CLASS-8
CONTINUED RATIO

CONTINUED RATIO

If three quantities are such that the ratio of the first two is a : b and the ratio of the last two is b : c, then the three quantities are said to be in the ‘Continued Ratio’ = a : b : c. here we can notice that in 1st ratio b is consequent and in 2nd ratio b is antecedent. It would be useful to remember that if the ratio of two quantities be a : b then the two quantities can be expressed as ak and bk. If the continued ratio of three quantities be a : b : c then the three quantities can be expressed as ak, bk and ck

There are some example are given below for your better understanding –

Example.1)  If  a : b = 3 : 5 and  b : c = 5 : 7, then find a : b : c = ?

Ans.)  If  a : b = 3 : 5 and  b : c = 5 : 7,

         then a : b : c = 3 : 5 : 7



Example.2)  x : y = 3 : 5 and  y : z = 15 : 22, then find x : y : z = ?

Ans.)  As per the given condition,  in 1st ratio x : y = 3 : 5 , here y = 5

but in 2nd ratio y : z = 15 : 22, y = 15

So,  15 ÷  5 = 3 ,

Now we have to multiply by 3 with 1st  ratio  x : y = 3 : 5

                           =  ( 3 X 3 ) : ( 5 X 3 ) =  9 : 15

And we have 2nd ratio which is  y : z = 15 : 22

Now, in above both the ratio now y = 15

So, now x : y : z = 9 : 15 : 22



Example.3) x : y = 24 : 21 and  y : z = 7 : 5, then find x : y : z = ?

Ans.) As per the given condition,  in 1st ratio x : y = 24 : 21 , here y = 21

but in 2nd ratio y : z = 7 : 5, y = 7

So,  21 ÷  7 = 3 ,

Now we have to multiply by 3 with 2nd  ratio  x : y = 7 : 5

                       =  ( 7 X 3 ) : ( 5 X 3 ) =  21 : 15

And we have 1st ratio which is  x : y = 24 : 21

Now, in above both the ratio now y = 21

So, now x : y : z =  24 : 21 : 15

                    =  8 : 7 : 5  ( via simplification )

Another way

As per the given condition, in 1st ratio x : y = 24 : 21, here y = 21

but in 2nd ratio y : z = 7 : 5, y = 7

If we simplify 1st ratio then we find x : y = 24 : 21 = 8 : 7

Now, we find in both 1st & 2nd ratio  y = 7

So, now we have x : y = 8 : 7 and y : z = 7 : 5

Now we can find that,  x : y : z =  8 : 7 : 5

 

Example.4) Arrange the given ratio in ascending order –

a) 4 : 5, 6 : 7 ,  b)  5 : 8, 5 : 11,   c) 7 : 11, 9 : 11

a)  4 : 5, 6 : 7   

Ans.) Since the ratio have different consequents and antecedents, we will change them into like fractions.

The LCM of the consequents 5 & 7  = 35

                   4          4 X ( LCM ÷ 5 )         4 X 7    

Now, 4 : 5 = ------- = ----------------- = ----------

                   5          4 X ( LCM ÷ 5 )         5 X 7


                                                        28

                              = --------- =  28 : 35

                                     35

 

                   6          6 X ( LCM ÷ 7 )         6 X 5       

Now, 6 : 7 = ------- = ----------------- = ---------

                   7          7 X ( LCM ÷ 7 )         7 X 5 

                          30

           = -------- =  30 : 35

                 35

 

In above-obtained ratio we can observe that, 30 > 28 where consequent for the both ratio is 35

So, we can say that   30 : 35  > 28 : 35 thus  6 : 7 >  4 : 5  (Ans.)

 

b)  9 : 8, 7 : 11

Ans.) Since the ratio have different consequents and antecedents, we will change them into like fractions.

The LCM of the consequents 8 & 11  = 88

                   9          9 X ( LCM ÷ 8 )        9 X 11      

Now, 9 : 8 = ------- = ----------------- = ----------

                   8          8 X ( LCM ÷ 8 )        8 X 11


                               99

              = -------- =  99 : 88

                    88

                                  

 

                    7          7 X ( LCM ÷ 11 )        7 X 8   

Now, 7 : 11 = ------- = ----------------- = ---------- 

                              11        11 X ( LCM ÷ 11 )            11 X 8


                                                    56 

                           = --------- =  56 : 88

                                  88

 

In above-obtained ratio we can observe that, 99 > 56 where consequent for both ratio is 88.

So, we can say that 99 : 88 > 56 : 88 thus  9 : 8 > 7 : 11  (Ans.)

 

 c)  7 : 11, 9 : 11

Ans.)  The ratio has the same consequences. So, the ratio with the greater antecedent will be greater. Thus,  9 : 11 > 7 : 11    (Ans.)


 

Example.5) There are some ratio has been given, such as 3 : 5, 4 : 7 and 3 : 2. Which of the following ratio is the biggest and which is the smallest ?

Ans.) Since the ratio have different consequents and antecedents, we will change them into like fractions.

The LCM of the consequents  5, 7, 2  = 70

                   3            3 X ( LCM ÷ 5 )        3 X 14   

Now, 3 : 5 = -------- = ----------------- = ---------- 

                             5            5 X ( LCM ÷ 5 )        5 X 14 


                                                   42

                           = --------- =  42 : 70

                                  70

                              

 

                    4            4 X ( LCM ÷ 7 )      4 X 10    

Now, 4 : 7 = --------- = ---------------- = --------- 

                    7            7 X ( LCM ÷ 7 )      7 X 10


                                40

               = -------- =  40 : 70

                     70


                                   

                   3           3 X ( LCM ÷ 2 )       3 X 35       

Now, 3 : 2 = -------- = ---------------- = ---------- 

                   2            2 X ( LCM ÷ 2 )      2 X 35


                       105

                  = -------- =  105 : 70

                        70                                      

In the above-obtained ratio we can observe that, 105 > 42 > 40 where consequent for the both ratio is 70.

So, we can say that 105 : 70 > 42 : 70 > 40 : 70 thus 3 : 2 >  3 : 5 > 4 : 7

So, the smallest ratio is = 4 : 7 and the biggest ratio is = 3 : 2  (Ans.)