Find Two Numbers by Geometric and Harmonic Mean Calculator

Welcome to the free online calculator to find two numbers by given geometric and harmonic mean. For example, the geometric mean of two numbers is 8, and their harmonic mean is 6.4. What are the two numbers?

The geometric mean corresponds to the central tendency of a set of numbers using the product of the values, and it contrasts with the arithmetic mean, which uses the sum of values. So, the geometric mean is computed as the Nth root of the product of n numbers. A geometric mean is only calculated for positive integers. The formula for the geometric mean is

Harmonic mean is the reciprocal of the average arithmetic mean of the reciprocals of each number in a given set of data.

The formula to calculate the harmonic mean is

Next, use the example at the beginning of the article to explain how to find two numbers based on the given geometric and harmonic mean.

The geometric mean of two numbers is 8, and their harmonic mean is 6.4. What are the two numbers?

Let the two numbers be x and y.

So, we can list the geometric and the harmonic mean of these two numbers.

√xy = 8………(1)

21x + 1y = 6.4………(2)

Square both sides of equation 1 at the same time.

√xy = 8

xy = 64………(3)

Simplify the second equation.

21x + 1y = 6.4

2xyx + y = 6.4………(4)

Substitute the third equation into the fourth equation.

2 * 64x + y = 6.4

x+y = 20………(5)

Now the problem becomes that the sum of two numbers is 20 and the product of the two numbers is 64, find the two numbers. It can be solved with find two numbers by sum and product calculator.

Continue to solve.

Simplify the 5th equation, let x be represented by y.

x + y = 20

x = 20 – y………(6)

Substitute x = 20 – y into the third equation and simplify.

xy = 64

(20 – y) y = 64

20y – y² = 64

y² – 20y + 64= 0

(y – 4) (y – 16) = 0

y = 4 or y = 16

According to the relationship between x and y, we can calculate the value of x.

x = 20 – y = 20 – 4 = 16

or

x = 20 – y = 20 – 16 = 4

So, the two numbers are 4 and 16.

Verify.

√4 * 16 = √64 = 8

214 + 116 = 2 * 4 * 164 + 16 = 12820 = 6.4

Correct.

Using the calculator is simple, follow these 3 steps:

Next, let’s look at some examples.