Problems on Ratio and Proportion

  1. 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 or consequent.
    Eg. The ratio 5 : 9 represents5with antecedent = 5, consequent = 9.
    9
    Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
    Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
  2. 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).
  3. 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.
  4. Comparison of Ratios:
    We say that (a : b) > (c : d)   ==>  a>c.
    bd
    Compounded Ratio:
    The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
  5. Duplicate Ratios:
    Duplicate ratio of (a : b) is (a2 : b2).
    Sub-duplicate ratio of (a : b) is (a : b).
    Triplicate ratio of (a : b) is (a3 : b3).
    Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).
    Ifa=c, thena + b=c + d.     [componendo and dividendo]
    bda - bc - d
  6. 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?
A.Rs. 460
B.Rs. 484
C.Rs. 550
D.Rs. 664
Answer: Option B
Explanation:
4A=2B
155
 A =
2x15B
54
 A =3B
2

A=3
B2
 A : B = 3 : 2.
 B's share = Rs.
1210 x2
= Rs. 484.
5
2. 
Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:
A.2 : 5
B.3 : 5
C.4 : 5
D.6 : 7
Answer: Option C
Explanation:
Let the third number be x.
Then, first number = 120% of x =120x=6x
1005
Second number = 150% of x =150x=3x
1002
 Ratio of first two numbers =
6x:3x
= 12x : 15x = 4 : 5.
52
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?
A.Rs. 500
B.Rs. 1500
C.Rs. 2000
D.None of these
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. 
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?
A.2 : 3 : 4
B.6 : 7 : 8
C.6 : 8 : 9
D.None of these
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).


140x 5x
,
150x 7x
and
175x 8x
100100100
 7x,21xand 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:
A.20 litres
B.30 litres
C.40 litres
D.60 litres
Answer: Option D
Explanation:
Quantity of milk =
60 x2litres = 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 + x2
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?
A.8 : 9
B.17 : 18
C.21 : 22
D.Cannot be determined
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).


120x 7xand
110x 8x
100100

42xand44x
55
 The required ratio =
42x:44x
= 21 : 22.
55
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?
A.Rs. 17,000
B.Rs. 20,000
C.Rs. 25,500
D.Rs. 38,000
Answer: Option D
Explanation:
Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.
Then,2x + 4000=40
3x + 400057
 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:
A.1.12
B.1.2
C.1.25
D.1.30
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:
A.20
B.30
C.48
D.58
Answer: Option B
Explanation:
Let the three parts be A, B, C. Then,
A : B = 2 : 3 and B : C = 5 : 8 =
5 x3
:
8 x3
= 3 :24
555
 A : B : C = 2 : 3 :24= 10 : 15 : 24
5
 B =
98 x15
= 30.
49
10. 
If Rs. 782 be divided into three parts, proportional to  :  : , then the first part is:
A.Rs. 182
B.Rs. 190
C.Rs. 196
D.Rs. 204
Answer: Option D
Explanation:
Given ratio =  :  :  = 6 : 8 : 9.
 1st part = Rs.
782 x6
= 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?
A.3 : 3 : 10
B.10 : 11 : 20
C.23 : 33 : 60
D.Cannot be determined
Answer: Option C
Explanation:
Let A = 2k, B = 3k and C = 5k.
A's new salary =115of 2k =
115x 2k
=23k
10010010
B's new salary =110of 3k =
110x 3k
=33k
10010010
C's new salary =120of 5k =
120x 5k
= 6k
100100
 New ratio
23k:33k: 6k
= 23 : 33 : 60
1010
12. 
If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number?
A.2 : 5
B.3 : 7
C.5 : 3
D.7 : 3
Answer: Option C
Explanation:
Let 40% of A =2B
3
Then,40A=2B
1003

2A=2B
53

A=2x5=5
B323
 A : B = 5 : 3.
13. 
The fourth proportional to 5, 8, 15 is:
A.18
B.24
C.19
D.20
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:
A.27
B.33
C.49
D.55
Answer: Option B
Explanation:
Let the numbers be 3x and 5x.
Then,3x - 9=12
5x - 923
 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?
A.50
B.100
C.150
D.200
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
100100100100

60x= 30    x =30 x 100= 50.
10060
Hence, the number of 5 p coins = (3 x 50) = 150.

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