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# Properties of Proportion. Geometric Mean

Ratio and Proportion Learning Material
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4.4Properties of Proportion. Geometric Mean

In this Math tutorial, you will learn:

• What are the properties of proportion?
• How do we use the properties of proportion to solve problems easier?
• How do we calculate the n-th proportional in a set of numbers participating in a proportion?
• What is geometric mean? Where does it differ from the arithmetic mean?
• How to extend the use of geometric mean if more than two numbers are present?

## Introduction

In the previous tutorial, we introduced and discussed proportion as a structure containing two or more equal ratios, and its types. Now we will look in more detail at the properties of proportion with the focus on "direct proportion" as the main type of proportion. This is because the inverse proportion is based on the direct proportion, we simply invert down the second ratio.

Another concept discussed in this tutorial is that of geometric mean - a useful tool related to proportions, especially in banking and finance. Moreover, a number of examples are provided with each concept or property to allow a better understanding by you, the reader.

## Properties of Proportion

First, it is worth highlighting that in all properties of proportion explained in the following part, we will assume the main proportion in the form

a:b = c:d

or

a/b = c/d

The following properties are true for direct proportion:

### Swapping the means in a proportion still provides a proportion

In symbols,

If a/b = c/d then a/c = b/d

For example, if

6/10 = 9/15

then

15/10 = 9/6

In the first case, both ratios have the value 3/5 when written in the simplest form, while in the second case the values of both ratios are 3/2 when written in the simplest form.

The advantage of this property is evident when the terms in each ratio of the original proportion are difficult to deal with. Thus, when we swap the means we obtain fractions that are easier to handle, so the operations become shorter.

### Example 1

Prove that the proportion 33:54 = 44:72 is true.

### Solution 1

First, we write the proportion in fraction form. We have

33/54 = 44/72

We cannot simplify the fractions expressing the individual ratios in the actual form. Thus, we can simplify by 3 the fraction on the left and by 4 that on the right. Therefore, we swap the means to obtain fractions that can be simplified. Hence, we obtain

33/44 = 54/72

Now, the task is easier, as the fraction on the left can be simplified by 11 while that on the right by 18 to obtain

3/4 = 3/4

Since the two final fractions representing the possible proportion are equal, the original proportion is true.

### Swapping the extremes in a proportion still provides a proportion

In symbols we write

If a/b = c/d then d/c = b/a

The purpose of using this property is again to make the operations simpler. Let's consider an example:

### Example 2

Check whether the fractions 7/24 and 21/72 are in proportion or not.

### Solution 2

One method could be simplifying both fractions by 3 and 7 respectively to obtain 3/8 in both cases. However, not everybody is able to figure out what is the GCF in either fraction (moreover, in the specific case, the terms of the first fraction are relatively prime, i.e. they cannot be simplified). Therefore, we use the second property of proportion to write

7/24 = 21/72
72/24 = 21/7

Both fractions give 3 when simplified (they express whole number, not improper fractions), so the original proportion is true.

### Inverting both ratios in a proportion still provides a proportion

In symbols, we write:

If a/b = c/d then b/a = d/c

This property is particularly useful when the inverted ratios give whole numbers. An example of the use of this method (inverting ratios) in practice is when dealing with parallel combination of resistors, when we have to invert the fractions in the final step to obtain the result, as in the example below.

### Example 3

Two resistors, R1 = 12 Ω and R2 = 18 Ω are connected in parallel in an electrical circuit. Calculate the equivalent resistance of the circuit. Recall that the equivalent resistance Req of a parallel combination is given by the formula

1/Req = 1/R1 + 1/R2

### Solution 3

Using the above formula, we write

1/Req = 1/R1 + 1/R2
1/Req = 1/12 + 1/18

Using the method of adding two fractions with different denominators, we obtain

1/Req = 1 × 3/12 × 3 + 1 × 2/18 × 2
1/Req = 3/36 + 2/36
1/Req = 5/36

Now, it is the time to explore the advantage the third property that ratios offer. Inverting down the two ratios (fractions) we obtain

Req/1 = 36/5
Req = 7.2 Ω

### Adding or subtracting the corresponding denominator to the numerator of each ratio in a proportion still provides a proportion

In symbols, we write

IF a/b = c/d THEN a ± b/b = c ± d/d

Let's try a numerical example to confirm this property.

### Example 4

Prove that the proportion 12:30 = 24:60 is true.

From the result obtained in (a) prove that 42:30 = 84:60

### Solution 4

First, we write the two ratios forming the proportion in fraction form. We have

12/30 = 24/60

We use the first property of ratios (swapping the means) to make the operations easier. Thus,

12/24 = 30/60

Simplifying the first ratio by 12 and the second one by 30, yields

12 ÷ 12/24 ÷ 12 = 30 ÷ 30/60 ÷ 30
1/2 = 1/2

Therefore, the original proportion is true.

Adding the denominators to the corresponding numerator of each ratio in the original proportion as the fourth property of proportion suggests, yields

12/30 = 24/60
12 + 30/30 = 24 + 60/60
42/30 = 84/60

Applying the second property of ratios (swapping the extremes), yields

60/30 = 84/42

Thus,

2 = 2

The last expression is a proof that the new ratio given in (b) is also true.

Another example with difference. The condition in this case is that the fraction must be improper; otherwise the difference becomes negative. To illustrate this property, let's take a known ratio, for example 4:3 = 8:6. Hence, we obtain

4/3 = 8/6
4 - 3/3 = 8 - 6/6
1/3 = 2/6

The last proportion is also true as when 2/6 is simplified by 2 gives 1/3.

Now, let's see an application of the above property in practice.

### Example 5

Two brothers are 8 and 11 years old respectively. After how many years the ratio of their ages will be 5:6?

### Solution 5

Let's denote the number of years to come until the condition above will be fulfilled by x. Thus, we can write

8 + x/11 + x = 5/6

Applying the cross product to remove fractions, we obtain

6 × (8 + x) = 5 × (11 + x)

Using the distributive property of addition, we obtain

6 × 8 + 6 × x = 5 × 11 + 5 × x
48 + 6x = 55 + 5x
6x - 5x = 55 - 48
x = 7

### Multiplying up and down one or both ratios in a proportion with the same number (they can be different in each side), provides a new proportion

In symbols,

IF a/b = c/d THEN a × m/b × m = c × n/d × n and a ÷ m/b ÷ m = c ÷ n/d ÷ n

The condition for this property to be true is that both m and n must not be zero.

In the previous examples, we dealt with this property silently but using division instead of multiplication because we wanted to make simplifications. However, since division is a multiplication by the inverse, it is obvious that this new property is also true. Let's consider an example to clarify this point.

### Example 6

Prove that the proportion 32:54 = 80:135 is true

Prove that 96:162 = 160:270 is also true

### Solution 6

We use the above property to confirm the ratio. We have

32/54 = 80/135

The first fraction is simplified by 2 while the second by 5. Hence, we obtain

32 ÷ 2/54 ÷ 2 = 80 ÷ 5/135 ÷ 5
16/27 = 16/27

It is easy to observe that the proportion

92/162 = 160/270

is obtained by multiplying up and down the first ratio by 3 and the second ratio by 2, i.e.

32 × 3/54 × 3 = 80 × 2/135 × 2
96/162 = 160/270

If we have the following proportion

a/b = c/d = e/f = k

then, we can express it as

a:c:e = b:d:f

### Example 7

The sides a, b and c of a triangle are in a ratio a : b : c = 5 : 9: 13 to each other and the perimeter (the sum of all side lengths) of the triangle is 108 cm. Calculate the length of each side.

### Solution 7

Applying the 6th property of proportions, we can write

a∶b∶c = 5∶9:13
a/5 = b/9 = c/13 = k

Thus,

a = 5k
b = 9k
c = 13k

Moreover, we have

a + b + c = 54 cm

Thus,

5k + 9k + 13k = 54 cm
27k = 54 cm
k = 2 cm

Therefore, the length of the three sides of the triangle are

a = 5 × 2 cm = 10 cm
b = 9 × 2 cm = 18 cm
c = 13 × 2 cm = 26 cm

Remark! We have silently used the 6th property of proportions too when explaining the definition of proportion in the previous tutorial.

## Calculating the n-th Proportional in a Set of Numbers

We already know that one of the forms of expressing a proportion is

a/b = c/d = e/f = ⋯

Usually we limit to the first four terms: a, b, c and d in most situations. Sometimes one of these terms is missing, so we have to find it. We call the missing term the n-th proportional of the number set given in the proportion. Thus, if a is missing, it is called the first proportional, when b is missing, we say it is the second proportional and so on.

### Example 8

What is the fourth proportional of 2, 3 and 4?

### Solution 8

We have a = 2, b = 3, c = 4 and d is missing. Thus, we have

a/b = c/d
2/3 = 4/d

Applying the cross product, we obtain

2 × d = 3 × 4
2 × d = 12
d = 12/2
d = 6

## The Geometric Mean

Before explaining what the geometric mean is, it is worth having a quick look at the concept of arithmetic mean, x ̅, which is a concept that represents the simple average of a set of values. Arithmetic mean of a set of values a1, a2, , an is calculated through the formula

x ̅ = a1 + a2 + ⋯ + an/n

For example, if four measurements taken give the values 20, 23, 19 and 22 respectively, their arithmetic mean is

x ̅= a1 + a2 + ⋯ + an/n
= 20 + 23 + 19 + 22/4
= 84/4
= 21

On the other hand, if we have a proportion that meets the condition

a/x = x/b

then, the number x is called the geometric mean of a and b.

From the above proportion, we can write

x × x = a × b
x2 = a × b
x = √a × b

For example, if

9/x = x/4

where a = 9 and b = 4, then, the geometric mean x of 9 and 4 is

x = √a × b
= √9 × 4
= √36
= 6

We can extend the meaning of geometric mean for more than two numbers. Thus, if we have a set of n numbers, from a1 to an, we obtain for their geometric mean x:

x = √n&a1 × a2 × × an

Geometric mean is a very important concept used in finance and banking. More specifically, the geometric mean is extremely useful when numbers in the series are not independent of each other or if numbers tend to make large fluctuations. In a set of measurements it helps determine the deflection from the true value of the quantity measured when the arithmetic mean (we will explain it below too) is not suitable to use. Let's consider an example which involves the calculation of geometric mean.

### Example 9

A rectangular container has the dimensions 25 cm × 8 cm × 5 cm. The container is melted down and the material is used to produce a cube with the same volume. What is the side length of the cube?

### Solution 9

From geometry, it is known that the volume of rectangular container (cuboid) is

V = a × b × c

where a, b and c are the dimensions of the cube (a = 25 cm, b = 8 cm and c = 5 cm). Therefore, the volume of container is

V = 25 cm × 8 cm × 5 cm
= 1000 cm3

The cube produced when the container is melted down must have the same volume (1000 cm3). Given that the volume of cube is x3, where x is the side length of the cube, we use the concept of geometric mean to calculate it. Hence,

x = ∛(25 cm × 8 cm × 5 cm)
= ∛(1000 cm3 )
= 10 cm

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