Students can find geometric mean questions here with in-depth explanations, which will help them to fully grasp the concept. As we all know, a mean is the average of the specified data value. A mean can be categorized into various types, including arithmetic mean, geometric mean, harmonic mean, and more. We have offered various geometric mean questions here that will help students understand the differences between different kinds of means. The problems are offered for students to practice, and they can compare their solutions to those given on our page. Click here to learn more about the geometric mean.

Geometric Mean Questions with Solutions

1. Find the geometric mean of 2, 6, 9, 5, 12.

Solution:

Given data values: 2, 6, 9, 5, and 12

We know that the formula to find the geometric mean is (a₁ × a₂ × … × aₙ)^1/n

Now, substitute the values in the formula, we get

Geometric Mean, GM = (2 × 6 × 9 × 5 × 12)^1/5

GM = (6480)^1/5

Thus, the 5th root of 6480 is 5.785.

Therefore, the geometric mean of 2, 6, 9, 5 and 12 is 5.79 (rounded to two decimal places).

2. Find the geometric mean of the data values: 3, 7, 21 and 15.

Solution:

Given data values: 3, 7, 21 and 15

Here, we have 4 data values, and hence n = 4.

Now, substitute the values in the geometric mean formula (a₁ × a₂ × … × aₙ)^1/n, we get

Geometric Mean = (3 × 7 × 21 × 15)^1/4

Geometric Mean = (6615)^1/4

Therefore, the fourth root of 6615 is 9.018

Therefore, the geometric mean of 3, 7, 21 and 15 is 9.02 (rounded to two decimal places)

3. Calculate the geometric mean of 11, 17, 28, 65, and 14.

Solution:

Given: 11, 17, 28, 65, 14

Here n = 5, as there are 5 data values.

Substituting the values in the geometric mean formula, we get

Geometric Mean = (11 × 17 × 28 × 65 × 14)^1/5

GM = (4764760)^⅕

Therefore, the 5th root of 4764760 is

GM = 21.657

Therefore, the geometric mean = 21.66 (rounded to two decimal places).

4. The arithmetic mean and the geometric mean of two positive integers are 10 and 8. Determine the two numbers.

Solution:

Let the two numbers be “a” and “b”.

From the given condition, we can write:

Arithmetic Mean: (a+b)/2 = 10 …(1)

Geometric Mean: √ab = 8 …(2)

From equation (1), we get

a +b = 20 …(3)

Squaring the both sides of equation (2), we get

(√ab)² = 8²

Thus, the above equation becomes:

ab = 64 ..(4)

As, “a” and “b” are the roots of the equation, we can write as:

x² – ( a + b )x + ab = 0 …(5)

Now, substitute the values (3) and (4) is equation (5)

x² – 20x + 64 = 0

(x – 16)(x -4) = 0

Hence, the values of x are x = 4 and x = 16.

It means, a = 4 and b = 16.

Hence, the two numbers are 4 and 16.

5. The sum of 2 numbers is six times their GM. Show that the numbers should be in the ratio (3+2√2) : (3− 2√2).

Solution:

Assume that the two numbers are x and y.

So, Geometric mean, GM = √(xy)

From the given condition, we can write:

x + y = 6√(xy) …(1)

To remove the square root on the right side of the given equation, take square on both sides.

(x+y)² = [6√(xy)]²

(x+y)² = 36(xy)

Also, (x -y)² = (x+y)² – 4xy

Now, plug the value (x+y)² = 36(xy) in the above formula,

(x -y)² = 36xy- 4xy

(x -y)² = 32xy

⇒ x- y = √32.√(xy)

⇒ x- y = 4√2.√(xy) …(2)

Now, add the equations (1) and (2),

2x = (6+4√2)√(xy)

Hence, x = (3 + 2√2)√(xy).

Now, by substituting the “x” value in equation (2), we will get the value of “y”.

(3 + 2√2)√(xy) + y = 6√(xy)

Hence, y = 6√(xy) – (3 + 2√2)√(xy)

y = [6 – 3 – 2√2]√(xy)

y = (3- 2√2)√(xy)

Hence, the ratio of x to y is:

x/y = [(3 + 2√2)√(xy)] / [(3- 2√2)√(xy)]

x/y = (3 + 2√2) / (3- 2√2)

Thus, the ratio of two numbers is (3 + 2√2) / (3- 2√2).

Hence, proved.

6. What is the geometric mean of 6 and 24?

Solution:

Given two data values are 6 and 24.

So, n = 2.

So, to find the geometric mean of 6 and 24, we have to take the square root for the product of 6 and 24.

Geometric Mean = √(6 × 24)

Geometric Mean = √(144)

Thus, the square root of 144 is 12.

Therefore, the geometric mean of 6 and 24 is 12.

7. Find the geometric mean and arithmetic mean of 4 and 5.

Solution:

Given data values: 4 and 5.

Finding Geometric Mean of 4 and 5:

Here, n = 2.

Geometric Mean = √(4 × 5)

GM = √20

GM = 4.47

Hence, the geometric mean of 4 and 5 is 4.47.

Finding Arithmetic Mean of 4 and 5:

Arithmetic Mean = (4 + 5)/2

AM = 9/2

AM = 4.5

Hence, the arithmetic mean of 4 and 5 is 4.5.

8. If the arithmetic mean and the harmonic mean of the set of observations are 9 and 49, respectively, then find the geometric mean.

Solution:

Given: Arithmetic Mean, AM = 9

Harmonic Mean = 49.

As we know, the relationship between AM, HM and GM is AM × HM = GM².

The above relation can be written as: GM = √(AM × HM)

Now, substituting the AM and HM values in the above formula, we get

Geometric Mean, GM = √(9 × 49)

Geometric Mean, GM = √441

So, the square root of 441 is 21, which is the required geometric mean.

Therefore, the geometric mean is 21, if AM = 9 and HM = 49.

9. Find the geometric mean of 11, 22 and 33.

Solution:

Given data values: 11, 22 and 33.

Here, n = 3.

Substituting the values in the geometric mean formula, we get

Geometric mean of 11, 22 and 33 = (11 × 22 × 33)^⅓

GM = (7986)^⅓

Hence, the cube root of 7986 is 19.99.

Therefore, the geometric mean of 11, 22 and 33 is 19.99.

10. Calculate the GM for the given set of observations: 45, 21, 77, and 68.

Solution:

Given data observations are 45, 21, 77, and 68.

Thus, Geometric Mean = (4948020)^1/4.

To find the geometric mean, we have to find the fourth root of 4948020.

GM = 47.16

Hence, the geometric mean of 45, 21, 77, and 68 is 47.16.

Explore More Articles:

• Average Questions
• Mean Questions
• Median Questions
• Mean, Median and Mode Questions
• Arithmetic Mean Questions
• Factorisation Questions

Practice Questions

Solve the following geometric mean questions:

1. Determine the geometric mean of 3, 6 and 9.
2. What is the geometric mean of 4 and 6.
3. Find the arithmetic mean and the geometric mean of 12, 18, 21, 72, and 65.

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