Find Two Numbers by Harmonic and Arithmetic Mean Calculator

Welcome to the free online calculator to find two numbers by given harmonic and arithmetic mean. For example, the harmonic mean of two numbers is 6.4, and their arithmetic mean is 10. What are the two numbers?

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

The arithmetic mean, also known as the average, refers to the sum of a set of numbers divided by the total number of numbers in the set.

The arithmetic mean formula is

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

The harmonic mean of two numbers is 6.4, and their arithmetic mean is 10. What are the two numbers?

Let the two numbers be x and y.

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

21x + 1y = 6.4………(1)

x + y2 = 10………(2)

Simplify the first equation.

21x + 1y = 6.4

2xyx + y = 6.4………(3)

Simplify the second equation.

x + y2 = 10

x + y = 20………(4)

Substitute the fourth equation into the third equation.

2xyx + y = 6.4

2xy20 = 6.4

2xy = 128

xy = 64………(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.

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

16 + 42 = 202 = 10

Correct.

Using the calculator is simple, follow these 3 steps:

Next, let’s look at an example.