Continued Proportion

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

Such as how &nbsp2&nbsp ratios are classed as in proportion if they are equal to each other in value overall.

As an example:

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


“3&nbsp is to &nbsp5“&nbsp &nbsp &nbsp as &nbsp &nbsp &nbsp”6&nbsp is to &nbsp10”

or

“3&nbsp out of &nbsp5” &nbsp &nbsp is the same ratios as &nbsp &nbsp “6&nbsp out of &nbsp10“.

In addition to this, there is also another concept to learn which is called &nbsp”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 &nbsp&nbsp 3 : 6 : 12 &nbsp&nbsp is in continued proportion.


6&nbsp is double &nbsp3,&nbsp while&nbsp 12&nbsp is double &nbsp6.


The ratio between successive quantities is the same.

“3&nbsp is to &nbsp6” ,&nbsp as&nbsp “6&nbsp is to &nbsp12”

General Case:

So if we have &nbsp3&nbsp quantities &nbsp&nbsp a , b , c,

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


Using proportion notation this looks like &nbsp => &nbsp&nbsp a : b :: b : c



Or if one was writing ratios in fraction form,

a : b &nbsp=&nbsp b : c &nbsp&nbsp can be written as &nbsp&nbsp \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 &nbsp a × c &nbsp=&nbsp b × b.


So &nbsp a × c &nbsp=&nbsp b².

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

Examples &nbsp &nbsp


(1.1)&nbsp

4 : 8 : 16 &nbsp &nbsp is in continued proportion.

4 × 16 = 64 &nbsp &nbsp , &nbsp &nbsp 8 × 8 = 64



(1.2)&nbsp

2 : 4 : 6 &nbsp &nbsp is NOT in continued proportion.

2 × 6 = 12 &nbsp &nbsp , &nbsp &nbsp 4 × 4 = 16

12 ≠ 16

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Continued Proportion, Further Examples

(2.1)&nbsp

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

Solution&nbsp &nbsp

So we have &nbsp&nbsp 2 : t :: t : 32.

Now, &nbsp&nbsp 2 × 32 &nbsp=&nbsp t ².

=> &nbsp&nbsp 64 &nbsp=&nbsp t ² &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp √64 &nbsp=&nbsp t

=> &nbsp&nbsp t &nbsp=&nbsp 8





(2.2)&nbsp

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

Solution&nbsp &nbsp

We have &nbsp&nbsp 3 : 9 :: 9 : s.

3 × s &nbsp=&nbsp 9².

=> &nbsp&nbsp 3s &nbsp=&nbsp 9² &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 3s &nbsp=&nbsp 81

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

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