Properties of Proportion. Geometric Mean - Revision Notes

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4.4Properties of Proportion. Geometric Mean


In these revision notes for Properties of Proportion. Geometric Mean, we cover the following key points:

  • 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?

Properties of Proportion. Geometric Mean Revision Notes

Proportions have a number of useful properties that help calculate easier any missing quantity involved in them. They are as follows:

Swapping the means in a proportion still provides a proportion

In symbols,

IF a/b = c/d THEN a/c = b/d

Swapping the extremes in a proportion still provides a proportion

In symbols,

IF a/b = c/d THEN d/c = b/a

Inverting both ratios in a proportion still provides a proportion

In symbols,

IF a/b = c/d THEN b/a = d/c

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

In symbols,

IF a/b = c/d THEN a ± b/b = c ± d/d

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.

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

Sometimes one of terms in a proportion is missing; in this case, we have to calculate it. We call the missing term the n-th proportional of the number set given in the proportion. It is calculated by using any of the properties of proportion.

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. It is calculated by the formula

x = √a × b

If the proportion is made by more than two fractions, then the geometric mean is

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.

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