and geometric is xy^1/2
Step 1: so x+y/2 =2(xy)^1/2
Step 2: (x+y)^2=(4(xy)^1/2)^2
Step 3: x^2 + 2xy + y^2 = 16xy
Step 4: x^2 -14xy + y^2=0
Step 5: in order to find the ratio for x/y, we divide everything by y^2
x^2/y^2 - 14x/y +1 = 0
Step 6: using the quadratic equation in (x/y) to find
x/y = (14+ 14^2 -4)/2 = (14+ 192)/2 = 13.93 or 0.07
Step 7: Already given x > y > 0, so the answer is d. 14
http://ohiorc.org/for/math/stella/problems/problem.aspx?id=441
ab
Arithmetic Means
n/( 1/(Xv1)+ 1/(Xv2)+... +1/(Xvn))
Find the mean when x= 18 and y = 30
A group of children take a school bus on a trip, it travels between 2 cities A and B. From a to B, the bus had an average speed of v1. O the way back, the average speed is v2. Express the average speed the trip takes.
A set of data may be defined as the sum of the values divided by the number of values in the set.
Solution
=
Let t1 and v1 represent the time and speed for the trip A to B and let t2 and v2 represent the trip from B to A.In both, the distance will be represented by s.
Step 1: Set up the equation
2(18)(30)
18 + 30
Step 2: Solve
1080
48
45 or 22.5
2
Agarwal, B.L. Programed Statistics. New Delhi: New Age International, 2007. Print. Page 40.
=
distance traveled s + s 2s 2s
time t1 + t2 s + s sv2 + sv1
v1 v2 v1v2
v1v2 2v1v2
s(v1 + v2) v1 +v2
(Xv1 +Xv2 +...+Xvn)/(n)
(2s)
=
Geometric Means
Agarwal, B.L. Programed Statistics. New Delhi: New Age International, 2007. Print.Page 37.
The nth root of the product of n values of a set of observations.
((Xv1) (Xv2) ... (Xvn))^(1/n)
Agarwal, B.L. Programed Statistics. New Delhi: New Age International, 2007. Print. Page 39.
Relationship Between the Three
Comparison
(Xv1 +Xv2 +...+Xvn)/(n)> ((Xv1) (Xv2) ... (Xvn))^(1/n) > n/( 1/(Xv1)+ 1/(Xv2)+... + 1/(Xvn))
Both arithmetic and geometric series we have learned in class they can be used in finding arithmetic and geometric means, because they are the series' averages. By using the number of terms the sequence has you can find the average pertaining to what kind of series it is.
Bibliography
- Agarwal, B.L. Programed Statistics. New Delhi: New Age International, 2007. Print.
- <http://ohiorc.org/for/math/stella/problems/problem.aspx?id=441>
- <http://www.teaching.martahidegkuti.com/shared/lnotes/4_collegealgebra/means/means.pdf>