# Continued Proportion

The  proportion  definition page introduced the topic of ratio and proportion.

Such as how  2  ratios are classed as in proportion if they are equal to each other in value overall.

As an example:

The ratios   3 : 5   and   6 : 10   are in proportion.

3  is to  5”      as      “6  is to  10
or
3  out of  5”     is the same ratios as     “6  out of  10“.

In addition to this, there is also another concept to learn which is called  “continued proportion”.

## What is Continued Proportion?

Three or more quantities are considered to be in continued proportion if the ratio between successive quantities is the same.

For example, the ratio    3 : 6 : 12    is in continued proportion.

6  is double  3,  while  12  is double  6.

The ratio between successive quantities is the same.

3  is to  6” ,  as  “6  is to  12

### General Case:

So if we have  3  quantities    a , b , c,

these quantities will be in continued proportion if    a : b  =  b : c.

Using proportion notation this looks like.

a : b :: b : c

Or if one was writing ratios in fraction form,

a : b  =  b : c    can be written as    \bf{\frac{a}{b}} = \bf{\frac{b}{c}}.

What is particularly important to note and remember is that when two ratios are in continued proportion,

then   a × c  =  b × b.

So   a × c  =  b2.

When these conditions are present,  b  is known as the “mean proportional”.
More information on the mean proportional can be seen on the fractions cross multiplication page  here.

Examples

(1.1)

4 : 8 : 16     is in continued proportion.

4 × 16 = 64     ,     8 × 8 = 64

(1.2)

2 : 4 : 6     is NOT in continued proportion.

2 × 6 = 12     ,     4 × 4 = 16

1216

## Continued Proportion, Further Examples

(2.1)

Find the value of  t  if the ratio     2 : t : 32     is in continued proportion.

Solution

So we have    2 : t :: t : 32.

Now,    2 × 32  =  t 2.

=>    64  =  t 2       ,       √64  =  t

=>    t  =  8

(2.2)

Find the value of  s  if the ratio     3 : 9 : s     is in continued proportion.

Solution

We have    3 : 9 :: 9 : s.

3 × s  =  92.

=>    3s  =  92       ,       3s  =  81

=>    s  =  \bf{\frac{81}{3}}       ,       s  =  27

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