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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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