Ratio is the relation which one quantity bears to another of the same kind. The ratio of two quantities a and b is the fraction a/b and we write it as a: b.
In the ratio a: b, we call a as the first term or antecedent and b, the second term or consequent.
Note: The multiplication or division of each term of a ratio by the same non- zero number does not affect the ratio.
Compound Ratio: – It is obtained by multiplying together the numerators for new numerator and denominators for new denominator.
Example 1. If the ratios are 4:3, 15:20, 2:6 and 3:5 find the compound ratio?
Example2. If we divide 4185 into two parts such that they are in ratio 7:2, then find the values of both the parts?
Sol 2. Let the actual variable be 7x and 2x.
So, the 1st part = 7 ×465=3255
The 2nd part = 2 ×465=930
Note:
The ratio of first , second and third quantities is given by
ac : bc : bd
If the ratio between first and second quantity is a:b and third and fourth is c:d .
Similarly, the ratio of first, second, third and fourth quantities is given by
ace : bce : bde : bdf
If the ratio between first and second quantity is a: b and third and fourth is c:d.
Proportion
Introduction:- Four quantities are said to be proportional if the two ratios are equal i.e. the A, B, C and D are proportion. It is denoted by “::” it is written as A : B : C : D where A and D are extremes and B and C are called means . Product of the extreme = Product of the means
Direct proportion: – The two given quantities are so related that if one quantity increases (or decreases) then the other quantity also increases (or decreases).
Example 1. If 5 pens cost Rs 10 then 15 pen cost?
Sol 1. It is seen that if number of pens increases then cost also increases. So,
5 pens: 15 pens:: Rs 10 : required cost
Inverse proportion: – The two given quantities are so related that if one quantity increases (or decreases) then the other quantity also decreases (or increases).
Example 2.If 10 men can do a work in 20 days then in how many days 20 men can do that work?
Sol 2. Here if men increase then days should decrease, so this is a case of inverse proportion, so
Rule of three: It Is the method of finding 4th term of a proportion if all the other three are given, if ratio is a:b :: c:d then ,
ALLIGATION
Introduction:-
The word allegation means linking. It is used to find:
1. The proportion in which the ingredients of given price are mixed to produce a new mixture at a given price.
2. The mean or average value of mixture when the price of the two or more ingredients and the proportion in which they are mixed are given.
Mathematical Formula:
For two ingredient:-
Example 1: If the rice at Rs 3.20 per kg and the rice at Rs 3.50 per kg be mixed then what should be their proportion so that the new mixture be worth Rs 3.35 per kg ?
Sol 1: CP of 1 kg of cheaper rice CP of 1 kg of dearer rice
Hence they must be mixed in equal proportion i.e. 1:1
Example 2: Find out the ratio of new mixture so that it will cost Rs 1.40 per kg from the given three kinds of rice costing Rs 1.20, Rs 1.45 and Rs 1.74?
Sol 2: 1st rice cost = 120, 2nd rice cost = 145 and 3rd rice cost = 174 paisa.
From the above rule: we have,
Therefore, three rice must be mixed in 39: 20: 20 ratios to have a new mixture of rice.
Questions
Level-I
..
A and B together have Rs. 1210. If of A’s amount is equal to of B’s amount, how much amount does B have?
Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?
The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?
Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit’s salary?
The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries?
the A, B, C and D are proportion. It is denoted by “::” it is written as A : B : C : D where A and D are extremes and B and C are called means .
of A’s amount is equal to 
of B’s amount, how much amount does B have?
: 
: 
, then the first part is:
A =
B
A =
A : B = 3 : 2.
B’s share = Rs.
Ratio of first two numbers =
B’s share = Rs. 2
7
The required ratio = 7
14
2 : 3 : 4.
litres = 40 litres.
20 + 
Quantity of water to be added = 60 litres.
The required ratio =
57(2
6
3
A : B : C = 2 : 3 :
B =
: 
: 
= 6 : 8 : 9.
1
New ratio
A : B = 5 : 3.
23(3
9
The smaller number = (3 x 11) = 33.
