Find Two Numbers by Geometric and Arithmetic Mean Calculator

Welcome to the free online calculator to find two numbers by given geometric and arithmetic mean. For example, the geometric mean of two numbers is 20, and their arithmetic mean is 25. 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

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 geometric and arithmetic mean.

The geometric mean of two numbers is 20, and their arithmetic mean is 25. What are the two numbers?

Let the two numbers be x and y.

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

√xy = 20………(1)

x + y2 = 25………(2)

Square both sides of equation 1 at the same time.

√xy = 20

xy = 400………(3)

Simplify the second equation.

x + y2 = 25

x + y = 50………(4)

Now the problem becomes that the sum of two numbers is 50 and the product of the two numbers is 400, find the two numbers. It can be solved with find two numbers by sum and product calculator.

Continue to solve.

Simplify the 4th equation, let x be represented by y.

x + y = 50

x = 50 – y………(5)

Substitute x = 50 – y into the third equation and simplify.

xy = 400

(50 – y) y = 400

50y – y² = 400

y² – 50y + 400 = 0

(y – 10) (y – 40) = 0

y = 10 or y = 40

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

x = 50 – y = 50 – 10 = 40

or

x = 50 – y = 50 – 40 = 10

So, the two numbers are 10 and 40.

Verify.

√10 * 40 = √400 = 20

10 + 402 = 502 = 25

Correct.

Using the calculator is simple, follow these 3 steps:

Next, let’s look at some examples.