# 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 a – b c – 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

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

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

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

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

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

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

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. 1.3

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

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

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

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

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

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

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

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.