Ratio
Introduction:
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 1^{st} part = 7 ×465=3255
The 2^{nd} 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
10 men: 20 men :: required days : 20 days
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:
 The proportion in which the ingredients of given price are mixed to produce a new mixture at a given price.
 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: 1^{st} rice cost = 120, 2^{nd} rice cost = 145 and 3^{rd} 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
LevelI
..  1. 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?  

2.  Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:  

3.  A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B’s share?  

4.  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?  

5.  In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quanity of water to be further added is:  
 
6.  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?  

7.  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?  

8.  If 0.75 : x :: 5 : 8, then x is equal to:  

9.  The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to the third is 5 : 8, then the second number is:  

10 .If Rs. 782 be divided into three parts, proportional to : : , then the first part is:  

Answers
 Answer:Option B
Explanation:
4  A  =  2  B  
15  5 
A =  2  x  15  B  
5  4 
A =  3  B 
2 
A  =  3  
B  2 
A : B = 3 : 2.
B’s share = Rs.  1210 x  2  = Rs. 484.  
5 
2 .Answer: Option C
Explanation:
Let the third number be x.
Then, first number = 120% of x =  120x  =  6x 
100  5 
Second number = 150% of x =  150x  =  3x 
100  2 
Ratio of first two numbers =  6x  :  3x  = 12x : 15x = 4 : 5.  
3 .Answer: Option C
Explanation:
Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x – 3x = 1000
x = 1000.
B’s share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.
4 .Answer: Option A
Explanation:
Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.
Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).
140  x 5x  ,  150  x 7x  and  175  x 8x  
100  100  100  
7x,  21x  and 14x. 
2  
The required ratio = 7x :  21x  : 14x 
2 
14x : 21x : 28x
2 : 3 : 4.
5 .Answer: Option D
Explanation:
Quantity of milk =  60 x  2  litres = 40 litres.  
3 
Quantity of water in it = (60 40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
Then, milk : water =  40  .  
20 + x 
Now,  40  =  1  
20 + x  2 
20 + x = 80
x = 60.
Quantity of water to be added = 60 litres.
6 .Answer: Option C
Explanation:
Originally, let the number of boys and girls in the college be 7x and 8x respectively.
Their increased number is (120% of 7x) and (110% of 8x).
120  x 7x  and  110  x 8x  
100  100  
42x  and  44x  
5  5  
The required ratio =  42x  :  44x  = 21 : 22  
7 .Answer: Option D
Explanation:
Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.
Then,  2x + 4000  =  40 
3x + 4000  57  
57(2x + 4000) = 40(3x + 4000)
6x = 68,000
3x = 34,000
Sumit’s present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.
8 .Answer: Option B
Explanation:
(x x 5) = (0.75 x 8) x =  6  = 1.20  
5  
9 .Answer: Option B
Explanation:
Let the three parts be A, B, C. Then,
A : B = 2 : 3 and B : C = 5 : 8 =  5 x  3  :  8 x  3  = 3 :  24  
5  5  5  
A : B : C = 2 : 3 :  24  = 10 : 15 : 24 
5  
B =  98 x  15  = 30.  
49 
10 .Answer: Option D
Explanation:
Given ratio = : : = 6 : 8 : 9.
1^{st} part = Rs.  782 x  6  = Rs. 204  
LevelII
11.  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?  
Answer: Option C
Explanation: Let A = 2k, B = 3k and C = 5k.
 
12.  If 40% of a number is equal to twothird of another number, what is the ratio of first number to the second number?  
Answer: Option C
Explanation:
A : B = 5 : 3. 
13.  The fourth proportional to 5, 8, 15 is:  
Answer: Option B
Explanation: Let the fourth proportional to 5, 8, 15 be x. Then, 5 : 8 : 15 : x 5x = (8 x 15)

14. 
Two number are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is:  
Answer: Option B
Explanation: Let the numbers be 3x and 5x.
23(3x – 9) = 12(5x – 9) 9x = 99 x = 11. The smaller number = (3 x 11) = 33. 
15. 
In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 1 : 2 : 3. If there is Rs. 30 in all, how many 5 p coins are there?  
Answer: Option C
Explanation: Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively.
Hence, the number of 5 p coins = (3 x 50) = 150. 