:: Ratio and Proportion
1. In a college, the ratio of the number of boys to girls is 8 : 5. If there are 160 girls, the total number of students in
the college is:. [Central Excise ]
100
260
250
416
Answer & Explanation
416
Answer:
Explanation: Let the number of boys and girls be 8x and 5x.
Total number of students = 13x = 13 x 32 = 416.
2. The ratio of two numbers is 3 : 4 and their sum is 420. The greater of the two numbers is :. [Railways 1991]
175
240
200
315
Answer & Explanation
240
Answer:
Explanation: Required number = (420 * (4/7)) = 240.
3. If a+b : b+c : c+a = 6 : 7 : 8 and a + b + c = 14, then the value of c is : . [C.B.I. 1993]
6
8
7
14
Answer & Explanation
14
Answer:
Explanation: (a/3) = (b/4) = (c/7) then a = 3k, b = 4k, c = 7k
a+b+c/c = 3k+4k+7k/7k = 14k/7k = 2
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4. If 15% of x is the same as 20% of y, then x : y is :. [C.B.I. 1990]
3:4
17 : 16
4:3
16 : 17
Answer & Explanation
4:3
Answer:
Explanation: 15% of x = 2O% of y ⇒ 15x/100 = 20y/100 ⇒ x/y = 4/3
5. A sum of Rs. 427 is to be divided among A, B and C such that 3 times A’s share, 4 tunes B’s share and 7 times
C’s share are all equal. The share of C is : (Assistant Grade 1994)
Rs. 84 .
Rs. 196
Rs. 147
Rs. 240
Answer & Explanation
Rs. 84 .
Answer:
Explanation: 3A = 4B = 7C = k,Then A = k/3, B = k/4 and C= k/7.
A : B : C = k/3 : k/4 : k/7 = 28:21 :12.
Cs share = Rs. [427 x (12/61)] = Rs. 84
6. An alloy is to contain copper and zinc in the ratio 9 :4. The zinc required (in kg) to be melted with 24 kg of
copper, is 7
10 and 2/3
9 and 2/3
10 and 1/3
9
Answer & Explanation
10 and 1/3
Answer:
Explanation: 9:4: 24:x ⇒ 9x = 4 * 24 ⇒ x = (4*24)/9 = 32/3 Kg. hence `0 and 1/3
7. Rs. 730 were divided among A, B, C in such a way that if A gets Rs. 3, then B gets Rs. 4 and if B gets Rs. 3.50
then C gets Rs. 3. The share of B exceeds that of C by : . [Central Excise ]
Rs. 30
Rs. 70
Rs. 40
Rs. 210
Answer & Explanation
Rs. 40
Answer:
Explanation: A:B = 3:4 and B:C = 7/2:3 = (8/7)*(7/2)*(8/7)*3 = 4:(24/7)
A : B : C = 3 :4: 24/7 = 21 : 28 : 24.
Bs share = Rs. [730 *(28/73)]= Rs. 280.
C’s share = Rs. [730 * (24/73)] = Rs. 240.
Difference of their shares = 40
8. If 0.4 : 1.4 :: 1.4 : x, the value of x is :. [Teachers Exam 1]
49
0.49
4.9
0.4
Answer & Explanation
4.9
Answer:
Explanation: 0.4 * x = 1.4 * 1.4 ⇒ x = (1.4*1.4)/0.4 = 4.9
9. What number should be subtracted from both the terms of the ratio 15 : 19 so as to make it as 3 : 4 ? (Assistant
Grade 1994)
3
6
5
9
Answer & Explanation
3
Answer:
Explanation: Let x be subtracted. Then,
(15 - x) / (19 - x) = 3/4 ⇒ 4(15 - x) = 3(19 - x) x = 3
10. If p : q = 3 : 4 and q : r= 8 : 9, then p : r is :. [Assistant Grade]
1: 3
2:3
3:2
1: 2
Answer & Explanation
2:3
Answer:
Explanation: p/r = (p/q) * (q/r) = (3/4) * (8/9) = 2/3 so p : q = 2:3
11. The salaries of A, B, and C are in the ratio of 1 : 2 : 3. The salary of B and C together is Rs. 6000. By what
percent is the salary of C more than that of A? (Bank P.O. 1992)
100%
300%
200%
600%
Answer & Explanation
600%
Answer:
Explanation: Let the salaries of A, B, C hex, 2x and 3x respectively.
Then,2x + 3x = 6000 = x = 1200. As salary = Rs. 1200, Bs salary = Rs. 2400,
and Cs salary Rs. 3600.
Excess of Cs salary over As=[(2400/1200)x100] = 200%.
12. An amount of money is to be distributed among F, Q and R in the ratio 3 : 5 : 7. If Qs share is Rs. 1500, what is
the difference between Ps and Rs shares? . [Bank P.O. 1994]
Rs. 1200
Rs. 1600
Rs, 1500
Rs. 1900
Answer & Explanation
Rs. 1200
Answer:
Explanation: Let P = 3x, Q = 5x and R = 7x.
Then, 5x = 1500 ⇒ x = 300. P=900,Q=1500 and R = 21OO.
Hence, (R - p) = (2100 - 900) = 1200
13. What number should be added to each of the numbers 8, 21, 13 and 31 so that the resulting numbers, in this
order form a proportion?. [U.D.C. 1993]
2
5
3
7
Answer & Explanation
5
Answer:
Explanation: (8+x)/(21+x) = (13+x)/(31+x)
Then, (8 + x)(31 + x) = (13 + x)(21 + x)
or39x + 248 = 34x + 273 or 5x=25 or x = 5.
14. The ratio of income of A to that of B is 5 : 4 and the expenditure of A to that of B is 3: 2. If at the end of the year,
each saves Rs, 800, the income of A is : . [Assistant Grade]
Rs. 1600
Rs. 2000
Rs. 1800
Rs. 2200
Answer & Explanation
Rs. 2000
Answer:
Explanation: Let the incone of A and B be 5x and 4x and. the expenditures of A and B be 3y
and 2y. Then, 5x—3y = 800 and 4x— 2y= 800.
On solving we get : x = 400. As income = 5x = Rs. 2000.
15. A profit of Rs. 30000 is to be distributed among A, B, C in the proportion 3 : 5 : 7. What will be the difference
between B’s and C’s shares?. [Bank P.O. 1993]
Rs. 2000
Rs. 10000
Rs. 4000
Rs. 14000
Answer & Explanation
Rs. 4000
Answer:
Explanation: Bs share = Rs. 30000 *(5/15) = Rs.10000.
C’s share = Rs. 30000 * (7/15) = Rs.14000,
Difference in Bs and Cs shares = Rs.4000.
16. Rs. 120 are divided among A, B, C such that A’s share is Rs. 20 more than B’s and Rs. 20 less than C’s. What is
B’s share.. [Railways 1991]
Rs. 10
Rs. 20
Rs. 15
Rs. 25
Answer & Explanation
Rs. 20
Answer:
Explanation: Let C = x. Then A = (x—20) and B = (x—40).
x + x - 20 + x - 40 = 120 Or x=60.
A:B:C = 40:20:60 = 2:1 :3.
Bs share = Rs. 120*(1/6) = Rs. 20.
17. If 7 : x = 17.5 : 22.5 , then the value of x is:. [Assistant grade]
9
6
7.5
5.5
Answer & Explanation
9
Answer:
Explanation: 7*22.5 = x*17.5 ⇒ x = 7 * 22.5/17.5 ⇒ x = 9.
18. X, Y and Z share a sum of money in the ratio 7 : 8 : 16. If Z receives Rs. 27 more than X, then the total money
shared was:. [Central Excise ]
Rs. 48
Rs. 279
Rs. 93
Rs. 558
Answer & Explanation
Rs. 93
Answer:
Explanation: Let X = 7x, Y = 8x & Z = 16x. Then, total money = 31x.
Now, Z - X = 27 so 16x—7x = 27 that is why x = 3.
Total money 31*x = Rs.93.
19. A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of
the speed of the dog to that of the hare is :. [U.D.C. 1993]
8:5
8:7
9:5
9:7
Answer & Explanation
9:5
Answer:
Explanation: Dog : Hare = (3*3) leaps of hare : 5 leaps of hare = 9 : 5.
20. The least whole number which when subtracted from both the terms of the ratio 6 : 7 to give a ratio less than
16 : 21, is. [Central Excise ]
2
4
3.
6
Answer & Explanation
3.
Answer:
Explanation: Let x is subtracted. Then, ((6 - x)/(7 - x)) < 16 / 21
21(6—x) < 16(7—x) ⇒ 5x > 14 = x > 2.8.
Least such number is 3.
21. The compounded ratio of (2 : 3), (6: 11) and (11 :2) is. [Assistant Grade]
1: 2
11: 24
2: 1
36: 121
Answer & Explanation
2: 1
Answer:
Explanation: required ratio = (2/3) * () * (6/11) * (11/2) = 2/1
22. A sum of money is to the divided among F, Q andR in the ratio of 2 : 3 : 5. If the total share of P andR together
is Rs 400 more than that of Q, what is R’s share in it ?. [Bank P.O. 1991]
Rs. 400
Rs. 600
Rs. 500
Rs. 750
Answer & Explanation
Rs. 500
Answer:
Explanation: Let P = 2x , Q = 3x and R=5x. Now P+R-Q = 400 2x+5x-3x = 400 hence x
=1OO R = 5x = 500.
23. A certain amount was divided between Salim and Rahim in the ratio of 4 : 3. If Rahim’s share was Rs. 2400, the
total amount was. [Bank P.O. 1988]
Rs. 5600
Rs. 9600
Rs. 3200
Rs. 16800
Answer & Explanation
Rs. 5600
Answer:
Explanation: Let S = 4x and R = 3x. Total amount = 7x.
Then, 3x = 2400 so x= 800.
Total amount = 7x = Rs. 5600
24. If three numbers in the ratio 3 : 2: 5 be such that the sum of their squares is 1862, the middle number will
be:. [Teacher’s Exam.]
7
21
14
35
Answer & Explanation
14
Answer:
Explanation: Let the numbers be 3x, 2x and 5x. Then,
9x + 4x + 25x =1862 ⇒ 38x = 1862 ⇒ x = 49 ⇒ x = 7.
middle number = 2x = 14.
25. Pencils, Pens and Exercise books in a shop are in the ratio of 10: 2 : 3. If there are 120 pencils, the number of
exercise books in the shop is:. [Assistant Grade]
36
72
48
400
Answer & Explanation
36
Answer:
Explanation: Let Pencils = 10x, Pens = 2x & Exercise books = 3x. Now, 10x = 120 hence x =
12.
Numberr of exercise books = 3x = 36.
1.
A and B together have Rs. 1210. If
amount does B have?
of A's amount is equal to
A.
Rs. 460
B.
Rs. 484
C.
Rs. 550
D.
Rs. 664
of B's amount, how much
Answer & Explanation
Answer: Option B
Explanation:
4
15
2
A =
5
2
A=
A=
A
B
=
5
3
2
3
B
15
x
4
B
B
2
A : B = 3 : 2.
2
B's share = Rs.
2.
1210 x
5
= Rs. 484.
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 & Explanation
Answer: Option C
Explanation:
Let the third number be x.
Then, first number = 120% of x =
120x
100
150x
Second number = 150% of x =
=
100
6x
Ratio of first two numbers =
:
5
3.
=
6x
5
3x
2
3x
2
= 12x : 15x = 4 : 5.
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 & Explanation
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 & Explanation
Answer: Option A
Explanation:
Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7xand
8x respectively.
Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).
140
100
7x,
150
x 5x
21x
2
,
100
175
x 7x
and
and 14x.
The required ratio = 7x :
14x : 21x : 28x
21x
2
: 14x
100
x 8x
2 : 3 : 4.
5.
In a mixture 60 litres, the ratio of milk and water 2 : 1. If the 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 & Explanation
Answer: Option D
Explanation:
2
Quantity of milk =
60 x
3
litres = 40 litres.
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.
40
Then, milk : water =
40
Now,
.
20 + x
1
=
20 + x
2
20 + x = 80
x = 60.
6.
Quantity of water to be added = 60 litres.
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 & Explanation
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
100
42x
5
110
x 7x
and
and
100
x 8x
44x
5
42x
The required ratio =
7.
5
44x
:
5
= 21 : 22.
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 & Explanation
Answer: Option D
Explanation:
Let the original salaries of Ravi and Sumit be Rs. 2x
2 and Rs. 3
3x respectively.
Then,
2x + 4000
3x + 4000
=
40
57
57(2x + 4000) = 40(3x
40(3 + 4000)
6x = 68,000
3x = 34,000
Sumit's present salary = (3x
(3 + 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 & Explanation
Answer: Option B
Explanation:
6
(x x 5) = (0.75 x 8)
9.
x=
= 1.20
5
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 & Explanation
Answer: Option B
Explanation:
Let the three parts be A, B, C. Then,
3
A : B = 2 : 3 and B : C = 5 : 8 =
A:B:C=2:3:
24
5
5x
5
3
:
8x
5
24
=3:
5
= 10 : 15 : 24
15
B=
98 x
49
= 30.
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 & Explanation
Answer: Option D
Explanation:
Given ratio =
:
:
= 6 : 8 : 9.
6
1st part = Rs.
782 x
23
= Rs. 204
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 & Explanation
Answer: Option C
Explanation:
Let A = 2k, B = 3k and C = 5k.
5
115
A's new salary =
100
115
of 2k =
110
B's new salary =
100
100
10
100
x 3k
=
x 5k
= 6k
10
33k
10
120
of 5k =
23k
k
New ratio
=
110
of 3k =
120
C's new salary =
100
23k
x 2k
100
33k
:
10
: 6k
= 23 : 33 : 60
12. If 40% of a number is equal to two-third
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 & Explanation
Answer: Option C
Explanation:
Let 40% of A =
40A
2
3
2B
=
100
3
2A
2B
=
5
3
A
2
5
x
=
B
3
2
Then,
B
5
=
3
A : B = 5 : 3.
13. The fourth proportional to 5, 8, 15 is:
A.
18
B.
24
C.
19
D.
20
Answer & Explanation
Answer: Option B
Explanation:
Let the fourth proportional to 5, 8, 15 be x.
Then, 5 : 8 : 15 : x
5x = (8 x 15)
(8 x 15)
x=
5
= 24.
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 & Explanation
Answer: Option B
Explanation:
Let the numbers be 3x
3 and 5x.
Then,
3x - 9
5x - 9
=
12
23
23(3x - 9) = 12(5x
12(5 - 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 & Explanation
Answer: Option C
Explanation:
Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively.
25x
Then, sum of their values = Rs.
60x
100
= 30
x=
100
30 x 100
60
10 x 2x
+
100
5 x 3x
+
100
60x
= Rs.
100
= 50.
Hence, the number of 5 p coins = (3 x 50) = 150.
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,
b we call a as the first term or antecedent and b, the second term
or consequent..
Eg. The ratio 5 : 9 represents
5
9
with antecedent = 5, consequent = 9.
Rule: The multiplication or division of each term of a ratio by the same non
non-zero
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:
a
We say that (a : b) > (c : d)
b
>
c
d
.
5. Compounded Ratio:
6. The compounded ratio of the ratios: (a
( : b), (c : d), (e : f) is (ace : bdf).
7. Duplicate Ratios:
Duplicate ratio of (a
( : b) is (a2 : b2).
Sub-duplicate
duplicate ratio of (a
( : b) is (a : b).
Triplicate ratio of (a
( : b) is (a3 : b3).
Sub-triplicate
triplicate ratio of (a
( : b) is (a1/3 : b1/3).
If
a
b
=
c
d
, then
a+b
a-b
=
c+d
c-d
.
[componendo and dividendo]
8. Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write, x
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x
1
y
.
y.