Ratio and Proportion
Ratio and Proportion
Important Formulas
 Ratio:The ratio of two quantities a and b in the same units, is the fraction 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 orconsequent.
Eg. The ratio 5 : 9 represents 5 with antecedent = 5, consequent = 9. 9
Rule: The multiplication or division of each term of a ratio by the same nonzero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
 Proportion:The equality of two ratios is called proportion.
If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d (b x c) = (a x d).
 Fourth Proportional:If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional:
Mean proportional between a and b is ab.
 Comparison of Ratios:
We say that (a : b) > (c : d) a > c . b d
Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
 Duplicate Ratios:Duplicate ratio of (a : b) is (a^{2} : b^{2}).
Subduplicate ratio of (a : b) is (a : b).
Triplicate ratio of (a : b) is (a^{3} : b^{3}).
Subtriplicate ratio of (a : b) is (a^{1/3} : b^{1/3}).
If a = c , then a + b = c + d . [componendo and dividendo] b d a – b c – d
 Variations:We say that x is directly proportional to y, if x = ky for some constant k and we write,x y.
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x 1 . y
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?  

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

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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.  
5  2 
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?  

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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.
View Answer
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.
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:  

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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.
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?  

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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.  
5  5 
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?  

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

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Answer: Option B
Explanation:
(x x 5) = (0.75 x 8) x =  6  = 1.20  
5 
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:  

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

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Answer: Option D
Explanation:
Given ratio = : : = 6 : 8 : 9.
1^{st} part = Rs.  782 x  6  = Rs. 204  
23 
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?  

View Answer
Answer: Option C
Explanation:
Let A = 2k, B = 3k and C = 5k.
A’s new salary =  115  of 2k =  115  x 2k  =  23k  
100  100  10 
B’s new salary =  110  of 3k =  110  x 3k  =  33k  
100  100  10 
C’s new salary =  120  of 5k =  120  x 5k  = 6k  
100  100 
New ratio  23k  :  33k  : 6k  = 23 : 33 : 60  
10  10 
12.
If 40% of a number is equal to twothird of another number, what is the ratio of first number to the second number?  

View Answer
Answer: Option C
Explanation:
Let 40% of A =  2  B 
3 
Then,  40A  =  2B 
100  3 
2A  =  2B  
5  3 
A  =  2  x  5  =  5  
B  3  2  3 
A : B = 5 : 3.
13.
The fourth proportional to 5, 8, 15 is:  

View Answer
Answer: Option B
Explanation:
Let the fourth proportional to 5, 8, 15 be x.
Then, 5 : 8 : 15 : x
5x = (8 x 15)
x =  (8 x 15)  = 24. 
5 
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:  

View Answer
Answer: Option B
Explanation:
Let the numbers be 3x and 5x.
Then,  3x – 9  =  12 
5x – 9  23 
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?  

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Answer: Option C
Explanation:
Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively.
Then, sum of their values = Rs.  25x  +  10 x 2x  +  5 x 3x  = Rs.  60x  
100  100  100  100 
60x  = 30  x =  30 x 100  = 50.  
100  60 
Hence, the number of 5 p coins = (3 x 50) = 150.