**Ratio:**

The ratio of two quantities a and b in the same units, is the fraction a/b 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 represents 5/9 with antecedent = 5, consequent = 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.

**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/b > c/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 (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/b = c/d, then a + b/a-b = c + d/c-d [componendo and dividendo]

**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

## MCQS

**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:

4/15 A = 2/5 B

A =(2/5 x 15/4)B

A = 3/2B

A/B = 3:23

A : B = 3 : 2.

B’s share = Rs. 1210 x 2/5 = 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: Option C**

Explanation:

Let the third number be x.

Then, first number = 120% of x = 120x/100 = 6x/5

Second number = 150% of x =150x/100 = 3x/2

Ratio of first two numbers = (6x/5 : 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: 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.

**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).

(140/100 x 5x) ,(150/100 x 7x) and (175/100 x 8x)

7x, 21x/2 and 14x.

The required ratio = 7x : 21x/2 : 14x

14x : 21x : 28x

2 : 3 : 4.

** 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 x 2/3) 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.

Then, milk : water =(40/20 +x)

Now(40/20 + x) = 1/2

20 + x = 80

x = 60.

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: 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).

=> (120/100x 7x) and (110/100 x 8x)

=> 42x/5 and 44x/5

The required ratio = 42x/5 : 44x/5 = 21 : 22.

**Salaries of Raiz and Salman are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Salman’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 Raiz and Salman be Rs. 2x and Rs. 3x respectively.

Then,2x + 4000/3x + 4000 = 40/57

57(2x + 4000) = 40(3x + 4000)

6x = 68,000

3x = 34,000

Salman’s present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.

**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 = (6/5) = 1.20

**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 x 3/5) : (8 x 3/5) = 3 : 24/5

A : B : C = 2 : 3 : 24/5 = 10 : 15 : 24

B = (98 x 15/49) = 30.

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