#### Ratio and Proportion

Ratio is the frequently used **term when we compare two or more similar quantities or units (i.e same kind of quantities)**. The ratio between x and y expressed as x:y

(or) x/y. Here x is called as antecedent and y is called as consequent.

Example: What is the ratio between 50 apples and 100 apples ( apples = same kind of quantities, but dierent number) Then the ratio between them 1:2 ( i.e it

means 1st quantity of apples is half of the 2nd quantity of apples (or) 2nd quantity of apples is the double of the 1st quantity).

TYPES OF RATIOS – RATIO AND PROPORTION

**1. DUPLICATE RATIO**

When two numbers are in ratio, then the ratio of their squares is called duplicate ratio.(Duplicate ratio for x, y will be x² : y²)

Ex: The duplicate ratio of 3 and 4 will be 3²:4² = 9:16

**2. SUB-DUPLICATE RATIO**

When two numbers are in ratio, then the ratio of their square roots is called Subduplicate ratio.

Ex: The Sub-duplicate ratio of x and y will be √x : √y.

**3.TRIPLICATE RATIO**

When two numbers are in ratio, then the ratio of their cubes is called triplicate

ratio. (The triplicate ratio of x and y will be x³ : y³)

Ex: The triplicate ratio of 2 and 3 will be 2³:3³= 8:27

**4. SUB-TRIPLICATE RATIO**

When two numbers are in ratio, then the ratio of their cube roots is called Subtriplicate ratio.(Sub-triplicate ratio for x, y will be ∛x : ∛y)

Ex: The Sub-triplicate ratio of 1 and 125 will be ∛1 : ∛125 = 1:5.

**5. INVERSE RATIO**

If the two numbers are in the ratio, then their antecedent and consequent are interchanged and the ratio is known as inverse ratio. The inverse ratio for x, y will be y: x.

Ex: The inverse ratio for 4:5 will be 5:4

**6.COMPOUND RATIO**

When two or more ratios are given, then respective antecedents are multipliedand as in same way respective consequents are also multiplied. If a:b, c:d, e:f are

three ratios, then their compound ratio will be ace : bdf .

**Methods to compare ratios. **

Here we explained as below.

**METHOD 1: **

CASE 1: A : B > C : D (WHEN AD > BC)

CASE 2: A : B < C : D (WHEN AD < BC)

CASE 3: A : B = C : D (WHEN AD = BC

**METHOD 2:**

If there is two ratios are given to compare, initially convert each ratio in such a way that both ratios have the same denominator, then the fraction which has greater

numerator will be the greater ratio.

Example: Find the greater ratio between 2 : 3 and 4 : 5?

Answer: Here we need to convert both denominators equal.

2 x 5/ 3 x 5= 10/15 and 4 x 3/ 5 x 3=12/15 [ LCM of 2/3 and 4/5 i.e, 3 and 5 is 15]

12/15 > 10/15

Hence 4:5 > 2:3

**METHOD 3:**

If two ratio are given for comparison, initially convert each ratio in such a way that both ratios have same numerator, then compare their denominators, the fraction

which is having lesser denominator will be greater ratio.

Example: Find the least fraction between 6/7 and 7/9?

Answer: Here we need to convert both numerators equal.

6 x 7/ 7 x 7 = 42/49 and 7 x 6/9 x 6 = 42/54 [ LCM of 6/7 and 7/9 i.e, 6 and 7 is 42]

42/49 > 42/54

6/7 > 7/9.

**PROPORTION**

An equality between two ratios is known as a proportion. If a/b = c/d or a: b = c:d, then we can say that a, b, c, and d are in proportion and can be written as a: b:: c :

d, here symbol:: represents proportion. a and d are called “Extremes” and b and c

are called as “Means”.

**RULES FOR PROPORTION – RATIO AND PROPORTION**

**RULE 1**

When a:b :: b:c, then c is called third proportional to a and b. Then the value of c will be as follows below.

a: b :: b:c => a : b= b:c

=> a x c = b x b => b² =ac ( cross multiplication)

c=b²/a

**RULE 2**

If a ; b :: c:d, then d is called the 4th proportional to a, b and c, d will be calculated as a:b::c:d

=> a: b =c:d

=> ad = bc

=> d = bc /a

**RULE 3 **

The mean proportional between a and b is √ab. If mean proportional is x, then a : x :: x: b

=> a b = x x

=> x² = ab

=> x = √ab

RATIO AND PROPORTION – MISCELLANEOUS CONCEPTS

**CONCEPT 1:** IF A:B= A:B AND B:C = M:N, THEN A:B:C = AM : MB: NB AND A:C = AM:BN

Example: If a : b = 5: 14 and b: c= 7:3, then nd a: b: c

Answer: a:b:c = (5 x 7) : (7 x 14): (14 x 3)

A:B:C = am : mb: nb

[ Here b common and value dierentiate in both given ratios, so divide ratios by bvalue 7]

a:b:c = (5 x 7)/7 : (7 x 14)/7: (14 x 3)/7

a ; b : c = 5 : 14 : 6

**CONCEPT 2:** IF A:B = A:B, B:C = C:D AND C:D = E :F, THEN A: B : C: D

= ACE : BCE : BDE : BDF

Example: If A : B= 1 : 3 and B: C =2 : 5, and C:D = 2: 3 then nd A: B: C : D

Answer: A:B = 1:3, B:C = 2:5, C:D = 2:3

A: B : C: D = ace : bce : bde : bdf

= (1x2x2) : (3x2x2) : (3x5x2) : (3x5x3)

= 4 : 12 : 30: 45.

**CONCEPT 3:** IF X IS DIVIDED IN A:B, THEN 1ST PART = AX/ A+B;

2ND PART = BX/A+B

Example : Divide 1111 in the ratio of 8:3.

Answer: According to formula x= 1111, a= 8, b=3.

1st part = ax/ a+b; 2nd part = bx/a+b

=> 1st part = 8/8+3 x 1111 = 808

=> 2nd part = 3/8+3 x 1111 = 303.

**CONCEPT 4 : **IF X IS DIVIDED IN A:B:C, THEN

1ST PART = AX / A+B+C

2ND PART = BX /A+B+C

3RD PART = CX /A+B+C

Example: Divide 2324 in the ratio of 35 :28: 20

Answer: According to formula

x = 2324, a = 35, b=28, c=20

1st part = ax / a+b+c => (35/35+28+20)2324 = 35/83 x 2324 =980

2nd part = bx /a+b+c => (28/83 x 2324) = 28 x 28= 784

3rd part = cx /a+b+c => (20/83 x 2324) = 20 x 28=560

Example: The sum of three numbers is 315. If the ratio between 1st and 2nd is 2:3 and the ratio between 2nd and 3rd is 4:5, then nd the 2nd number.

Answer: 1st number : 2nd number = 2:3 = (2 x 4) : (3 x 4) = 8:12

2nd number : 3rd number = 4:5 = (4 x 3):(5 x 3) = 12:15

1st number : 2nd number : 3rd number = 8:12: 15

2nd number = 12/ 8+12+15=108.

**Concept 5:** The incomes of two persons are in ratio of a:b and their expendituresare in the ratio of c:d. If each of them saves Rs x, then their incomes are given by [ x (d-c)/ ad-bc] a and [x(d-c)/ad-bc] b respectively And their expenditures are given by [x (b-a)/ad-bc] c and [ x(b-a)/ad-bc] D respectively.

Example: The ratio of incomes of Rajesh and Gagan is 4:3 and the ratio of their expenditures is 3:2. If each person saves Rs 2500/-, then nd their incomes and

Expenditures.

Answer: According to formula here a = 4, b=3, c=3, d=2 and x = Rs 2500/-

Income of Raman = [ x (d-c)/ ad-bc] a = 2500(2-3)/8-9 x 4 = Rs 10000/-

Income of Gagan =[x (d-c)/ ad-bc] b = 2500(2-3)/8-9 x 3 = Rs 7500/-

Expenditure of Raman = [x (b-a)/ad-bc] c = 2500(3-4)/8-9 x 3 =Rs 7500/-

Expenditure of Gagan = [x (b-a) ad-bc] d = 2500(3-4)/8-9 x 2 = Rs 5000/-