 ## NOTES ON RATIO AND PROPORTION

### Ratio & Proportion

→ 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 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 : ySome important rules of proportion:-Rule 1:- If x : y :: y : z, then z is called third proportional to and yx : 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, yi.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 = mxRule 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 is subtracted from the numbers, then the ratio becomes m : n. The two numbers will be

### Partnership

→ 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 activitiesEx) 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 ## SHARE

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