I this Chapter followig topics will be covered Itroductio verages ad its applicatios Mixtures ad llegatios Suppose there is a group havig so may values, imagie if we represet that etire group by a sigle value, this leads to cocept of averages. The basic formula for the average of umbers 1,,, is verage ( ) 1... So total of umbers ( Solved out problems ). Example 1: The average of 16, 18 ad 0 is 16 18 0 Solutio: 54 18 It meas if umbers are replaced by 18 there will ot be ay chage i total of umbers. 16 + 18 + 0 54 18 + 18 + 18(16+) +18 + (0 ) 0 16 + 18 18 + 0 18 0 18 Fid the umber, which adds to above umbers so that average will icrease by. 18 + 0 18 + 0 18 + 0 So 4 th umber will be, 0 + 6 (Here, 0 is the ew average ad times added to above umbers). Or after addig oe value total of four umbers will be 0 4 80. Before, addig it total of three umbers is 18 54. Fourth umber is, 80 54 6. Example : The average of two umbers is 7.5 ad square root of their product is 6. What are the umbers? (S.S.C 01) 10 ad 5 B 1 ad C 9 ad 6 D 1 ad a b Solutio: 7.5 ad ab 6 ll the optios give average of 7.5 but Optio B satisfies secod coditio i.e. 1 6. Weighted verage Suppose there are two sectios ad B. verage marks of sectio is 60 ad average marks of sectio B is 80. Fid the average marks of etire studets i the two sectios. Here, two separate sectios we have, verage marks Total marks of etire class Total umber of studets Without kowig umber of the studets i each class, we caot solve above problem. ssume there are 0 studets ad 0 studets i sectio ad B respectively, So total marks i a sectio 60 0 Total marks i a sectio B 80 0 60 0 800 verage of combied class 0 0 7. So wheever we have two (or) more groups, whose idividual averages are kow, we eed to fid average (weighted average) of all the groups together. ssume there are total groups, each group havig x1, x, x... x elemets with averages 1,.. respectively. Weight average is x x x... x 1 1 x x x... x 1 Example : 0 boys with average weight 50 kgs ad 0 girls with a average weight 60. Fid the average weight of whole class? 050 060 Solutio: 56 0 0 Example 4: The average of a batsma i 15 iigs is 0, i the ext iigs he scored 6 rus, his ew verage is?
Solutio: Method 1: Total rus 150+6 New verage (15 1) 16 Method : Take the differece betwee ew score ad old verage i.e. 6 0 Now this is spread over 16 iigs so 16, ew verage is 0 +. pplicatios of verages Numbers: The sum of 1 st atural umbers is 1, verage is 1 1. verage of 1 st 1 atural umbers is 1 1 11. verage of 1 st 50 atural umbers? swer: 5.5 We kow that the sum of 1 st eve 1 cosecutive atural umbers is 1 verage is 1 The sum of 1 st odd cosecutive atural umbers is, verage is Example 5: Fid the average of 1 st 50 eve umbers ad 1 st 50 odd umbers? Solutio: Eve umber average is 50 + 1 51. 1 st 50 odd umbers average is 50 I Time ad Speed Total distace travelled verage speed. Total time take Problem 8: perso travels at 60kmph o the jourey from P to Q ad returs at 40 km ph. fid his average speed for the jourey? Solutio: Let the distace betwee P ad Q is d. Total distace travelled is d + d d Total time d d take is (time d/s) 60 40 d 48. d d 60 40 Short-cut: If the distace is costat (or) same, verage speed is ab (a ad b are speeds). a b I Mixtures: Example 6: 6 kg of rice costig rs.1 mixed with 8 kg of rice costig rs.10 fid the average cost of the mixture? Solutio: Cost of 6 kg rice 1 6 7 Cost of 8 kg rice 8 10 80 Total 14 kg of rice rs.15 verage 15 14 rs.10.85. ges If the average age of a group of persos is y years the after years their average age is y years, years ago average age would have bee y years. Example 7: The average age of 19 childre i a school is 0. Whe teacher also icluded with age 55, what is the ew average age? Solutio: Method 1: 190+55 New average is 1.5. 19 1 Method : Differece is 55 0 5. This 5 spread over 0 people 5/0 1.5. verage icreases by 1.5, 0 + 1.5 1.5. Example 8: The average age of 0 studets is 1 years fid the age of teacher, if he icluded average is icreased by 0.5 years? Solutio: Method 1: ge of the teacher is 1 1.5 0 1 7.5 years. Method : ge of the teacher is 1.5 + 0 0.5 1.5 + 15 7.5 years. llegatios ad Mixtures Itroductio The allegatio rule is very useful i problems o weighted averages ad mixtures. Just recollect weighted averages formula, if there are two
groups havig x1elemets with average 1 ad x elemets with average respectively, weighted average is x1 1 x x x 1 x 1 x x 1 1 x x x 1 1 x 1 x 1 bove equatio is called the allegatio equatio. Graphical represetatio, 1 : 1 Example 9: Two sectios i a school havig average of 0 marks ad 0 marks respectively, o combiig them the resultat group have average marks of 6. Fid the ratio of the studets i two sectios? x1 0 6 Solutio: 4 x 6 0 6 Graphical represetatio, 0 0 6 4 : 6 : Example 10: O combiig the two classes studets havig average marks 65 ad 80 respectively, the overall average obtaied is 70. Fid the umber of studets i the secod class if first class has 50 studets? Solutio: Ratio of the studets i two classes : 1. The, studets i the secod class are (½) 50 5. pplicatios of llegatios I verage price of goods: perso buys 0 kg of rice at rs.40 kg ad 10 kg of rice at rs.5 what is the average price? 0 10 x 5 40 x, x 5. I Percetages: sales ma makes a profit of 40% by sellig 0% of his articles ad a profit of 5% by sellig 10% of his articles. Fid et percetage profit? 0 10 x 5 x 5% 40 x I verage speed: car travels at 40 km/hour for 0 km ad 5 km/hour for 10 km. fid the average speed of the car i etire jourey? swer: 5 km/hour I Mixtures: Milk cotaier cotaiig 0 liters is mixed with 40% of water ad milk cotaier B cotaiig 10 liters is mixed with 5% of water. It we combies both the mixtures fid the percetage of water i fial mixture? 0 10 x 5 x 5% 40 x I Profit ad Loss: How may kgs of rice costig rs.6 per kg must be mixed with 0 kgs of rice costig rs.4. per kg, so that 40% gai may be obtaied by sellig the mixture at rs.7 per kg? Solutio: You eed to cosider cost price ot sellig price so Sellig price 140% X cost price. Cost price 7/1.4 rs.5 6 4. 0.8 : 1 So, there are i the ratio 0.8: 1 x: 0. 0.8 1, 8 10 x, x 4 0 5
Mixture of Liquids: Glass cotais milk ad water i the ratio : 4, Glass B cotais milk ad water i the ratio 4: 1, i what ratio they should mix so that total mixture have milk ad water i the ratio :? Solutio: Milk i glass /7, milk i glass B 4/5 ad milk i a mixture is /5. /7 4/5 x x y 4 / 5 / 5 / 5 / 7 /5 : y 1/ 5 6 / 7 6 Simple Iterest: perso has rs.5000. He ivests a part of it at % per aum ad rest of the amout at 8% per aum at simple iterest. His total icome i years is rs.900. Fid the sum iterested at differet rates of iterest? Solutio: Rs.900 gaiig i years, it meas i 1 year it 900/ 00. PTR 900, R 6%. 100 % 8% 6% 8-6 : 6- : Therefore, amout o %, (/5) 5000 000. mout o 8%, /5 5000 000. Example 11: liters of milk are draw from a cask cotaiig 0 liters of milk ad it replaced by water. The above process is repeated times. 1. Fid the fial quatity of milk left i the cask.. Fid the ratio of milk to water fially. Solutio: 1. Milk left i the cask is Capacity x 1 fractio of milk withdraw Number of process times. 01 1 0.1 0. Ratio of milk to water is, 0 0 0.9 th milk left after operatio th water left after operatio Milk becomes 0.79 (or 7.9%) times of total quatity ad remaiig is water that equals 1 0.79 0.71 (or 7.1%) times of total quatity of cask. 1 0 11 0 Practice set -1 0.9 1 0.9 0.79 0.71. 1. The average marks of all the studets i three sectios, B ad C i a school are 0, 40 ad 50 marks respectively. The umber of studets i these three sectios is 40, 60 ad 80 respectively. Fid the average marks of etire class? 6. 5 B 40 C 4. D 5. The average of a batsma i 15 iigs is 0, after he scored some rus i 16 th iigs, average is icreased by rus, fid the rus i 16 th iigs? 78 B 64 C 80 D 96. The average of a batsma after 0 iigs was 5 rus. If after 1 st iigs, average icreased by rus, fid rus i 1 st iigs? 110 B 10 C 14 D 115 4. The average age of jury of 5 is 40. If a member aged 5 resigs ad a ma aged 5 becomes a member, the the average age of the ew jury is (S.S.C 01) 0 B 8 C 40 D 4 5. 50 boxes with equal weights were loaded i a ship. 5 more boxes each weightig 105 kg were later added, makig the average weight of all the 55 boxes as 95 kg. The weight of each of the 50 boxes first loaded is (S.S.C 01) 95 kg B 9 kg C 98 kg D 94 kg 6. motor car goes to a place at a speed of 0 km/hr ad returs from there at a speed of 0 km/hr. The average speed (i km/hr) of the car i the etire jourey is (S.S.C 01) 4 B 4.5 C 6 D 5
7. Two kids of apples which cost rs.0 per kg ad rs.0 per kg are mixed i the ratio : 4 the fid the cost of the mixture per kg? 6 B 5.71 C 5.50 D 4.75 8. The temperature recorded i a city for the first days is o c, o 4 th day4 o c, o 5 th day 8 o c, o 6 th day 0 o c fid the average temperature of all the six days?.5 B C 1. D 9. The average age of 9 studets is 1 years, whe 4 years old teacher icluded, fid ew average? 1.5 B 14 C 1.5 D 1 10. The average age of 9 studets is 1. If the age of the teacher is also icluded the average age of the class becomes 1.6. Fid the age of teacher? 45 B 44 C 48 D 50 11. The average age of a group of 0 boys is 1 years. Whe ew boys joi them, the average icreases by ¼ year. What is the average age (i years) of the two ew boys? (S.S.C 01) 14 B 16 C 1 D 1 1. The average age of Navee, Kittu, viash, Mittu ad Siddu is 0 years. The average age of Mittu ad Kittu is 5 years. What is the average age of Navee, viash ad Siddu? 1. B C D 4 1. The average age of wife ad husbad, who were married 8 years ago, was 0 years. Icludig a child preset average age of the family is 1. Preset age of child? 6 B C 5 D 7 14. Five years ago average age of the family of five members was 7 years. Icludig a child preset average age is same today. What is the age of child ow? B C 4 D 5 15. The average age of ram ad his two childre is 17 years ad the average age of ram s wife ad same childre is 16 years. If the age of ram is years, the age of his wife is (i years) (S.S.C 01) 1 B C 5 D 0 16. The average age of ad B is 0 years. If is to be replaced by C, the average would be 19 years. The average age of C ad is 1 years. The ages of, B ad C i order (i years) are (S.S.C 01), 18, 0 B, 0, 18 C 18,, 0 D Noe 17. The average age of studets i a class is 0 years. If 4 ew studets with a average 4 years joied i the class, ow average of the etire class becomes 7 years. How may studets were there iitially? B 4 C 6 D 10 18. The average of three umbers, B ad C is 7, ad average of B, C ad D is 4 if D is equal to 1/ rd of, what are the values of ad D? 9, B 1.5, 4.5 C 1, 4 D Noe 19. Three years ago, the average age of, B ad C was 7 years, ad that of B ad C five years ago was 0 years. s preset age is (S.S.C 01) 40 Years B 48 Years C 0 Years D 5 Years 0. If 0 liters of 40% milk solutio is mixed with 0 liters of 5% milk solutio, the fid the cocetratio of milk i the resultig solutio (i %)? 5% B % C 4% D 6% 1. The average marks i a class of 0 studets are 4. What will be the average marks of the rest of studets, if the average marks of the 0 studets are 45? B 5 C 8 D 6. perso was asked to calculate the arithmetic mea of 10 positive itegers, each of which had digits. By mistake, he iterchaged the digits, say x ad y, i oe of these 10 positive itegers. s a result, his aswer for the arithmetic mea was 1.8 more tha what it should have bee. The y x is: B C 4 D Noe. The average of first three umbers is double of the fourth umber. If the average of all the four umbers is 1, fid the 4 th umber. (S.S.C 01) 16 B 48/7 C 0 D 18/7 4. The mea of 100 items was 46. Later o it was discovered that a item 16 was misread as 61 ad aother item 4 was misread as 4. It was also foud that the umber of items was 90 ad ot 100. The what is the correct mea? (S.S.C 01) 5 B 5.7 C 50 D 50.7
5. perso divides his total route of jourey ito three equal parts ad decides to travel the three parts with speeds of 40, 0 ad 15 km/hr respectively. Fid his average speed durig the whole jourey? B 0 C 4 D 8 6. There were 5 studets i a hostel. If the umber of studets icreases by 7, the expeces of the mess icrease by Rs.4 per day while the average expediture per head dimiishes by Rs.1. Fid the origial expediture of the mess? 40 B 400 C 480 D 540 llegatios 7. Two verities of sugar at rs.5 per kg ad rs.6 per kg are mixed together i the ratio1: fid the average price of the mixture? 5. B 6.66 C 6 D 5.66 8. I a club, the average age of the members is 0 years, the average age of the male members is 4 years ad that of the female members is 6 years. The percetage of the male members is (S.S.C 01) 0% B 40% C 50% D 60% 9. I a automated plat assembly lie, the rate of rejectio of compoets was 10% o July 1 st ad 6% o July d. The combied rate of rejectio for the two days was 9%. The ratio of productio volumes o July 1 st ad o July d is: 1: B : 1 C : 1 D 1: 0. There are some cows ad hes. If there are total 88 heads ad 6 legs. Fid the differece betwee cows ad hes? 6 B 5 C 5 D 8 1. I a school with 600 studets, the average age of the boys is 1 years ad that of girls is 11 years. If the average age of the school is 11 years ad 9 moths, the the umber of girls i the school is (S.S.C 01) 50 B 50 C 450 D 150. Two vessels cotai milk ad water i the ratio : ad 7:. Fid the ratio i which the cotets of the two vessels have to be mixed to get a ew mixture i which the ratio of milk ad water is : 1? (S.S.C 01) 4: 1 B 1: 4 C : 1 D 1:. From a vessel cotaiig 100 l of wie, 10 l are draw out ad a equal amout of water is added. From the mixture, 10 l is agai draw out ad same quatity of water is added. What is the fial ratio of water ad wie? 81: 100 B 81: 19 C 19: 81 D Noe 4. 8 liters are draw from a cask full of wie ad is the filled with water. This operatio is performed three more times. The ratio of the quatity of wie ow left i cask to that of the water is 16: 65. How much wie did the cask hold origially? 18 liters B 4 liters C liters D 4 liters 5. Tea worth of rs.16/kg ad rs.15/kg are mixed with third variety i the ratio 1: 1:. If the mixture is worth of rs.15/kg, fid the cost of third variety? 10.5 B 17.5 C 175.5 D 150 6. verage salary of all the workers i a workshop is rs.8000. The average salary of 7 techicias is rs.1000 ad the average salary of the rest is rs.6000. The total umber of workers i the workshop is? 14 B 7 C 1 D Noe 7. total amout of Rs.04 was collected from boys ad girls of class of 150 studets. If each boy cotributes Rs.1 ad each girl cotributes Rs.1.50, the fid the umber of girls i the class? 4 B 64 C 105 D 108 Key Practice set -1 1. C.. D 4. B 5. D 6. 7. B 8. C 9. D 10. 11. B 1. 1. D 14. 15. D 16.
17. B 18. B 19. 0. C 1. D.. B 4. D 5. C 6. 7. D 8. C 9. B 0. D 1. D. D. C 4. B 5. C 6. C 7. D Practice set 1. The average score of Sehwag i 10 iigs was 77 rus. I the 11 th iigs he had scored zero rus. The average score of Sehwag decreased by after 11 th iigs: () 7.7 (B) 7 (C) 6 (D) 5. shopkeeper mixes two verities of sugar costig rs 5/Kg ad rs 0/Kg i a certai ratio such that the cost of the mixture is rs /Kg the fid the ratio i which the types of sugar were mixed? : B : C 1: 1 D : 1. Cosider a sequece of seve cosecutive itegers, the average of first five itegers is. The average of all seve itegers is: () (B) +1 (C) + (/7) (D) Noe 4. The average age of 10 studets i a class is icreased by year whe two studets aged 1 year ad 14 year are replaced by girls. Fid the average age of the two girls? 5 B 46 C D 5. The average of 9 umbers is 11. If each of these 9 umbers is multiplied by 5 ad the 5 is added to each of these resultat umbers, the the ew average is: () 45 (B) 0 (C) 55 (D) 60 6. verage percetage scored by Nadii i her seve semesters is 78. How much she must score i her eighth semester so as to have her overall average as 80? 80 B 9 C 94 D Noe 7. Varu mixed 50 Kgs of sugar costig 0 rs/kg with 70 Kgs of sugar costig rs/kg. Fid the average cost of the mixture? 9 B 0 C 7 D Noe 8. The average score of Sehwag after 48 iigs is 48 ad i the 49 th iigs Sehwag scores 97 rus. I the 50 th iigs the miimum umber of rus required to icrease his average score by rus tha it was before 50 th iigs: () 99 (B) 149 (C) 151 (D) 150 9. Whe a studet weighig 45 kgs left a class, the average weight of the remaiig 59 studets icreased by 00g. What is the average weight of the remaiig 59 studets? 56.8 B 57 C 56 D 59 10. The average of all the prime umbers ad composite umbers upto 100 is: () 51 (B) 50.50 (C) 49.50 (D) 55 11. verage cost of 5 apples ad 4 magoes is rs.6. The average cost of 7 apples ad 8 magoes is rs.48. Fid the total cost of 4 apples ad 4 magoes 1044 B 088 C 1056 D 110 1. I a village the average age of people is 4 years. But after the verificatio it was foud that the age of a perso had bee cosidered 0 years less tha the actual age, so the ew average, after correctio, icreased by 1, the value of is: () 19 (B) 0 (C) 1 (D) 1. The average price of 10 books is Rs.1 while the average price of 8 of these books is Rs.11.75. Of
the remaiig two books, if the price of oe book is 60% more tha the price of the other, what is the price of each of these two books? 10 ad 8 B 10 ad 16 C 8 ad 14 D Noe 14. The average of all the perfect squares upto 100 is: () 1000 (B) 100 (C) 8.5 (D) 85 15. There are two vessels ad B i which the ratio of milk ad water are as 5: ad 8:7 respectively. Two gallos are draw from vessel ad gallos from vessel B, ad are mixed i aother empty vessel. What is the ratio of milk ad water i it? 106: 69 B 10: 7 C 89: 86 D Noe 16. The average temperature of all 7 days of the week is 0, that of the first days is 8 ad of the ext days is 1. What is the temperature o the last day of the week? B C 7 D 6 17. The average age of 0 studets is 0 years. Whe the average age of class teacher also icluded, the average age of the whole class icreases by 1 year. The age of the class teacher is: () 1 (B) 60 (C) 61 (D) 65 18. The average weight of 15 persos is icreased by kg whe i place of a perso havig weight 70 kg aother ew perso is icluded. The weight of the ew perso is: 100 kg B 98 kg C 10 kg D oe 19. The average of 7 cosecutive odd umbers if the smallest of those umbers is deoted by k: () k + 7 (C) k + 6 (B) k + 5 (D) 7k 0. The average of a, b ad c is 79 ad the average of a adc is also 79. The the value of b is: () 0 (B) 79 (C) -79 (D) 100 1. The average of 6 observatios is 1. ew seveth observatio is icluded ad the ew average is decreased by 1. The seveth observatio is: 5 B 7 C 6 D 11. The average of marks obtaied by 10 cadidates i a certai examiatio is 5. If the average of passed cadidates is 9 ad that of the failed cadidates is 15. The umber of cadidates who passed the examiatio is: 150 B 10 C 110 D Noe. The average weight of liquid i 100 bottles is 500 gm. The total weight of all the bottels is 0 kg. The average weight of a bottle with liquid is: ().65 kg (B) 0.7 kg (C) 70 ml (D) Noe 4. The average age of three childre of Satosh is 1 years ad their ratio of ages is : 4: 5. The average age of the yougest ad eldest child is if he had oly childre: () 10 (B) 4 (C) 9 (D) 1 5. The average weight of 0 cars is 180 kg. If a old Car is removed from this group of four wheelers, the ew average becomes 178 kg. The weight of the removed Car is: () 0 (B) 18 (C) 18 (D) 15 6. The average of ay 5 cosecutive odd atural umbers is k. If two more such umbers, just ext to the previous 5 umbers are added, the ew average becomes: () k + (C) k + 4 (B) k + 5 (D) k + 7. The average icome of, B, ad C is Rs.1, 000 per moth ad the average icome of B, C ad D is Rs. 15, 000 per moth. If the average salary of D is twice that of, the the average salary of B ad C is (i Rs): () 1, 500 (C) 18, 000 (B) 8, 000 (D) 9, 000 8. The average of the first five prime umbers greater tha 0 is:
1.01 B.0 C.00 D Noe 9. The average mothly salary of all employees i a factory is rs.600 ad that of 16 officers is rs.000. If the average mothly salary of workers is rs.550, the umber of workers, leavig aside officers, i the factory is? 744 B 768 C 84 D Noe 0. The average of 7, 14, 1, 8, 77 is: () 11 (B) 66 (C) 4 (D) 66 1. The average weight of all the 11 players i Idia cricket team is 50 kg. If the average weight of first six lightest weight players is 49 kg ad that of the six heaviest players is 5 kg. The average weight (i kgs) of the player which lies i the sixth positio i the list of players whe all the 11 players are arraged i the order of icreasig or decreasig weights: () 55 (B) 56 (C) 51 (D) 5. Two vessels ad B cotai milk ad water mixed i the ratio 8: 5 ad 5: respectively. The ratio i which these two mixtures be mixed to get a ew mixture cotaiig milk ad water i the ratio 9: 4? : 7 B : 5 C 5: D 5: 7. Two classes took the same test. Oe class of 0 studets made a average grade of 80%, the other class of 0 studets made a average grade of 70%. The average grade for all studets i both classes is: 75% B 74% C 78% D 79% 1. B 1. B 14. C 15. 16. 17. C 18. 19. C 0. B 1.. D. B 4. D 5. C 6. 7. 8. B 9. D 0. C 1. B.. B Wishig you a good luck for all the aspirats of competitive exams. Regards SPOORTHY STUDY CIRCLE Hyderabad Mob: 900097690 Practise Set - KEY 1. B. B. B 4. C 5. D 6. C 7. C 8. B 9. B 10. 11. B