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AVERAGE

Examples

1. A batsman has a certain average runs for 16 innings. In the 17th inning he made a score of 85 runs thereby his average is increased by 3. What is his average after 17th inning?Examples

Sol: The average for 17th inning has been increased by 3.
 The total increase in the runs for 17th inning = 17 × 3 
                                                                        = 51
 But the batsman scores 85.
 Average runs in his 16th innings = 85 – 51 
                                                     = 34.
 Hence the average of runs after 17th innings = 34  + 3 
                                                                         = 37

 

2.  A man has 7 children. When their average age was 12 years, the child who was 6 years of age, died. What was the average age of surviving children 5 years after the death of the above child?
Sol: Average age of 7 children  = 12 years
       Total age of 6 children  =  12 × 7 = 84 years
       Total age of 6 children after the death of a child aged 6 years = 84 – 6 = 78
       Hence   the average age of the surviving children =  
       After 5 yrs. = 13 + 5 = 18 yrs.

 

3.  If the weights of 5 students of a class are 49.6 kg, 39.8 kg, 45.2 kg and 24.6 kg respectively then what is their average weight?
Sol: Total weight of 5 students = 49.6 + 39.8 + 40.8 + 45.2 + 24.6
                                                 = 200 kg.
     ∴    Their average weight = 
                                           = 40 kg.

 

4.  The average temperature for Monday, Tuesday and Wednesday was 36oC. The average temperature for Tuesday, Wednesday and Thursday was. If the temperature for Thursday was 37oC, what was the temperature on Monday?
Sol:   Average temperature for Monday, Tuesday and Wednesday = 36oC
              Total temperature for Monday, Tuesday and Wednesday = 36 × 3 
                                                                          = 108oC
        ∴   Average temperature for Tuesday, Wednesday and Thursday = 38oC
           ∴    Total temperature for Tuesday, Wednesday and Thursday = 38 × 3 
                                                                            = 114oC  
             ∴    Total temperature for Tuesday and Wednesday only = 114 – 37 
                                                                       =  77oC
             ∴    Temperature for  Monday only = 108 – 77 
                                                = 31oC

 

5.  A train covers the first 16 km at a speed of 20 km per hour another 20 km at 40 km per hour and the last 10 km at 15 km per hour. Find the average speed for the entire journey.

 

6.  A vehicle travels from A to B at the speed of 40 km/hr, but from B to A at the speed of 60km/hr. what is its average speed during the whole journey?
Sol: Let the distance from A to B be x km
      
       

7.  The average age of a class of 40 boys is 16.95 years. A new boy joins the class and the average age now is 17 years. What is the age of the new boy?
Sol: The average age of 40 boys  = 16.95 years
             Total are of 40 boys  = 16.95 × 40
                                        = 678 years
        The average age of 41 boys  = 17 years
              Total age of 41 boys  = 17 × 41
                                       = 697 years
             Age of the new boy  = 697 – 678
                                      = 19 years
                                                       

The average of the given quantities is calculated by dividing the sum of the quantities by their number.

If the average of ‘x’ quantities is ‘a’ and average of ‘y’ quantities is ‘b’, then average of x and y is

  
Weighted Average
Let x1, x2, x3, …..xn be the quantities and w1, w2, w3, ….wn be the weights attached to them then Weighted Average

 

 SOLVED EXAMPLES

Example 1. A cricketer has completed 18 innings and his average is 26.5 runs. How many runs must he make in his next innings so as to raise his average to 27?
(1) 36           (2)46            (3)38            (4) 40            (5) None of these
Explanation:
(1)   Formula = n(y – x) + y
        n = 18 
        x = 265 
        y = 27 
        = 18(27 – 26.5) + 27 
        = 9 + 7 
        = 36

 

Example 2. The average of 13 numbers is 30. The average of 1st 7 of these numbers is 32 and the last 7 of these numbers is 22. Find the 7th number.
(1) 15         (2) 16          (3) 12          (4) 20          (5) None of these
Explanation:
(3)    Total number = 13 × 30 = 390 
          1st 7 numbers = 7 × 32 = 224 
          Last 7 number = 7 × 22 = 154  
          7th number = 390 – (224 + 154) 
          = 12

 

Example 3. The average of marks obtained by 65 candidates in a certain examination is 25. If the average marks of passed candidates is 27 and that of the failed candidates is 14, what is the no. of candidates who passed the examination?
(1) 53            (2) 64            (3) 55            (4) 70            (5) None of these
Explanation:
(3)  Passed candidates = x 
        Failed candidates = 65 – x 
        Total marks ∴   25  ×  65 = 27x + (65 – x)14 
      ⇒ 1625 = 27x + 910 – 14x 
      ⇒ 13x = 715  
      ⇒ x = 55 

 

Example 4. In a class there are 30 students whose average is decreased by6 months, when 4 students aged 16, 17, 18 and 19 years respectively are replaced by the some no. of students. Find the average of the new students.
(1) 16.25               (2) 15.25               (3) 16               (4) 15.75               (5) None of these



Example 5. The average age of 11 persons in a committee is increased by 2 years when three men aged 32 years, 33 years and 34 years are substituted by three women. Find the average age of these three women.
(1) 40 1/3        (2) 40          (3) 41 1/3        (4) 41 years          (5) None of these



Example 6. The average of 40 numbers is 405. If each of the numbers is divided by 15. Find the average of new set of numbers.
(1) 27          (2) 28          (3) 21          (4) 26          (5) None of these

Explanation:

405/15 = 27

 

Example 7. The average age of 80 boys in a class is 15. The average age of a group of 15 boys in the class is 16 and the average age of another 25 boys in the class is 14. What is the average age of remaining boys in the class. (approx.)
(1) 20          (2) 15          (3) 18          (4) 21          (5) None of these
Explanation:
(2)    Remaining boys average age = x
        ⇒ 15 × 16 + 25 × 14 + 40x = 80 × 15 
        ⇒ 240 + 350 + 40x = 1200 
        ⇒ 40x = 1200 – 590 
        x = 610/40 
         x = 15. 25

 

Example 8. A batsman in his 12th innings makes a score of 63 runs and there by increases his average score by ‘2’. The average of his score after 12th innings is.
(1) 39        (2) 41         (3) 37         (4) 45         (5) None of these
Explanation:
(2)    Average score = x till 11 innings 
         Total runs = 11x 
         Total runs after 12th inning = 11x + 63 
         Average after 12th inning = ( x + 2) 
         Total run = 12 ( x + 2) 
         12 ( x + 2) = 11x + 63 ⇒ x = 63 – 12 × 2 = 39 
         Average of his score = 39 + 2 = 41

 

Example 9. A certain amount was to be distributed among the A, B and C in the ratio 2 : 3 : 4 but was erroneously distributed in the ratio 7 : 2 : 5 as a result of this ‘B’ received Rs. 40 less. What is the actual?
(1) 210         (2) 270         (3) 230         (4) 280         (5) None of these

Example 10. Two number are respectively 20% and 50% more than a third number. The ratio of the two numbers is.
(1) 2 : 5         (2) 4 : 5        (3) 3 : 5        (4) 6 : 7        (5) None of these

 PRACTICE EXERCISE

1. The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that  of the failed candidates is 15. What is the number of candidates who passed the examination?
(1) 90        (2) 85         (3) 100         (4) 120         (5) None of these
Ans: 100

 

2. The average salary of all workers in workshop is 8000. The average salary of 7-technicians is Rs. 12000 and the average salary of the rest is Rs. 6000. The total number of workers in the work shop is.
(1) 20         (2) 21         (3) 23         (4) 22         (5) None of these
Ans: 21

 

3. In a competitive examination, the average marks obtained was 45. It was later discovered that there was some error in computerization and the marks of 90 candidates had to be changed from 80 to 50, and the average come down to 40 marks. The total number of candidates appeared in the examination.
(1) 500         (2) 600         (3) 540         (4) 580         (5) None of these
Ans: 540

 

4. The average weight of A, B, C is 84 kg. If D joins the group the average weight of the group becomes 80 kg. If another man ‘E’ who weights is 3kg more than D replace A, then average of B, C, D and E becomes ‘79’ kg. What is the weight of ‘A’
(1) 64        (2) 72         (3) 75         (4) 100         (5) None of these
Ans: 75

 

5. 1/3rd of certain journey is covered at the rate of 25kmph. 1/4th at the rate of 30 kmph and the respect 50 kmph. Find the average speed for the hole journey .
(1) 33.33 kmph         (2) 36 kmph         (3) 42 kmph         (4) 27 kmph         (5) None of these
Ans: 33.33 kmph        

 

6. Deepak’s opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Deepak and he thinks that Deepak’s weight is greater than 60 kg but less than 70 kg. His mother’s view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Deepak.
(1) 67 kg         (2) 68kg         (3) 69 kg         (4) 70 kg         (5) None of these
Ans: 67 kg        

 

7. The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that of the failed candidates is 15. What is the number of candidates who passed the examination?
(1) 90         (2) 85          (3) 100          (4) 120          (5) None of these
Ans: 100

 

8. The average age of Raju & Aditya is 35 years. If Raghu replaces Raju. The average age becomes 32 years and if Raghu replaces Adithya then the average becomes 32 years. If the average age of Naveen and Ramu behalf of the average age of Raju, Aditya and Raghu. Then the average age of all the people-?
(1) 28          (2) 32          (3) 35          (4) 30          (5) None of these
Ans: 28

 

9. The average salary is being paid to all it’s employees by a company is 15,500. The average salary of the senior employees is 18000 per month and the average salary of the junior employees is 12000 per month. If there are only two levels of employees viz junior and senior levels, then what fraction of the total employees is the too junior level employees are
(1) 7/5    (2) 7/12       (3) 5/12       (4) 5/7         (5) None of these
Ans: 7/12

 

10. The average age of ‘8’ persons in a committee is increased by 2 years when two men aged 35 years and 45 years are substituted by two women. Find the average age of these two women.
(1) 48           (2) 36          (3) 42          (4) 29          (5) None of these
Ans:  48

 

The average of the given quantities is calculated by dividing the sum of the quantities by their number.
Average = Sum of the quantities / Number of quantities 

Weighted average: Let x1, x2, x3, ...... xn be the
quantities and w1, w2, w3, ...... wn be the weights attached to them
then Weighted Average = w1x1 + w2x2 + w3x3 + ........ + wnxn / w1 + w2 + w3 + ........ + wn 

 

MODEL QUESTIONS
1. The average of seven numbers is 18. The average of first three numbers is 14 and the average of last three numbers is 19. What is the middle number?
A) 9           B) 27           C) 32           D) 35

2. The average of 17 results is 60. If the average of first 9 results is 57 and that of the last 9 results is 65. What is the value of 9th result? (SSC-CGL – 2017)
A) 39         B) 78         C) 117         D) 156

3. In a class, the average age of 23 boys is 12 years and the average age of 17 girls is 11 years. What is the average age of the whole class?
A) 11.5 years      B) 11.245 years     C) 11.575 years      D) 11.525 years

4. In a class, the average score of thirty students on a test is 69. Later it was found that the score of one of the students was wrongly read as 88 instead of 58. The actual average score is (SSC-CGL – 2019)
A) 71           B) 78           C) 65           D) 68

5. The average of 23 numbers is 28. Later, it was pointed out that value of one the numbers was wrongly taken as 78 instead of 87. What is the correct average?
A) 31.25     B) 27.15     C) 28.39     D) 32.65

6. Out of 6 numbers the sum of the first 5 numbers is 7 times the 6th number. If their average is 136, then the 6th number is.... (SSC-CGL – 2019)
A) 102         B) 96         C) 84         D) 116

7. A cricketer’s average for 11 matches is 25 runs. In the next four matches he scored 72, 34, 51 and 48 runs. Find his new average for the 15 matches?
A) 39          B) 32          C) 36          D) 28

8. The average age of a class of 24 students is 11 years. If the teacher’s age is also included, the average increases by one year. Find the age of the teacher?
A) 35           B) 39           C) 34           D) 36

9. The average age of six members of a family is 20 years. If the age of the servant is included, then the average age increase by 25%. What is the age of the servant (in years)? (SSC-CGL – 2017)
A) 30           B) 55           C) 50           D) 35

10. The average weight of 8 persons increases by 1.5 kg when a person weighting 65 kg is replaced by a new person. What could be the weight of the new person?
A) 55 kg     B) 57.5 kg     C) 75 kg    D) 77 kg

11. The average of twelve numbers is 45.5. The average of the first four numbers is 41.5 and that of the next five numbers is 48. The 10th number is 4 more than the 11th number and 9 more than the 12th number. What is the average of the 10th and 12th number? (SSC-CGL – 2019)
A) 46.5         B) 46         C) 47.8         D) 47

12. 5 kg sugar is purchased at Rs.10 per kg, 6kg at Rs.11 and 9kg at Rs.12 per kg. What is the weighted average price of sugar per kg?
A) 8.8        B) 10.8        C) 11.2        D) 11.8

13. The average of 12 numbers is 9. If each number is multiplied by 2 and added to 3, the average of the new set of numbers is (SSC-CGL – 2016)
A) 9           B) 18           C) 21           D) 27

14. In a class, average height of all students is 'a' cms. Among them, average height of 10 students is 'b' cms and the average height of the remaining students is 'c' cms. Find the number of students in the class. (Here a > c and b > c) (SSC-CGL – 2016) 


15. The average weight of some students in a class is 68.5 kg. If four new students of 72.2 kg, 70.8 kg, 70.3kg and 66.7 kg are enrolled in the class, the average weight of students increases to 300 g. Initially how many students were there in the class? (SSC-CGL – 2018)
A) 11          B) 26          C) 21          D) 16

 

ANSWERS: 1-B; 2-B; 3-C; 4-D; 5-C; 6-A; 7-B; 8-D; 9-B; 10-D; 11-A; 12-C; 13-C; 14-D; 15-D.

Explanations
1.
The total of seven numbers = 7 × 18 = 126
The total of first 3 and last 3 numbers is = 3 × 14 + 3 × 19 = 99
∴ Middle number is (126 − 99) = 27

Shortcut: 3 × (−4) + 3 × (+1) = −12 + 3 = −9
∴ Middle number = 18 + 9 = 27

2. The total of 17 numbers = 17 × 60 = 1020
The total of first 9 and last 9 numbers is = 9 × 57 + 9 × 65 = 1098
∴ Middle number is (1098 − 1020) = 78

Shortcut:  9 × (−3) + 9 × (+5) = −27 + 45 = +18
∴ Middle number = 60 + 18 = 78

3. Total age of 40 students = (23 × 12 + 17 × 11) = 463
Average age = 463/40 = 11.575 years

Shortcut: Let all the students average be 11 years
Extra age by all boys = 23 × 1 yr. = 23 years
Extra average for all 40 students = 23/40 = 0.575
∴ Actual Average = 11 + 0.575 = 11.575

4. The Correct Average 

Shortcut: 
Additional score taken = 88 − 58 = 30
Additional average = 30/30 = 1
∴ New average is 69 − 1 = 68 

5. Correct Average 

Shortcut: Value less taken = 87 − 78 = 9
Average less taken = 9/23 = 0.39
∴ New average is 28 + 0.39 = 28.39

6. Let the 6th number be ‘X’ ∴ 7x = (136 × 6) − x
⇒ x = 102

7. Total runs in 11 matches = 11 × 25 = 275 runs
Total runs in the next 4 matches
= 72 + 34 + 51 + 48 = 205 runs 

= 32 runs

Shortcut: Let the average of all 15 matches be 25 Extra runs in 4 matches
= (72 − 25) + (34 − 25) + (51 − 25) + (48 − 25) = 105
Extra Average = 105 /15 = 7
∴ New Average = 25 + 7 = 32

8. Total of 24 students is 24 × 11 = 264 years
Total age with teacher is 25 × 12 = 300 years
∴ Teacher’s age is 300 − 264 = 36 years

Shortcut: Teacher’s age = 11 + (25 × 1) = 36 years

9. Total age of the family is 6 × 20 = 120 years
Average age after servant’s inclusion
= 125% of 20 = 25 years
Total age with servant = 7 × 25 = 175 years
∴ Servant’s age is 175 − 120 = 55 years

Shortcut: Servant’s age = 20 + (7 × 5) = 55 years

10. Let the average weight of 8 persons be ‘x’ years and the age of the new person be ‘y’ years

Shortcut: 65 + 8 × 1.5 = 77 kg

11. The total of 12 numbers = 12 × 45.5 = 546
The total of first 4 and next 5 numbers is
= 4 × 41.5 + 5 × 48 = 406
Total of 9th, 10th & 11th numbers = 546 − 406 = 140
Let the 10th number be x
∴ x + x − 4 + x − 9 = 140
⇒ x = 51 

Shortcut: 4 × (− 4) + 5 × (+ 2.5) = − 16 + 12.5 = − 3.5
∴ Total of 9th, 10th & 11th numbers
= (45.5 × 3) + 3.5 = 140
Let the 10th number be x
∴ x + x − 4 + x − 9 = 140
⇒ x = 51 

= 46.5


12. Weighted average 

Shortcut: Let the assumed average be Rs. 10
Additional average 

 

13. New average 

Shortcut: New average = (9 × 2) + 3 = 21

14. Let the number of students be ‘x’ 

 

15. Let the number of students be ‘x’ 

⇒ 68.5x + 280 = 68.8(x + 4)
⇒ 68.5x + 280 = 68.8x + 275.2
⇒ 0.3x = 4.8 

Some more

Average is defined as the ratio of summation of all the data to the number of units present in the list.

Suppose, we have given with ‘n’ number of values such as x1, x2, x3..... xn. The average or the mean of the given data will be equal to

Model Questions

1. The average income of 40 persons is Rs.4200 and that of another 35 persons is Rs.4000. The average income of the whole group is:

2. A cricketer whose bowling average is 24.85 runs per wicket, takes 5 wickets for 52 runs in next inning and thereby decreases his average by 0.85. The number of wickets taken by him till the last match was?
A) 64      B) 85      C) 80         D) 96

Explanation: 5 × 24.85 − 52 = 72.25
The number of wickets taken by him

3. The average of first three numbers is thrice the fourth number. If the average of all the four numbers is 5, then find the fourth number.
A) 4.5       B) 5           C)           2 D) 4
Explanation: Let fourth number = x

and average of first three numbers = 3x
x + 3 × 3x = 20 ⇒ x = 2
                                                                         Ans: C
4. The average of a collection of 20 measurements was calculated to be 56 cm. But later it was found that a mistake had occurred in one of the measurement which was recorded as 64 cm., but should have been 61 cm. The correct average must be:
A) 53 cm         B) 54.5 cm      C) 55.85 cm          D) 56.15 cm

Correct Average = 56 − 0.15 = 55.85 cm
                                                                  Ans: C
5. The average of 8 numbers is 20. The average of first two numbers is  and that of the next three is  If the sixth number be less than the seventh and eighth numbers by 4 and 7 respectively, then the eighth number is:
A) 18       B) 22          C) 25            D) 27
Explanation: Sum of 8 numbers = 160
Sum of 6th, 7th, 8th numbers
                              = 160 − (31 + 64) = 65

  8th number = 18 + 7 = 25                              Ans: C

6. The average of runs of a cricket player of 10 innings was 32. How many runs must he make in his next inning so as to increase his average of runs by 4?
A) 76       B) 70           C) 4           D) 2
Explanation: 32 + 11 × 4 = 76
                                                                         Ans: A
7. The average of marks of 28 students in Mathematics was 50. 8 students left the school, then this average increased by 5. What is the average of marks obtained by the students who left the school?
A) 50.5          B) 37.5              C) 42.5            D) 45
Explanation:
Average increases for 20 students
 20 × 5 = 100
Average marks who left the school

                             Ans: B

8. A librarian purchased 60 story books for his library. But he found that he could get 4 extra books by spending Rs.336 more and then the overall average price per book would be reduced by Rs.1. The previous average price of each book was?
A) Rs.84             B) Rs.83           C) Rs.68             D) Rs.100
Explanation:
Let the average price of book is Rs.x/book
60x + 336 = 64(x − 1)
64x − 60x = 400
x = 100 (Initial book price)
                                                    Ans: D
9. A batman has a certain average of runs for 12 innings. In the 13th inning he scores 96 runs there by increasing his average by 5 runs. What will be his average after 13th inning?
A) 28         B) 32            C) 36           D) 42
Explanation: average after 13th inning = 96 − (12 × 5) = 36
                                                                                               Ans: C
10. The average revenues of 7 consecutive years of a company is Rs.79 lakhs. If the average of first 4 years is Rs.74 lakhs and that of last 4 years is Rs.86 lakhs, what is the revenue for the 4th year?
A) Rs.87 lakhs               B) Rs.89 lakhs
C) Rs.85 lakhs               D) Rs.83 lakhs

Explanation: Revenue of fourth year
= (4 × 74 + 4 × 86 − 7 × 79)
= (640 − 553) = 87 Lakhs
                                                    Ans: A
11. In the first 30 overs of a cricket match, the run rate was 5.2 runs/over. What is the required run rate in the remaining 20 overs to reach the target of 280 runs?
A) 6.8           B) 7.4       C) 6.2           D) 5.6
Explanation: 30 × 5.2 = 156

        Ans: C

12. A batsman scores 87 runs in the 21st match of his career. His average runs per match increases by 2. What was his average
before the 21st match?
A) 45        B) 46       C) 44         D) 43
Explanation: 21 × 2 = 42
Average before 21st match = 87 − 42 = 45
            (By using equality concept)
                                                                  Ans: A
13. The average weight of 12 crewmen in a boat is increased by 1/3 kg. When one of  the crewmen whose weight is 55 kg is replaced by a new man. What is the weight of that new man?
A) 58 kg          B) 60 kg            C) 57 kg            D) 59 kg
Explanation: Weight of new man

       Ans: D

14. The average age of 120 students in a group is 13.56 years. 35% of the number of students are girls and the rest are boys. If the ratio of the average age of boys and girls is 6 : 5, then what is the average age (in years) of the girls?
A) 14.4         B) 10       C) 11.6       D) 12
Explanation: Let the average of boys and girls are 6x and 5x respectively
78 × 6x + 42 × 5x = 120 × 13.56
678x = 1627.2
x = 2.4
average age of girls = 5x = 5 × 2.4 =12
                                                                      Ans: D

15. The average of 33 numbers is 74. The average of the first 17 numbers is 72.8 and that of the last 17 numbers is 77.2. If the 17th number is excluded, then what will be the average of the remaining numbers (correct to one decimal place)?
A) 72.9      B) 73.4          C) 71.6           D) 70.8
Explanation: 17th number
= (17 × 72.8 + 17 × 77.2) − 33 × 74
= 2550 − 2442
= 108
Average of remaining numbers

            Ans: A

16. The average of thirteen numbers is 47. The average of the first three numbers is 39 and that of next seven numbers is 49. The 11th number is two times the 12th number and 12th number is 3 less than the 13th number. What is the average of 11th and 13th numbers?
A) 56.5       B) 56      C) 54.5        D) 57
Explanation: 11th, 12th & 13th numbers sum
= 13 × 47 − (3 × 39 + 7 × 49)
= 611 − 460 = 151

17. From a class of 42 boys, a boy aged 10 years goes away and in his place, a new boy is admitted. If on account of this change, the average age of the boys in that class increases by 2 months, the age of the newcomer is:
A) 19 years                   B) 17 years
C) 10 yrs. 6 month       D) 12 yrs. 2 month

Explanation: The age of new boy
= 10 years + 42 × 2 months
= 10 years + 7 years
= 17 years (by using equality concept)
                                                                     Ans: B
18. The average marks obtained by 150 students in an examination is 40. If the average marks of passed students is 60 and that of the failed students is 20, then what is the number of students who passed the examination?
A) 25      B) 50        C) 75       D) 100

Posted Date : 28-11-2022

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