RATIO, PROPORTION AND UNITARY METHODS
Ratio : Ratio is defined as the relation between two quantities of similar type. If x and y are two numbers (y ¹ 0). Then
x is called the ratio of x and y. This is y
written as x : y. Ratio of 4 : 3 is written as 4/3. Similarly ratio of 2 and 3 is written as 2/3.
Points about ratio :
- If Greatest Common Factor (G.C.D.) of x and y is 1, then x : y is said to be in the simplet form.
- Ratio does not have any units i.e. it is without any dimensions.
- The two quantities being used in the ratio must be expressed in the same unit. For example length should be in metres, cms or mms. Time should be in hours, minutes or
- Ratio is
Incommensurable : If the ratio of two quantities can not be expressed as the ratio of two integers it is said to be incommesurable. As an example the ratio of the side of a square to its diagonal is
1 : Ö2
Properties :
- If both the quantities x and y of a ratio are multiplied or divided by the same quantity, the result does not
- Two or more ratios can be compared by making their denominator
- Convert each ratio having the same second term and compare the first
RATIOS OF DIFFERENT KINDS
- A ratio x : y is called a ratio of greater inequality or less inequality or the equality according as x > y, x < y or x =
- Compounded ratio of x : y and z : t is xz : yt. here x and y are called antecedents
z and t are called consequents x and t are called extremes
z and y are called medians or middle terms.
- Squared ratio of x : y is x2 : y2.
- Cubed ratio of x : y is x3 : y3.
- Under-root or square root ratio of x : y is Öx : Ö
- Cube root ratio of x : y is x1/3 : y1/3.
- Reciprocal ratio of x : y is 1: 1or y :
x y
Proportion : x,y,z and t are said to in propotion
if x
= z
and it is written
y t
as x : y : z : t, then x and t are called the extreme terms and y and z are called the means or the middle terms. We say, x,y,z, t are in proportional, the fourth term t, is called the fourth proportional to x,y,z. If x : y : : z : t, then we have the following three proportions.
- x : z : : y : t
- y : x : : t : z
- z : x : : t : y
Important points about proportions
- Product of middle terms = Product of extreme
- The mean proportional between x and y is Ö x C
DIFFERENT KINDS OF PROPORTIONS
Direct proportion : In two quantities if increase in one causes a corresponding increase in the other, it is called the direct proportion.
Inverse proportion : In two quantities if an increase in one causes a corre- sponding decrease in the other, it is called the inverse proportion.
Continued proportion : When the first is to the second as the second is to the third, as the third is to the fourth, and so on, are equal they are said to be in continued proportion i.e.
x =
y = z
= t =
u =………
y z t u m
the quantities x,y,z,t,u,m are said to be in continued proportion.
Theorems :
- Four quantities are in proportion if and only if, product of the extreme terms is equal to the product of middle terms and
- If three quantities are in continued proportion then the product of the extreme terms is equal to the square of the middle
- Fundamental theorem : If three quantities are in continued proportion then the ratio of first to third is the squared ratio of the first to
Proporties of proportions :
- convertendo : If x : y : : z : t, then x : (x – y) : : z : (z – t).
- Invertendo : If x=
z Þ
y = t .
y t y z
- Alternendo : If x= z Þ
x = y .
y t z t
- Componendo : If x=
z Þ x+y =
z+t .
y t y t
- Dividendo : If x= z
Þ x-y = z-t .
y t y t
6. Componendo and Dividendo : If
x = z Þ x+y = z+t .
y t x-y z-t
Unitary method : This method is used to solve the problems of work and men. For example, if a work can be done by 10 men how much work will be done by 25 men. To solve this problem we have to use unitary method as follows
10 men can do 1 work
1 men can do 1 work 25 men can do 25 work
10 10
= 2.5 work.
