The document discusses various topics related to ratios and proportions including: - Definitions of ratios, proportions, extremes, means - Types of ratios like duplicate, triplicate, compound - Finding fourth proportionals and mean proportionals - Comparing and computing ratios - Applications to partnership problems and mixtures/alligation problems It provides examples of ratio and proportion word problems along with step-by-step solutions.

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2. Ratio & Proportion Content

3. SUB TOPICS RATIO AND PROPORTION PARTNERSHIP ALLIGATION OR MIXTURE 4

4. Ratio : The ratio of two quantities x and y in the same units is represented as x : y such that x : y = x / y here , x is called antecedent and y as consequent. Note - if we multiply or divide each term of a ratio by some non-zero number; the ratio remains the same. E.g. - 3 : 7 = ( 3 *5 ) : ( 7 * 5 ) = 15 : 35 RATIO AND PROPORTION

5. Proportions : The equality of two ratios is termed as proportion. Let p : q = r : s  p : q : : r : s Here p, q, r, s are said to be in proportion . Also , p and s are known as extremes . q and r are known as means . Note - product of extremes = product of means so , p : q = r : s  ( p * s ) = ( q * r ) RATIO AND PROPORTIONS

6. Fourth proportional : let p : q = r : s ; s is said to be the fourth proportional . Third proportional : let p : q = q : r ; r is said be the third proportional of p and q . Mean proportional : if r is a number such that r =  pq Here , r is called as mean proportion of p and q . Comparison of rati os : ( p : q ) < ( r : s )  ( p / q ) < ( r / s ) . RATIO AND PROPORTIONS

7. Duplicate ratio of p : q = p² : q² Sub – duplicate ratio of p : q = √ p : √ q Triplicate ratio of p : q = p³ : q³ Sub – triplicate ratio of p: q = p1/3 : q1/3 Componendo and dividendo : let p/q = r/s then, ( p + q ) / ( p - q ) = ( r + s ) : ( r –s ) . Compounded ratio of x : p , y : q , z : r = xyz : pqr RATIO AND PROPORTIONS

8. Eg: If x : y = 3 : 5 and y : z = 6 : 7 , then what is the ratio of x : y : z ?

9. Solution : Here, x : y = 3 : 5 and y : z = 6 : 7 = ( 6 * 5 / 6 ) : ( 7 * 5 / 6 ) = 5 : (35 / 6) x : y : z = 3 : 5 : (35/6) = 18 : 30 : 35

10. Eg: The ratio of three numbers is 2 : 5 : 7 and their multiplication is 8750 . Find the sum of these numbers .

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

12. Whenever two or more persons run a business together, they are called partners and the deal among them is called partnership . Ratio of profits / losses: - Let Ram and Shyam invest Rs. X and Rs. Y in a partnership for 1 year, then Ram´s share of profit / loss : Shyam´s share of profit / loss = X : Y -Let Ram invest Rs. X for p months and shyam invest Rs. Y for q months , then Ram´s share of profit / loss : Shyam´s share of profit / loss = Xp : Yq PARTNERSHIP

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

14. SOL : Let Mohan joined the enterprises for x months. ( 35,500 * x ) : ( 71,000 * 12 ) = 1 : 3 x = ( 71,000 * 12 ) / ( 35,500 * 3 ) = 2 * 4 = 8 months.

15. When two quantities of different cost prices are mixed in some ratio to find an other mixture to sell, then the selling price of mixture is called mean price Rule of alligation says - Quantity of cheaper : Quantity of dearer = ( C.P. of dearer - Mean Price ) : ( Mean Price - C.P. of cheaper ) Alligation or mixture

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

17. SOL: Let the price of mixed variety of wheat is Rs. X per kg. By the rule of alligation , 3 : 5 = ( 15 - x ) : ( x - 10 ) 3 * ( x - 10 ) = 5 * ( 15 - x ) 3x + 5x = 75 + 30 8x = 105 x = 105 / 8 x = Rs. 13.125 / kg.

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