CONTINUED PROPORTION
When three quantities x, y, z are said to be in Continued Proportion or Continued Ratio then the equation would be x : y = y : z
The term y is to be considered such as Mean Proportional between x & z while the term z is called Third Proportional to x & y.
x y
In this case , --------- = --------- , xz = y²
y z
if y² = xz , then y = √xz also x = y²/z and z =y²/x
Example.1) Find the continued proportion where the numbers are 8, 16, 32.
Ans.) This is to be remembered, In continued proportion, all the quantities must be of the same kind and in the same unit
Yes, the above number belongs to continued proportion, because
8 1
8 : 16 = ---------- = ----------
16 2
16 1
16 : 32 = ---------- = ----------- (Ans.)
32 2
Example.2) If in continued proportion 4, y², 16 then find the y
As per the given condition, the numbers are 4, y², 16
We let x =4, z = 16 , and we have to find y² = ? ,
As per the continued proportion rules x : y = y : z
x/y = y/z , So y² = xz
where, y² = 4 X 16 , y = √16 = 4
y² = 64
y = √64 = 8
So, the continued proportion would be 4, 8, 16. (Ans.)
Example.3) Find the third proportional where the other numbers are 5,10
We let x =4, y = 16 , and we have to find z = ? ,
As per the continued proportion rules x : y = y : z
x/y = y/z , so y² = xz
x y
=> ---------- = ----------
y z
5 10
=> --------- = ----------
10 z
=> 5 z = 10 X 10
10 X 10
So, z = --------------- = 20
5
So, the continued proportion would be 5, 10, 20. (Ans.)
4) Find the mean proportion where the other number is 13, 52
As per the given condition, the numbers are 13, y², 52
We let x =13, z = 52 , and we have to find y² = ? ,
As per the continued proportion rules x : y = y : z
x/y = y/z , so y² = xz
where, y² = 13 X 52 ,
y² = 13 x 13 x 4
y = √13 X 13 X 2 X 2
= 13 X 2 = 26 (Ans.)