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# Smart Tricks to Solve Partnership, Ratio and Proportion Questions

### 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 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 ratio
For 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 ratio
For 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
Ex) calculate 3rd proportional of 18 and 36 Let 3rd proportion be x
18 : 36 ∷ 36 : x
Let mean proportion be x
Then 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 = 1150
23x = 1150
∴ x = 50
∴ 1st part = 14x = 700
2nd 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) Rs
expenditure of Aman = (3x – 3000) Rs
income of manan = 2x = 10000 Rs
income of Aman = 3x = 15000 Rs
and expenditure of manan = 10000 – 3000 = 7000 Rs
expenditure 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

### 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 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      :     B
6000        8000
× 12          × 7
72             56
∴ 18         14
∴ 9           7
∴ A’s share = 16000 × 9/16 = 9000 rs
B’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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