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RatioRatio & & Proportion Proportion

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SUB TOPICS

- RATIO AND PROPORTION

- PARTNERSHIP

- ALLIGATION OR MIXTURE

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RatioRatio : The ratio of two quantities x and y in the same : The ratio of two quantities x and y in the same units is represented as x : y units is represented as x : y such that such that x : y = x / y x : y = x / y

here , x is called antecedent and y as consequent. here , x is called antecedent and y as consequent.

NoteNote - if we multiply or divide each term of a ratio by some - if we multiply or divide each term of a ratio by some non-zero number; the ratio remains the same.non-zero number; the ratio remains the same.

E.g. - 3 : 7 = ( 3 *5 ) : ( 7 * 5 ) E.g. - 3 : 7 = ( 3 *5 ) : ( 7 * 5 )

= 15 : 35 = 15 : 35

RATIO AND PROPORTION

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Proportions Proportions : The equality of two ratios is termed as : The equality of two ratios is termed as proportion.proportion.

Let p : q = r : s Let p : q = r : s ⇔⇔ p : q : : r : s p : q : : r : s

Here p, q, r, s are said to be in proportion .Here p, q, r, s are said to be in proportion .

Also , p and s are known as extremes .Also , p and s are known as extremes . q and r are known as means . q and r are known as means .

NoteNote - product of extremes = product of means - product of extremes = product of means

so , p : q = r : s so , p : q = r : s ⇔⇔ ( p * s ) = ( q * r ) ( p * s ) = ( q * r )

RATIO AND PROPORTIONS

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Fourth proportional Fourth proportional : let p : q = r : s ; s is said to : let p : q = r : s ; s is said to be the fourth proportional .be the fourth proportional .

Third proportional Third proportional : let p : q = q : r ; r is said be : let p : q = q : r ; r is said be the third proportional of p and q .the third proportional of p and q .

Mean proportional Mean proportional : if r is a number such that : if r is a number such that r = r = √√ pq pq Here , r is called as mean proportion of p and q . Here , r is called as mean proportion of p and q .

Comparison of ratiComparison of ratios : os : ( p : q ) < ( r : s ) ( p : q ) < ( r : s ) ⇔⇔ ( p / q ) < ( r / s ) . ( p / q ) < ( r / s ) .

RATIO AND PROPORTIONS

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Duplicate ratio Duplicate ratio of p : q = p² : q²of p : q = p² : q²

Sub – duplicate ratio Sub – duplicate ratio of p : q = of p : q = √√p : p : √√q q

Triplicate ratioTriplicate ratio of p : q = p³ : q³ of p : q = p³ : q³

Sub – triplicate ratio Sub – triplicate ratio of p: q = p1/3 : q1/3 of p: q = p1/3 : q1/3

Componendo and dividendo Componendo and dividendo : let p/q = r/s: let p/q = r/s

then, ( p + q ) / ( p - q ) = ( r + s ) : ( r –s ) . then, ( p + q ) / ( p - q ) = ( r + s ) : ( r –s ) .

Compounded ratio Compounded ratio of x : p , y : q , z : r = xyz : pqrof x : p , y : q , z : r = xyz : pqr

RATIO AND PROPORTIONS

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Eg:Eg:If x : y = 3 : 5 andIf x : y = 3 : 5 and y : z = 6 : 7 , then y : z = 6 : 7 , then

what is the ratio of what is the ratio of x : y : z ? x : y : z ?

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Solution Solution : Here,: Here,

x : y = 3 : 5 and x : y = 3 : 5 and

y : z = 6 : 7 y : z = 6 : 7 = ( 6 * 5 / 6 ) : ( 7 * 5 / 6 ) = ( 6 * 5 / 6 ) : ( 7 * 5 / 6 ) = 5 : (35 / 6) = 5 : (35 / 6)

x : y : z = 3 : 5 : (35/6) x : y : z = 3 : 5 : (35/6) = 18 : 30 : 35 = 18 : 30 : 35

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Eg:Eg:The ratio of three numbers is The ratio of three numbers is 2 : 5 : 7 and their multiplication 2 : 5 : 7 and their multiplication is 8750 . is 8750 .

Find the sum of these Find the sum of these numbers .numbers .

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SOLSOL:Let the numbers are 2a, 5a, 7a :Let the numbers are 2a, 5a, 7a then ,then , 2a * 5a * 7a = 8750 2a * 5a * 7a = 8750 70 a³ = 8750 70 a³ = 8750 a³ = 8750 / 70 a³ = 8750 / 70 a³ = 125 a³ = 125 a³ = 5³ a³ = 5³ a = 5 a = 5 Sum of numbers = 2a + 5a + 7aSum of numbers = 2a + 5a + 7a = 14a = 14a = 70 = 70

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Whenever two or more persons run a business together, they are Whenever two or more persons run a business together, they are called partners and the deal among them is called partnership .called partners and the deal among them is called partnership .

Ratio of profits / losses:Ratio of profits / losses:

- Let Ram and Shyam invest Rs. X and Rs. Y in a partnership for - Let Ram and Shyam invest Rs. X and Rs. Y in a partnership for 1 year, then1 year, then

Ram´s share of profit / loss : Shyam´s share of profit / lossRam´s share of profit / loss : Shyam´s share of profit / loss = X : Y = X : Y

-Let Ram invest Rs. X for p months and shyam invest Rs. Y for q -Let Ram invest Rs. X for p months and shyam invest Rs. Y for q months , then months , then

Ram´s share of profit / loss : Shyam´s share of profit / lossRam´s share of profit / loss : Shyam´s share of profit / loss = Xp : Yq = Xp : Yq

PARTNERSHIP

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EXEX::Dinesh began an enterprises withDinesh began an enterprises withRs. 71,000 and he was joined Rs. 71,000 and he was joined afterward by Mohan with Rs. 35,500.afterward by Mohan with Rs. 35,500.For how much time did Mohan join if For how much time did Mohan join if the profits at the end of the year are the profits at the end of the year are divided in 1 : 3 between Mohan and divided in 1 : 3 between Mohan and Dinesh.Dinesh.

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SOLSOL: Let Mohan joined the : Let Mohan joined the enterprises for x months. enterprises for x months.

( 35,500 * x ) : ( 71,000 * 12 ) = 1 : 3( 35,500 * x ) : ( 71,000 * 12 ) = 1 : 3

x = ( 71,000 * 12 ) / ( 35,500 * 3 ) x = ( 71,000 * 12 ) / ( 35,500 * 3 ) = 2 * 4 = 2 * 4 = 8 months. = 8 months.

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When two quantities of different cost prices are mixed in When two quantities of different cost prices are mixed in some ratio to find an other mixture to sell, then the some ratio to find an other mixture to sell, then the selling price of mixture is called selling price of mixture is called mean pricemean price

Rule of alligation Rule of alligation says - says -

Quantity of cheaper : Quantity of dearerQuantity of cheaper : Quantity of dearer

= ( C.P. of dearer - Mean Price ) : ( Mean Price - C.P. of = ( C.P. of dearer - Mean Price ) : ( Mean Price - C.P. of cheaper ) cheaper )

Alligation or mixture

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EX:EX: The cost of type A wheat is The cost of type A wheat is Rs. 10 per kg and the cost of Rs. 10 per kg and the cost of type B wheat is Rs. 15 per kg. If type B wheat is Rs. 15 per kg. If both are mixed in the ratio 3 : 5, both are mixed in the ratio 3 : 5, then what is the price of the then what is the price of the mixture wheat per kg.?mixture wheat per kg.?

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SOL:SOL: Let the price of mixed variety of Let the price of mixed variety of wheat is Rs. X per kg.wheat is Rs. X per kg.

By the rule of alligation ,By the rule of alligation ,

3 : 5 = ( 15 - x ) : ( x - 10 ) 3 : 5 = ( 15 - x ) : ( x - 10 ) 3 * ( x - 10 ) = 5 * ( 15 - x ) 3 * ( x - 10 ) = 5 * ( 15 - x ) 3x + 5x = 75 + 30 3x + 5x = 75 + 30 8x = 105 8x = 105 x = 105 / 8 x = 105 / 8 x = Rs. 13.125 / kg. x = Rs. 13.125 / kg.

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The endThe end

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