Ratio and Proportion

Ratio and Proportion

Important Formulas



  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 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 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 .
    b d

    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).






















    If a = c , then a + b = c + d .     [componendo and dividendo]
    b d ab cd


  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

View Answer


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:
A. 2 : 5 B. 3 : 5
C. 4 : 5 D. 6 : 7

View Answer


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?
A. Rs. 500 B. Rs. 1500
C. Rs. 2000 D. None of these

View Answer


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

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:
A. 20 litres B. 30 litres
C. 40 litres D. 60 litres

View Answer


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?
A. 8 : 9 B. 17 : 18
C. 21 : 22 D. Cannot be determined

View Answer


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?
A. Rs. 17,000 B. Rs. 20,000
C. Rs. 25,500 D. Rs. 38,000

View Answer


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:
A. 1.12 B. 1.2
C. 1.25 D. 1.30

View Answer


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

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:
A. Rs. 182 B. Rs. 190
C. Rs. 196 D. Rs. 204

View Answer


Answer: Option D


Explanation:


Given ratio = : : = 6 : 8 : 9.
















1st 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?
A. 3 : 3 : 10 B. 10 : 11 : 20
C. 23 : 33 : 60 D. Cannot be determined

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 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

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:
A. 18 B. 24
C. 19 D. 20

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:
A. 27 B. 33
C. 49 D. 55

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?
A. 50 B. 100
C. 150 D. 200

View Answer


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.


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