# Average

Average means a number which expressing the central or typical value of the given data.

The average of some numbers or some quantities is the sum of quantities divided by the number of quantity.

### Average = $\frac{sum of observation}{number of observation}$

Suppose we have n observation $x_{1},x_{2},x_{3}$,…………………….,$x_{n}$, then

### Average = ($x_{1}+x_{2}+x_{3}$+………………+$x_{n}$)/n

Sum of observation = average $\times$ number of observation

For example: Find the average of five numbers 10,12,14,16,18 .

Average = $\frac{10+12+14+16+18}{5}$

= $\frac{70}{5}$ = 14

Weighted average: Suppose we have two or more groups whose individual averages are known. Then to find the combined average of all the elements of all the groups, we use the weighted average.

Suppose we have m groups as shown in Table below

Example: The average price of 15 books from one shop is 10 and average price of 10 books from another shop is 30. Find the average price of all the books.

Solution: Average = $\frac{15\times10+30\times10}{15+10}$

=  $\frac{450}{25}$ = 18

## Some important formulae of Average

Example: The average age of a family of 6 members is 21. Find the average age of this family after 7 years if no member is excluded or included.

Solution: Average = 21 + 7 =28

Example: A man travels at 60 km/hr on the journey from A to B and returns at 100 km/hr. Find his average speed during the whole journey.

Solution: Average speed = $\frac{2\times60\times100}{100+60}$

= $\frac{2\times60\times100}{100+60}$ = 75 km/hr.

Question: The average of  6 numbers is 15. The average of the first two numbers is 16 and the average of the last three numbers is 13. What is the third number?

Solution:  sum of 6 numbers =$15\times6=90$

sum of first two numbers = $16\times2=32$

sum of last three numbers = $13\times3=39$

Sum of these five numbers = 32 + 39 = 71

Third number = 90 – 71 = 19

## Few questions of average

Question 1: The average of two numbers A and B is 20, B and C is 35 and C and A is 30. What is the value of A?

Solution:  A+B =  $2\times20=40$

B+C =  $2\times35=70$                                  (1)

C + A = $2\times30=60$

2(A+B+C) = 40 + 70  + 60 = 170

A + B + C = 85                                                                        (2)

Subtract equation (1)  from (2),we get

A = 85 – 70 = 15

Question 2: The average score of boys in an examination in a school is 71 and that of girls is 73. The average score of the school is 71.8 . find  the ratio of the number of boys to that of the girls that appeared in the examination.

Solution: Let the number of boys be x

And the number of girls be y

Sum of marks of all boys = 71x

Sum of marks of all girls = 73y

Sum of marks of all students in an examination = 71.8(x+y)

71.8(x+y)=71x + 73y

71.8x + 71.8y = 71x + 71y

.8x = 1.2y

$\frac{x}{y}=\frac{1.2}{.8}$

$\Rightarrow$  x : y = 3 : 2

Question 3: The average weight of  40 students in a class is 30 kg. If, however the weight of teacher be included, the average increases by 1 kg. What is the weight of teacher?

Solution: Let the weight of teacher be x.

Sum of weight of 40 students = $40\times30$ = 1200

After included the weight of teacher, the weight of whole class = 1200+x

$41\times31$ = 1200 + x

1271 = 1200 + x

x = 71

Alternatively

Since initially the average weight is 30 kg, but after including teacher the average weight increases by 1 kg meaning 41  kg weight is increasing of the whole class.

So the weight of teacher is 41 + 30 = 71

Question 4: 35 oranges and 75 apples were purchased for Rs 480. If the price per orange was Rs 3, then the average price of apples were ?

Solution: Total price of oranges = $35\times3$ = 105

Total price of apples =480 – 105  = 375

Average price per apples = $\frac{375}{75}$ = 5

Question 5: The average of 7 numbers is 26. If the average of first four numbers is 27 and the average of last four numbers is 25. What is the fourth number?

Solution: Suppose the seven numbers are $x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}$

Sum of seven numbers = $26\times7$= 182

Sum of first 4 numbers $x_{1}+x_{2}+x_{3}+x_{4}$= $27\times4$ = 108

Sum of last 4 numbers $x_{4}+x_{5}+x_{6}+x_{7}$= $25\times4$ = 100

$x_{1}+x_{2}+x_{3}+x_{4}+x_{4}+x_{5}+x_{6}+x_{7}$ = 108 + 100 = 208

Fourth number, $x_{4}$ = 208 – 182 = 26

Question 6: Five years ago, the average age of A and B is 13. Now the average age of  A,B and C is 17. Find the age of C after 10 years from now.

Solution: Five years ago, sum of the age of A and B is $2\times13$ = 26

Now, sum of the present age of  A and B is 26+10 = 36

Now, sum of the age of  A, B and C is $17\times3$ = 51

The present age of C is 51 – 36 = 15

After 10 years, the age of C will be 15 + 10 = 25

Question 7: The average age of some students of a school is 11, and average age of 20 teachers is 33. If average age of both the groups of students and teacher is 13. Find the number of students.

Solution: Let the number of boys be x

Average age of teachers is 20 more than the average of whole groups. It means that the age of whole group increases by $20\times20$ = 400

Average age of students is 2 less than that of whole group.

It means that the age of whole group decrease $20\times x$ = 20x

so, 400 = 2x

x = $\frac{400}{2}$ = 200

Question 8: Find the average of  the sequence 5, 12, 19, 26, ………………..705.

Solution: This is an arithmetic sequence with first term 5 and last term 705.

Average = $\frac{5+705}{2}$

= $\frac{710}{2}$ = 355

## Problems for Practice

1. The total age of A,B and C is 70 more than the age of C. what is the average age of A and B ?

a) 30                            b) 33                      c) 36                       d) 35

2. The average of 40 numbers is 12. If the average of first 13 number is 9, average of second 13 numbers is 13 and average of third 13 numbers is 12. What is the 40th number ?

a) 25                         b) 13                       c) 38                         d) 24

3. If the average of A,B and C is 50, average of B and C is 70, average of  A and C is 65. What is the average of A and B ?

a) 10                          b) 60                      c) 55                        d) 15

4. The average score of a school in an examination is 75. If the average score of boys is 69, and the ratio of the numbers of boys to the girls is 5 : 3 . what is the average score of girls ?

a) 85                          b) 80                         c) 81                           d) 78

5. A man travels from point A to B in such a way that, he travels half distance at a speed of 80 kmph and half distance at a speed of 20 kmph. And then returns from B to A in the same manner i.e. half at a speed of 60 kmph and half at a speed of 40 kmph. Find his average speed during the whole journey.

a) 37.5                       b) 36.4                      c) 38.4                          d) 39.4

6. The average of 31 numbers is 27. If three numbers 31, 25 and 46 are discarded, the average of the remaining numbers is

a) 25.5                      b) 24.5                      c) 26.25                            d) 26

7. The average marks obtained by a student in 7 papers is 35 and in the 8th paper he obtains  75 marks. Find his new average.

a) 37                       b) 40                          c) 39                            d) 42

8. The average age of a family of 7 members is 25. The age of  two youngest members, namely rahul and tarun are 4 and 7 respectively. Find the average age of family at the birth of tarun .

a) 20                      b) 20.5                          c) 21                            d) 21.5

9. A man travels one-fourth part of a journey at a speed of 8 kmph, next three – fifth part at a speed of 6km h and remaining part at a  speed of 10 kmph. What  is the average speed of a man during his whole journey ?

a) 7.8                      b) 6.23                     c) 6.83                            d) 7.2

10. The average age of a school of 35 teachers is 32. If the age of the principal be included then the average increases by 4 months. Find the age of principal.

a) 44                       b) 47                         c) 45                                d) 42

11. Find the average of the numbers 11,22,33,44,……………..1111.

a) 500                  b) 561                           c) 582                          d) 523

12. The average age of four boys is 12. If their ages are in the ratio 2 : 3 : 4 : 7 . What is the age of the youngest boy?

a) 5                        b) 8                             c) 6                                 d) 4

13. The average of 13 numbers is 15. If every number is multiplied by 9, then the average of new numbers.

a) 15                      b) 24                        c) 135                         d) 120

14. Find the average of odd numbers upto 999.

a) 400                    b) 600                            c) 500                           d) 700

15. The average of the following numbers _

2,2,2,2,2,2,2,6,6,6,6,6,6,6,10,10,10,10,10,10,10,14,14,14,14,14,14,14,18,18,18,18,18,18,18,22,22,22,22,22,22,22,26,26,26,26,26,26,26,30,30,30,30,30,30,30

a) 120                       b) 112                            c) 12                         d) 16

16. Find the average of first 50 multiples of 32.

a) 820                      b) 956                     c) 326                        d) 816