These are the main questions on Average topic. Important for Bank, SSC, Railways, Teacher's Exam, MAT, CAT & all other aptitudes and Govt. job exams.
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AVERAGE
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Sum of observations AVERAGE = Number of observations For example. If we have three values 1 5, 17 and 19, then the average of these three values will be (15+17+ 19) / 3 If we are finding the average of an AP (arithmetic progression), Then the average can be calculated directly by T + T MST 2 & it can also be directly calculated by middle term. For example. If we have again three values 15, 17 and 19, then the average of these three values will be (15+19) /2 — 17 (it is also the middle value)
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Example: Find the average of first 50 natural numbers. Solution: The average of first 50 natural number i,e. (1 ,2,3, 50) TI+T MST 2 Example: The average of all odd Solution: The average of all odd TI+T MST 2 1+50 2 = 51/2 = 25.5 numbers upto 100 is numbers upto 100 is 1 + 99 = 100/2 = 50 2
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Example: Find the average of first 20 multiples of 7. Solution: 1st multiple of 7 is 7 & 20th multiple is = The average of first 20 multiples of 7 TI+T MST 2 7 + 140 = 147/2 2 1 40 = 73.5 Example: The average of five consecutive odd numbers is 59. What is the difference between the highest and lowest number? Solution: Now the series is again AP as these are consecutive odd numbers so middle term is average -59 so 5 numbers are 55, 57, 59, 61, 63 So the difference between highest & lowest number = 63 - 55
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Example: The average of four consecutive even numbers is 29. The least number of these numbers is Solution: Now the series is AP as these are consecutive even numbers There are 2 middle terms and we know the average of these 2 is 29 so middle 2 numbers are 28 & 30 So all 4 numbers are 26, 28, 30, 32 Hence the least number is 26 Example. There are two sections A and B of a class consisting of 36 and 44 students respectively. If the average weight of section A is 40kg and that of section B is 35kg, find the average weight of the whole class? Solution: Total weight of(36+44) students=(36x40+44x35)kg =2980kg. Therefore weight of the total class=(2980/80)kg =37.25kg
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Example. The average marks of three batches of 55, 60 and 45 students respectively is 50, 55 and 60, then the average marks of all the students is? Solution: Total marks of(55+60+45) students =8750 Therefore average marks of the whole class =(8750/1 60) - 54.68 Example : The average of 25 results is 1 8. The average of 1 st 12 of them is 14 & that of last 12 is 17.Find the 1 3th result. Solution: Clearly 1 3th result=(sum of 25 results)-(sum of 24 results) 450-(1 = 78
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Example. Nine persons went to a hotel for taking their meals 8 of them spent Rs.12 each on their meals and the ninth spent Rs.8 more than the average expenditure of all the nine. What was the total money spent by them? Solution: Let the average expenditure of all nine be Rs.x Then or 8x=1 04 or 3. Total money spent = 9x=Rs.(9>
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CONCEPT OF NEW COMER: Example. The average age of a class of 39 students is 1 5 years. If the age of the teacher be included, then the average increases by3 months. Find the age of the teacher. Solution. Total age of 39 persons = (39 x 1 5) years - 585 years. Average age of 40 persons= 15 yrs 3 months — 61 /4 years. Total age of 40 persons = {(61 /4 )x 40} years= 610 years. Age of the teacher = (610 - 585) years=25 years. Shortcut to find age of new comer = old average +(change in average) xnew number of persons = 15 year +3month x40 = 1 5year + 40/4year = 1 5year + 10 year = 25 year
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Example. A batsman makes a score of 87 runs in the 1 7th inning and thus increases his avg. by 3. Find his average after 1 7th inning. Solution. let average runs till 16th inning = x & average runs till 1 7th inning = x +3 Total runs till 16 innings — 16x After 17 innings, total runs - = 17(x+3) or 16x +87 = + 51 or x = 87-51 -36 so his average after 1 7th inning = 36+3 - 39
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CONCEPT OF REPLACEMENT: Example. In a class there are 20 boys whose average age is decreased by 2 months when 1 boy of 18 years age is replaced by a new boy. Find the age of new boy? Solution. If boy 1 is replaced by boy2 Shortcut to find age of repalced boy = age of boyl +(change in average) xnumber of boys 18 year +(-2month )x20 1 8year - 40 month = 18 year - 3 years 4 months = 14 years 8months
Discussion
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