RATIO, PROPORTION & PARTNERSHIP

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ABOUT DISHA PUBLICATION One of the leading publishers in India, Disha Publication provides books and study materials for schools and various competitive exams being continuously held across the

1 ABOUT DISHA PUBLICATION One of the leading publishers in India, Disha Publication provides books and study materials for schools and various competitive exams being continuously held across the country. Disha's sole purpose is to encourage a student to get the best out of preparation. Disha Publication offers an online bookstore to help students buy exam books online with ease. We, at Disha provide a wide array of Bank / Engg./ Medical & Other Competitive Exam books to help all those aspirants who wish to crack their respective different levels of Bank / Engg./ Medical & Other Competitive exams. At Disha Publication, we strive to bring out the best guidebooks that students would find to be the most useful for all kind of competitive exam. RATIO, PROPORTION & PARTNERSHIP RATIO The ratio of two terms x and y is denoted by x : y. Compound ratio : The compound of a : b and c : d is a c, i.e., ac : bd. b d The duplicate ratio of x : y is x : y. The triplicate ratio of x : y is x : y. The subduplicate ratio of x : y is x : y. The subtriplicate ratio of x : y is x: y. Inverse ratio Inverse ratio of x : y is y : x. PROPORTION When two ratios are equal, the four quantities composing them are said to be in proportion. If a c, then a, b, c, d are in proportions. b d If four quantities are in proportion, the product of the extremes is equal to the product of the means. Let a, b, c, d be in proportion, then a c Þ ad bc. b d If three quantities a, b and c are in continued proportion, then a : b b : c \ ac b b is called mean proportional. SHORT CUTS (i) If a b c K K K K+ K + K +... K..., then a + b + c +... c For example: If P Q R, then find P + Q + R 7 R Sol. P, Q, R 7 Then P + Q + R R 7 a a a a an (ii) If... K, then a a a a a a : a n + n + (K) n A number added or subtracted from a, b, c & d, so that they ad - bc are in proportion ( a+ d) -( b+ c) For example : When a number should be subtracted from,, & so that they are in proportion. Find that number Sol. Req. No. ( ) ( ) If X part of A is equal to Y part of B, then A : B Y : X. For example: If 0% of A 0% of B, then find A : B. Sol. A : B 0% : 0% When X th part of P, Y th part of Q and Z th part of R are equal, then find A : B : C. Then, A : B : C yz : zx : xy PARTNERSHIP If the period of investment is the same for each partner, then the profit or loss is divided in the ratio of their investments. If A and B are partners in a business, then Investment of A Profit of A Investment of B Profit of B Loss of A or Loss of B If A, B and C are partners in a business, then Investment of A : Investment of B : Investment of C Profit of A : Profit of B : Profit of C, or Loss of A : Loss of B : Loss of C

2 EXERCISE. The fourth proportional to, 8, is: (a) 8 (b) (c) 9 (d) 0. Given that carat gold is pure gold, 8 carat gold is gold and 0 carat gold is 6 gold, the ratio of the pure gold in 8 carat gold to the pure gold in 0 carat gold is : (a) : 8 (b) 9 : 0 (c) : (d) 8 :. A sum of money is to be distributed among A, B, C, D in the proportion of : : :. If C gets ` 000 more than D, what is B s share? (a) ` 00 (b) ` 00 (c) ` 000 (d) None of these. The sum of three numbers is 98. If the ratio of the first to the second is : and that of the second to the third is : 8, then the second number is : (a) 0 (b) 0 (c) 8 (d) 8. The ratio of number of ladies to gents at a party was :, but when ladies and gents left, the ratio became :. How many people were originally present at the party? (a) 6 (b) 9 (c) (d) None of these 6. In an express train, the passengers travelling in A.C. sleeper class, First class and Sleeper class are in the ratio ::7, and rate for each class is in the ratio : :. If the total income from this train is `, 000, find the income of Indian Railways from A.C. sleeper class. (a) `,000 (b) ` 0,000 (c) `,000 (d) ` 0, In a mixture of litres, the ratio of milk and water is :. How much water must be added to make the mixture ratio :? (a) 7 litres (b) litres (c) litres (d). litres 8. The monthly income of two persons are in the ratio of : and their monthly expenditures are in the ratio of 7 : 9. If each saves ` 0 a month, then what are their monthly incomes? (a) ` 00, ` (b) ` 00, ` 0 (c) ` 00, ` 7 (d) ` 00, ` Salaries of A, B and C were in the ratio : : 7, respectively. If their salaries were increased by 0%, 60% and 0% respectively, what will be the new ratio of the their respective new salaries? (a) : : 7 (b) : 6 : 7 (c) : : 8 (d) 9 : 6 : 0. Out of a total amount of `,898, B receives 0 % more than A and % more than C. What is B s share? (a) ` 90 (b) `,860 (c) `,00 (d) `, 0. In a dairy farm, 0 cows eat 0 bags of husk in 0 days. In how many days one cow will eat one bag of husk? (a) (b) 0 (c) 0 (d) 80. In a camp, there is a meal for 0 men or 00 children. If 0 children have taken the meal, how many men will be catered to with the remaining meal? (a) 0 (b) 0 (c) 0 (d) 0. If 0.7 : x : : : 8, then x is equal to: (a). (b).0 (c). (d).0. There are 0 doctors and nurses at a hospital. If the ratio of doctors to nurses is : 7, then the nurses at the hospital are (a) 0 (b) 60 (c) 00 (d) 0. A bag contains ` 6 in the form of one rupee, 0 paise and paise coins in the ratio of : :. The number of 0 paise coins is : (a) 96 (b) (c) (d) 6. If A : B : C : :, then A : B : C is equal to : B C A (a) : 9 : 6 (b) 8 : 9 : (c) 8 : 9 : 6 (d) 8 : 9 : 7. A sum of money is to be distributed among A, B, C, D in the proportion of : : :. If C gets ` 000 more than D, what is B s share? (a) ` 00 (b) ` 00 (c) ` 000 (d) None of these 8. A began a business with ` 00 and was joined afterwards by B with ` 00. If the profits at the end of year was divided in the ratio : then B joined the business after : (a) month (b) months (c) 6 months (d) 7 months 9. on dividing ` 8 among A, B and C such that four times A s share is equal to times B s share which is equal to seven times C s share. Then share of B is: (a) (b) 96 (c) 0 (d) 0. If : 8 :: x :, then find the value of x. (a) 8 (b) 6 (c) (d). The ages of Aarzoo and Arnav are in the ratio of : respectively. After 7 years the ratio of their ages will be 0:. What is the difference in years between their ages? (a) years (b) 7 years (c) 6 years (d) years. The ratio of the radius of two circles is :. Find the ratio of their areas. (a) : (b) 8 : 0 (c) : 0 (d) :

3 . The sum of two numbers is 60 and their difference is 6. What is the ratio of the two numbers? (a) : 0 (b) 9 : (c) : 9 (d) 0 :. One litre of water was mixed to litres of sugar solution containing % of sugar. What is the percentage of sugar in the solution? (a) (b) (c) 6 (d) Insuffficient data. A and B enter into partnership with capitals in the ratio :. At the end of 0 months A withdraws, and the profits now are divided in the ratio of : 6. Find how long B remained in the business? (a) 9 months (b) 8 months (c) 6 months (d) 7 months 6. When 0% of one number is added to a second number, the second number increases to its four-thirds. What is the ratio between the first number and the second number? (a) : (b) : (c) : (d) Data inadequate 7. A and B started a business by investing `,000 and ` 0,000 respectively. B left the business after months and C joined the business with a sum of `,000. The profit earned at the end of the year is ` 8,. What is B's share of profit? (a) ` (b) `,000 (c) `,60 (d) Cannot be determined 8. A and B rent a pasture for 0 months. A puts in 00 cows for 8 months. How many cows can B put in for the remaining months, if he pays half as much as A? (a) 00 (b) 600 (c) 800 (d) There are two numbers such that the sum of twice the first number and thrice the second number is and the sum of thrice the first number and twice the second number is 7. Which is the larger number? (a) (b) 6 (c) 8 (d) 0. Populations of two villages X and Y are in the ratio of : 7 respectively. If the population of village Y increases by 000 and the population of village X remains unchanged the respective ratio of their populations becomes : 6. What is the population of village X? (a) 6,,000 (b) 6,7,000 (c) 8,7,000 (d) 9,00,000. In a class of students the ratio of boys and girls is :. How many more boys are to be added to make the ratio :? (a) 6 (b) (c) (d) None of these. Find the ratio of the perimeter of the triangle and a square if the each side of triangle and square is equal. (a) : (b) : (c) : (d) Cannot be determined. The total number of students in a school is 0. If the number of girls in the school is 9, then what is the ratio of the total number of boys to the total number of girls in the school? (a) : 7 (b) : 9 (c) 7 : (d) 9 : - Ninad, Vikas and Manav enter into a partnership. Ninad invests some amount at the beginning. Vikas invests double the amount after 6 months and Manav invests thrice the amount invested by Ninad after 8 months. They earn a profit of `, 000 at the end of the year. What is Manav s share in the profit? (a) ` ]000 (b) ` ]000 (c) ` ]000 (d) ` 9]000. In a mixture of milk and water the proportion of water by weight was 7%. If in the 60 gms of this mixture gm. water was added, what would be the percentage of water? (a) 7% (b) 80% (c) 90% (d) 00% 6. The largest and the second largest angles of a triangle are in the ratio of :. The smallest angle is 0% of the sum of the largest and the second largest angles. What is the sum of the smallest and the second largest angles? (a) 0 (b) 08 (c) 00 (d) 0 ANSWER KEY (b) 7 (c) (b) 9 (b) (a) (d) (b) 8 (d) (d) 0 (b) 6 (c) (b) (c) 9 (d) (b) (c) 7 (c) (c) (b) 0 (b) 6 (d) (d) 8 (b) (b) (c) (c) 7 (c) (c) 9 (c) (b) 6 (d) (b) 8 (d) (a) 0 (a) 6 (d)

4 . (b) Let the fourth proportional to, 8, be x. (8 ) Then, : 8 : : : x Þx (8 ) Þx.. (b) 8 carat gold pure gold 8 carat gold 0 carat gold pure gold 0 carot gold 6 6 Required ratio 8 : 0 9 : 0. (c) Let the shares of A, B, C and D be ` x, ` x, ` x and ` x respectively. Then, x x 000 Þ x 000 \ B s Share ` x ` 000. (b) A : B : : 0 : and B : C : 8 : 8 : Therefore, A : B : C 0 : : Let the numbers be 0x, x and x. Then, 0x + x + x 98 or 9x 98 or x Þ Second number x 0. (c) Let number of ladies x and, number of gents x x- Now, Þx- 6 x- x- Þ x \ Total number of people originally present (d) Let number of passengers x, x, 7x and Rate y, y, y Now, since income Rate Number of passengers \ Income of passengers xy, 8xy, xy \ Income in ratio : 8 : \ Income from A.C. sleeper class 8, ` 0, (c) Quantity of milk 6 litres Quantity of water 9 litres Let x litres of water be added. 6 Then, 9 + x Þ x or x or x litres 8. (d) Let the income of two persons be ` x and ` x and their expenses be ` 7y and ` 9y. Therefore, x 7y 0... (i) and x 9y 0... (ii) From (i) and (ii), we get x 00 and y 0 The income of the two persons are ` 00 and ` 00, respectively. HINTS & SOLUTIONS 9. (d) New ratio of their respective salaries : : :8 : 9 :6 : 0. (b) Given A + B + C (i) 0 Also B A and B C \ By (i), B + B + B Þ B `, (c) Let the required number of days be x. Less cows, More days (Indirect Proportion) Less bags, Less days (Direct Proportion) Cows :0 ü ý ::0:x Bags 0: þ \ 0 x 0 0 Þ x 0.. (b) There is a meal for 00 children. 0 children have taken the meal. Remaining meal is to be catered to 0 children. Now, 00 children º 0 men æ0 ö \ 0 children º ç 0 men 0men. è00 ø. (b) 6 (x ) (0.7 8) Þ x.0.. (d) 7 Number of nurses 0 0. (b) Let the no. of one rupee, 0 paise and paise coins be x, x and x respectively. According to question, x x 8x + 6x + x x Þ 6 \ x 8 \ Number of 0 paise coins 8 6. (d) Let A x, B x and C x. Then, A x B x C x, and B x C x A x A B C Þ : : : : 8:9:. B C A 7. (c) Let the shares of A, B, C and D be ` x, ` x, ` x and ` x respectively. Then, x x 000 Þ x 000 \ B s Share ` x ` (d) Let B joined after x months. Then, 00 : 00 ( x) : 00 or 00 ( - x) Þ -x ( )

5 or - x or x 7 months 9. (b) times A s share times B s share 7 times C s share A 's share B's share C's share (dividing by LCM of, and 7, i.e., 0) \ A : B : C : 8 : 0 \ Share of A ` Share of B 8 8 ` 96 8 Share of C 0 8 ` (b) : 8 : : x : Þ 8 x Þ x 6 8. (c) Let the present ages of Aarzoo and Arnav be x and x years respectively. x or, x + 7 or, 60x + 0 x + 6 or, 60x - x 6-0 or, 7x \ x 7 Difference between their ages x - x x 6 years. (d) Ratio of their areas : :. (c) The required ratio of the numbers or : (a) New % of sugar in ( + ) litre solution % ( + ). (a) Initially A s investment x and B s investment x Let B remain in the business for n months. Þ x 0 : x n : 6 \ x 0 6 x n Þ n 9 6. (c) Let the numbers be y and x respectively x + 0% of x y x y or, x y x y or, or, x 7. (c) Ratio of equivalent capitals of A, B and C for month 000 : 0000 : : 0 : 7 8 : 0 : Sum of the ratios æ 0 ö \ B's share ` ç 8 è ø `60 8. (b) Suppose B puts in x cows. The ratio of A s and B s rents :+ : : Then, 00 8 or, x cows. x 9. (c) Let the first number be x and the second number be y. x + y...(i) x + y 7...(ii) By equation (i) (ii), we have 6x + 9y 6x y 8 7 Þ y 7 Þ y From equation (i), x + 96 Þ x 96 Þ x 8 \ Larger number 8 0. (a) Let the population of the village X be x. and that of village Y 7x. x 7x Þ 80x 7x Þ x 6000 \ x (d) The number of boys 8 boys The number of girls 7 girls Now, when 9 more boys are added the new ratio becomes 7 : 7 : \ 9 more boys are to be added.. (b) Let each side of a triangle and each side of a square be x unit. Then, perimeter of the triangle x and perimeter of the square x \ Required ratio x : x :. (c) Total number of students 9 \ Number of boys \ Required ratio 7 : 9 7 :. (b) Ratio of profit : 6 : : : \ Manav s share 000 ` 000. (b) In 60 gms mixture proportion of water 60 7 gms 00 Total proportion of water in new mixture + 60 gms. \ Percentage of water % (d) 0 Smallest angle ( + ) 00 \ Ratio of angles : : \ Sum of smallest and second largest angles 80 ( + ) ( + + )

Ratio, Proportion & Partnership Examples with Solutions

Ratio, Proportion & Partnership Examples with Solutions RATIO Ratio is strictly a mathematical term to compare two similar quantities expressed in the same units. The ratio of two terms x and y is denoted by x:y. In general, the ratio of a number x to a number