ANSWERS AND EXPLANATIONS

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1 ANSWERS AND EXPLANATIONS EXERCISE. (a) Given,? x? x??. (c) Let the numbers be x and x. Now, difference of numbers 0 i.e. x x 0 x 0 Larger number 0 0. (d) Suppose the first number is x and the second number y. Therefore, 0% of x of y x y 0. (a) Let the salaries of A and B be 9x and x. 9x 7 x 00 salary of B 00 ` (a) Reqd ratio? : 7? : 8? 0 : 7 0 : 8 7 : :. (d) Total number of students in the school 89 Number of girls Number of boys 89 - Required ratio : : 7. (d) Share of Urmila in dividend 78 ` (c) Let the present ages of Sushama and Karishma be x and 7x respectively. x 8 8 7x 8 9 or x + x x Required ratio 9 : (e) Let the number of girls x Number of boys. x Required ratio. x : x : 9 : 0 (d) According to the question, A B 7... (i) 0 A 8 and... (ii) 7 B From equations (i) and (ii), we cannot find the earning of A and B.. (d) Ratio of the amounts received by A, B and C 7 : 8 : Sum of the ratios Sum received by Pinku 7 00 ` Rinku 8 00 `00 Tinku 00 ` According to the question, On adding `00 to the share of each one, the required ratio 00 : 800 : 8 : 9 : 7. (c) Total number of students 9 Number of boys Required ratio 7 : 9 7 :. (e) Ratio of the capital of Rinku and Pooja Rinku's share ` (b) Let the present ages of Richa and Shelly be x and 8x years. According to the question, After 0 years, x 0 7 8x 0 0 or, x x + or, x 0x 70 or, x 0

2 RATIO AND PROPORTION x 0 Shelly's present age 8x 8 0 years. (d) The sum of money is not known.. (a) Let the age of woman be x years and that of her daughter be x years. According to the question, x + x or, x 8 or, x 8 8 Daughter's age 8 years 7. (c) Let the number of girls be x Number of the boys x Required ratio x. x : 0 8. (a) Let the present ages of Smita and Kavita be x and 8x years respectively According to questions, x 7 8x 7 9 or, x + 8 7x + or, x 7x 8 or, x or, x 7 Kavita's present age 8x 8 7 years 9. (a) Let the ages of man and his son be x and x respectively. x + x 8 8x 9 x 9 8 Son's age years 0. (d) Difference in age 9 ( )(9 8) years. (e) Let Arun s present age be x years. Deepak s present age (x + ) years x? 7? x?? 7 7? 7x? x?? 9 8? x?? Deepak s present age + years. (d) Data is given in ratio. So age can t be determined.. (a) Let number of students in Arts and Commerce were x and x respectively. x 8 x x 0x 0 x 0 0 Number of students in Arts 0 0. (d) Ratio of capital 0000 : : Sarita s share 8000 ( ) ` 00. (c) Let the original number of boys and girls be x and x respectively. x? 8? x? 7? x? x?? 78? x?? Number of boys + 8. (a) Let the present ages of P and Q be x and x respectively. (x + ) (x + ) x x x P s present age yrs. 7. (a) Present age of Meena 88 (0 ) years 8. (b) Ratio of the salaries of Sumit and Rajan

3 9. (a) : : Rajan's salary `? 89 8? or? 8 89 or? 9 or?? 0. (b) Let the present age of father and son be 7x and 7x respectively. 7x 7x x 7x 8 x Father s present age 7 years.. (c) Required amount 80 ( 0) (9 ) 80 ` 708. (d) Smallest angle 0 ( ) Ratio of angles : : Sum of smallest and second largest angles 80 ( ) ( ) (a) Required number of gold coins {90 ( 0)} 7 (0 7 8) (b) Let the present ages be x and x respectively. x x 7 0x 8x x years. (b) Let the fourth proportional to, 8, be x. : 8 : : : x x (8 ) x (8).. (b) (x ) (0.7 8) x (d) Given the ratio : : :8:9. st part ` 78 ` (d) Number of nurses 0 0 EXERCISE. (e) Suppose the salaries of A, B and C were 00k, 00k and 700k respectively. After increment salary of A 00k + 0% of 00k 0k B 00k + 0% of 00k 800k C 700k + 0% of 700k k Hence, new ratio of the respective salaries of A, B and C 0k : 800k : k 9 : :. (b) Let, weight of sugar costing `.7 per kg x kg x (x + 7).7x x x 7 x 00 kg. (c) Ratio of the amount is r P r r 0% Pr P. (d) All the given options are wrong. Let the passengers in A.C. Sleeper Class, st Class and Sleeper Class be x, x and x respectively and the fares in these classes be y, y and y respectively. Hence the incomes from these classes are xy, 8xy, xy respectively. x Difference in age x x? Required income 000 9? ` 0

4 RATIO AND PROPORTION. (d) The number of police involved??? 8? Required number of supporters 8? (b) Let actual distance be x km. : :: 0 : x x 0 x 0 7. (c) Number of girls 8. (b) 8 carat gold 80 km 0 0 pure gold 8 carat gold 0 carat gold pure gold 0 carot gold Required ratio 8 : 0 9 : 0 9. (b) Let the no. of one rupee, 0 paise and paise coins be x, x and x respectively. According to question, x x 8x x x x x 8 Number of 0 paise coins 8 0. (d) Let A x, B x and C x. A x B x C x, and B x C x A x A B C : : : : 8:9 :. B C A. (c) Let the shares of A, B, C and D be ` x, ` x, ` x and ` x respectively. x x 0 x 0 B s Share ` x ` 000. (c) Let A k, B k and C k. A s new salary of k k k 0 B s new salary 0 0 of k k k 0 C s new salary New ratio 0 0 of k k k k k : : k : : (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 8,000 ` 0, 000. (a) Let the ratio be x : (x + 0). x 7x x 80 x. (x 0) 7 Required ratio :.. (c) Total age of boys ( ) years 7 years Ratio of their ages : : 7. Age of the youngest boy 7 years years. (c) Let enlarged breadth be x inches. : :: : x 8 x x inches 8 7. (c) Originally, let the number of boys and girls in the college be 7x and 8x respectively. Their increased number is (0% of 7x) and (0% of 8x) 0 0 i.e. 7x and 8x x x i.e. and Required ratio x : x :. 8. (c) Quantity of milk Quantity of water? litres 9 litres Let x litres of water be added. 9 x x or x or x litres 9. (e) Let number be divided in ratio x : y. Then First part? x x? y, second part? y x? y

5 Now,? x?? y???? x? y?? x? y? 0. (c) For 9 kg zinc, mixture melted (9 + ) kg. For 8.8 kg zinc, mixture melted kg kg. 9. (b) Let there be x men at the beginning. Less men, More days Men : x : x Days : : 9 : 9::x : (x ) (x ) 9x x 90 x. ` 7 Weavers :8 : : : x Days :8 (Direct Proportion) 88 x 88 x. 7. (a) Let the required days be x. More men, less days (Indirect proportion) More size, more days (Direct Proportion) Men 0 :: : x size x 0 0. x days. (b) Let A s share ` x, B s share ` x and C s share ` 7x. (a) Now, 7x? (x? x)?? x? 70 A s share ` x ` V V and C C () V V C [From ()] V / where V denoted for vanilla and C for chocolate. V + 9V 8V + 9V V. (d) Let the required price be ` x. Less toys, Less cost (Direct Proportion). : ::.7 : x x (.7) x (.7) x Approximate price of toys ` 0.. (a) Let the required men be x. More hours, less men (Indirect proportion) More days, less men (Indirect proportion) Hours : 8 ::x :8 Days 8 : x 88 x 8. (d) Let the required number of mats be x. More weavers, More mats (Direct Proportion) More days, More mats 8. (c) Let the required number of days be x. Less cows, More days Less bags, Less days (Direct Proportion) Cows :0 :: 0 : x Bags 0: 0 x 0 0 x (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 0 men 0 0 children 0 men 0 men (b) Let the required number of days be x. Less persons, More days More working hrs per day, Less days Persons 0:9 : : : x Working hrs / day : 9 0 x 9 x x. 0. (b) Initially, let there be x men having food for y days. After days, x men had food for (y ) days. x Also, x men had food for y days. x : x :: (y ) : y x y x(y ) y y 0 y 0 days

6 8 RATIO AND PROPORTION. (c) In days, men set tiles tiles Area of floor Number of tiles Area of each tile ft. (a) 000 men taking 900 gms per head have provision for days. Less ratio per head, more men Less days, more men Ratio 80 : 900 :: 000 : x Days 0 : 80 0 x x 00 strength of reinforcement men. (b) Let the required number of days be x. 8 men 7 boys men 7 boys 7 men and boys boys boys Now, More boys, less days :7 :: : x x 7 7 x days But work times Required days days. (b) Let the required number of hours be x. Speeds of working of first and second type of men are and. More work, More time (Direct Proportion) Less speed, More time Work : : : : x Speed : x x 7. EXERCISE. (d) Let, the number of passengers travelling by I and II class be x and 0x and, fares of I and II class be y and y. Revenue is x y + 0x y ` xy xy Amount collected from the II class passengers 0xy 0 ` 0.. (b) Weight of dry grapes without water 90 0 kg Let weight of fresh grapes be x kg. According to question, 0 x x kg 0. (c) Let the questions with right answer be x. Questions with wrong answer 90 x Marks obtained 87 x (90 x) 87 x 80 + x 87 7x x 8 Questions with wrong answers (d) Fare after reduction. st nd rd Ratio of revenue st nd rd

7 9 0 8 Ratio of revenue of all three classes 0 : : 0880 Collection for st class Rs 0. (b) A : B : : 0 : and B : C : 8 : 8 : Therefore, A : B : C 0 : : Let the numbers be 0x, x and x. 0x + x + x 98 or 9x 98 or x Second number x 0. (c) Let number of ladies x and, number of gents x x Now, x x x x Total number of people originally present (b) Let Son s share ` S; Daughter s share ` D; and Wife s share ` W. Also, S : W W : D : S : W : D 9 : : then S 9x, D x and 9x x 0,000 x ` 0 Total worth of the property (9 + + ) x x 0 `,0 8. (a) Let number of each type of coin x. x +.0 x +. x.7x x 0 coins 9. (b) Let S denotes the shirts and T denotes the ties. We have, S + T By hit and trial, S 0, T 0 + Ratio of shirts to ties 0 : : 0. (a) Let A s share be ` x, B s share be ` y. C s share ` [7 (x + y)] Now, x : y 7 : 7 (x y) 9 : : x : y 7 : 80 (x y) : : x 90 x ` Also y y Rs C s share Rs[7 ( )] Rs. (d) Let the income of two persons be ` x and ` x 9 and their expenses be ` 7y and ` 9y. Therefore, x 7y 0... (i) and x 9y 0... (ii) From (i) and (ii), we get x and y 0 The income of the two persons are ` 00 and ` 00, respectively.. (b) Let income of A ` x, income of B ` x and expenditure of A ` y, expenditure of B ` y Now, saving income expenditure x y x y 00 x y and y 00 x 00 A s income `. (b) Let A s share Rs x, B s share ` x and C s share `.x Given x? 0? x? 8 Total amount ` (c) Gold in C units units. 9 8 Copper in C units units. 9 8 Gold : Copper 7 : 7 :.. (c) Let the three containers contain x, x and x litres of mixtures, respectively. x Milk in st mix. x litres litres. Water in st mix. x x x litres litres. Milk in nd mix. x litres x litres. Water in nd mix. (x x) litres x litres. x Milk in rd mix. x litres litres. 7 7 x 0x Water in rd mix. x litres litres. 7 7 Total milk in final mix. x x x x litres litres. 7 Total water in final mix. x 0x 0x x litres litres. 7

8 0 RATIO AND PROPORTION Required ratio of milk and water x 0x : 7 :.. (b) Let the fixed amount be ` x and the cost of each unit be ` y. 0y + x (i) and 0 y+ x (ii) On subtracting (i) from (ii), we get 80y 0 y. Putting y in (i), we get : x (800 0) 80. Fixed charges ` 80, Charge per unit `. Total charges for consuming 00 units ` ( ) ` (b) Given A + B + C 898 (i) Also 0 B A and B C By (i), B B B B Rs, (c) Since, A : B : and B : C : A : B : C : : A s share 70 Rs (a) Let the basic salary of A be ` x and that of B be ` y. 80 Now, x y x : y : 0. (a) Let the number of male and female participants be x and x respectively. Therefore total no. of participants are x. During the tea break, the number of male participants are (x ) x... (i) and the number of female participants are (x ) x Now, x x... (ii) x x x. Therefore, the total number of participants are.. (b) Number of males 0 Number of females Amount distributed among males and females 7 80% ` 0 Let the wage paid to a male be ` x and that to a female be ` x. Therefore, 0 x + x 0 0x + 0x 0 x Wage received by a feamale labourer ` 8. (c) Let x pairs of brown socks were ordered. Let P be the price of a brown pair. Therefore, price of the black pair of sock P Now, P + Px. (Px + 8P) P Px (Px 8P) 8P Px Px P Px P x Required ratio :. (b) Let the required length be x metres. More breadth, Less length More depth, Less length More days, More length (Direct Proportion) Breadth 0:0 Depth :0 : : : x Days 0:0 0 0 x 0 0 x x (a) ( + 00 ) men can finish the work in day. i.e., 00 men can finish the work in day men can finish the work in days The work would be days behind the schedule.. (d) Remaining work Remaining time 0 days More work, more men (Direct Proportion) Less days, more men More hours, Less men

9 Work : Days : 0 : : 0 : x Hrs 9 : 8 9 x 080 x 0 Additional men to be employed 0 0 men. (c) After days : 0 students had provision for 0 days. Now, let 7 students had provision for x days. more persons, less days 7 : 0 : : 0 : x 7 x 0 0 x 00 days Again, after 0 days : 7 students had provision 00 for 0 90 days. Let ( 7 ) 0 students had provision for y days Less persons, more days 0 : 7 : : 90 : y 90 0 y 7 y 9 days 7. (b) More machines, less hours Less days, more hours More amount of coal, more hours ( D i r e c t Proportion) Less efficiency, more hours Machine : Days : 8 :: : x Amount of coal 9,000 :,000 Efficiency 0.8 : 0.9 9, x 8, x hrs 8. (a) Let men left the work after x days. after x days, 0 men should completed work in (0 x) days but, men completed work in ( 0 x ) days. 0 : : : (0 x) : (0 x) (0 x) 0 (0 x) x Alternate : x days Since, 0 men completed the work in 0 days. man completed the work in 00 days. Work of One day of a man th part. 00 Suppose men left the work after x days. work done by 0 men in x days + work done by men in (0 x) days 0 x (0 x) x + 00 x 00 x x days 9. (b) Let M denotes man and B denotes boy. (M + B)'s day's work i.e.?? M B 0 0 Ratio of their skill 8 i.e.? 8 M B Let efficiency of a man of days work x i.e. Now, Now, x M?? B 0 8 x???? M B B 8 x?? x? M and 8

In the ratio a : b, a and b are called the terms of ratio, `a' is the antecedent and `b' is the consequent.

Ratio: The number of times one quantity contains another quantity of the same kind is called ratio of the two quantities. The ratio of a to b is written as a : b a b a b In the ratio a : b, a and b are