→ Ratio of two Quantities is the number of times one quantity contains another quantity of same type.→ The ratio between a and b can be represented as a : b, where **a** is called **antecedent **and **b **is called **consequent**.i.e a/b or a : b**→ Different types of ratio are explained as below:-****(1) Duplicate Ratio :-** If the numbers given are in ratio, then the ratio of their squares is called duplicate ratio.For ex. 2:3=4:9**(2) Sub-Duplicate Ratio:- **If given two numbers are in ratio, then ratio of their square roots is called sub-duplicate ratio For ex 25 : 36 = 5 : 6**(3) Triplicate Ratio:-** If the given numbers are in ratio, then ratio of their cubes is called triplicate ratioFor ex- 4 : 5 = 64 : 125**(4) Sub- triplicate Ratio:-** If the given numbers are in ratio, then ratio of their cube roots is called sub-triplicate ratioFor ex- 125 : 343 = 5 : 7**(5) Inverse Ratio:-** If given numbers are in ratio then their antecedent and consequent are interchanged and the ratio obtained is called inverse ratio*** Proportion:**– An equality of two ratios is called the proportion.→ If L/M=x/y or L : M = x : y, then we can say that L, M, x, y are in proportion and can be written as L : M :: x : y**Some important rules of proportion:-****Rule 1:- **If **x : y :: y : z**, then z is called third proportional to **x **and **y**x : y ∷ y : z ⇒ x : y = y : z⇒ x × z = y × y⇒ y² = xz

**Rule 2:- **If **L : M :: x : y**, then y is called the 4th proportional to **L, M **and **x, y**i.e L : M ∷ x : y⇒ L : M = x : y⇒ L × y = Mx

Some examples are as follow:-**Ex)** calculate 3rd proportional of 18 and 36 Let 3rd proportion be x18 : 36 ∷ 36 : x

Let mean proportion be xThen 18 : x ∷ x : 8⇒ x² = 144⇒ x = 12* Some important rules of ratio:-**Rule 1:-** If two given ratio are **l : m** and **x : y**, then(i) **l : m > x : y if ly > mx**(ii) **l : m < x : y, if ly < mx**(iii) **l : m = x : y, if ly = mx****Rule 2:- **If two ratios are given, convert each ratio in such a way that both ratios have same denominator, then compare their numerators, the fractions with greater numerator will be greater

**Ex)** Divide 1150 in the ratio 14 : 9→ Let 1st part be 14x and 2nd part be 9x∴ 14x + 9x = 115023x = 1150∴ x = 50∴ 1st part = 14x = 7002nd part = 9x = 450

**(Ex)** The ratio of income of manan and Aman is in ratio 2 : 3 and their expenditure is in ratio 7 : 12. If each of then saves 3000 Rs, then find their income and expenditure?→ Let income of manan and Aman be 2x and 3x expenditure of manan = (2x – 300) Rsexpenditure of Aman = (3x – 3000) Rs

income of manan = 2x = 10000 Rsincome of Aman = 3x = 15000 Rsand expenditure of manan = 10000 – 3000 = 7000 Rsexpenditure of Aman = 15000 – 3000 = 12000 Rs

**Ex) **If two numbers are in ratio 3 : 4. If 15 is added to both the numbers, then the ratio becomes 7 : 9. Find the greatest number?Let the two numbers be 3x and 4x

∴ 9(3x + 15) = 7 (4x + 15)∴ 27x + 135 = 28x + 105∴ x = 30∴ greater number = 4x= 120*** Faster approach:-**Here x =3, y = 4, m = 7, n = 9 and a = 15∴ Two numbers are

*** Type 4:- **Two numbers are in ratio **x : y** and **a **is subtracted from the numbers, then the ratio becomes **m : n. **The two numbers will be

→ When two or more persons makes a group and invest money for running a certain business and after certain time receive profit in the ratio of their invested money and time period, then such a group is called partnership**→ It is of 2 types:-****(1) Simple partnership:-** If all partners invest their different capitals for the same time period or some capital for different time period.**(2) Compound partnership:-** If all partners invest their different capitals for different time period.*** Partners are also of 2 types:-****(1) Working partner:-** A partner who not only invests money, but also takes part in business activities and gets some salary for that**(2) Sleeping partner:-** A partner who only invests money, and not take part in business activities**Ex)** P and Q starts a business by investing 6000 Rs and 9000 Rs respectively. Find the ratio of their profit after 1 yrs?

**Ex)** A start a business with 6000 rs and B joins the business 5 month later with 8000 rs. Find the share of A and B if total profit is 16000 Rs after 1 year? A : B6000 8000× 12 × 7 72 56∴ 18 14∴ 9 7∴ A’s share = 16000 × 9/16 = 9000 rsB’s share = 16000 × 7/16 = 7000 rs→ Different types of questions asked nowadays in the exams are as follow:-**Type 1:-** If **a : b : c** is the ratio of investment and **x : y : z** is the ratio of profits, then the ratio of time period of investment is given by

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