Question

Level 1
1. Find the arithmetic and also geometric mean of the values 10,25and 30​

Answer 1

Step-by-step explanation:

First we multiply them: 2 × 18 = 36

Then (as there are two numbers) take the square root: √36 = 6

In one line:

Geometric Mean of 2 and 18 = √(2 × 18) = 6

It is like the area is the same!

geometric mean 2x18 = 6x6

Example: What is the Geometric Mean of 10, 51.2 and 8?

First we multiply them: 10 × 51.2 × 8 = 4096

Then (as there are three numbers) take the cube root: 3√4096 = 16

In one line:

Geometric Mean = 3√(10 × 51.2 × 8) = 16

It is like the volume is the same:

geometric mean 10x51.2x8 = 16x16x16

Example: What is the Geometric Mean of 1, 3, 9, 27 and 81?

First we multiply them: 1 × 3 × 9 × 27 × 81 = 59049

Then (as there are 5 numbers) take the 5th root: 5√59049 = 9

In one line:

Geometric Mean = 5√(1 × 3 × 9 × 27 × 81) = 9

I can't show you a nice picture of this, but it is still true that:

1 × 3 × 9 × 27 × 81 = 9 × 9 × 9 × 9 × 9

Example: What is the Geometric Mean of a Molecule and a Mountain

length continuum

Using scientific notation:

A molecule of water (for example) is 0.275 × 10-9 m

Mount Everest (for example) is 8.8 × 103 m

Geometric Mean = √(0.275 × 10-9 × 8.8 × 103)

= √(2.42 × 10-6)

≈ 0.0016 m

Which is 1.6 millimetres, or about the thickness of a coin.

We could say, in a rough kind of way,

"a millimetre is half-way between a molecule and a mountain!"

Another cool one:

Example: What is the Geometric Mean of a Cell and the Earth?

A skin cell is about 3 × 10-8 m across

The Earth's diameter is 1.3 × 107 m

Geometric Mean = √(3 × 10-8 × 1.3 × 107)

= √(3.9 × 10-1)

= √0.39

≈ 0.6 m

A child is about 0.6 m tall! So we could say:

"A child is half-way between a cell and the Earth"

So the geometric mean gives us a way of finding a value in between widely different values.