**AVERAGE Questions with solution**

**AVERAGE : **average of some numbers or some quantities is the sum of quantities divided by the number of quantity.

The formula for the average of n numbers x1,x2,x3,……..,xn is

(x1+x2+x3+…….Xn)/n

Sum of total quantities = average * number of quantities

**Example** **: **Find the average of five numbers 10,12,14,16,18 .

Average =10+12+14+16+18/5

=70/5

=14

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**WEIGHTED AVERAGE : **suppose you 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 weighted average.

Thus if we have m groups with averages A1,A2,…..,Am and having n1,n2,…..,nm elements respectively, then the average of all the elements of all the groups is given by weighted average i.e.

Weighted average = (n1A1+n2A2+……………+nmAm)/(n1+n2+……………+nm)

**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.

Average = 15*10 + 10*30/15+20

=150+300/25

=450/25

=18

**PROPERTIES**

- If the average age of a group of persons is x years today then after n years their average age will be x+n and also n years ago their average would have been x-n.

**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 exclude or include

Average = 21+7 =28

- Average speed = total distance/total time
- If a man travels at x kmph. From point A to B and returns at y kmph then average speed for the whole journey will be

2x*y/x+y

**Example :** a man travels at 60 kmph on the journey from A to B and returns at 100 kmph. Find his average speed for the journey.

Average = (2*60*100)/(60+100)

= 12000/160

= 75

**SOME IMPORTANT FORMULAE RELATED TO AVERAGE**

- Average of first n natural numbers = (n+1)/2
- Average of squares of first n natural numbers = (n+1)(2n+1)/6
- Average of cubes of first n natural numbers = n(n+1)2/4
- Average of even natural numbers upto n = (n/2)+1
- Average of first n even natural numbers = n+1
- Average of odd numbers upto n = (n+1)/2
- Average of first n natural odd numbers = n
- Average of any a,b,c,…..,n consecutive numbers = (a+n)/2
- Average of an arithmetic sequence with first number (a) and last number (l) = (a+l)/2

**WORKED -OUT PROBLEMS**

**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*6=90

sum of first two numbers = 16*2=32

sum of last three numbers = 13*3=39

third number = 90-32-39

= 90-71

= 19

**The average age of the family of five members is 24. If the present age of youngest members is 8 years, then what was the average age of the family at the birth of the youngest member?**

** Solution **sum of the age of five members = 24*5=120

At the birth of youngest member the age of each member was 8 less than present age.

Sum of age of five members at the birth of youngest member = 120-8*5

= 120-40

= 80

Average age at the birth of youngest member = 80/5

= 16

**Alternatively **

Average age of the family at the birth of youngest member means the average age of the family before 8 years

Average age of family is 24 years today, so 8 years ago average age of family would have been 24-8=16.

**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 =40

B+C =70 (1)

C+A =60

2(A+B+C) = 170

A+B+C = 85 (2)

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

A= 85-70=15

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**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

x/y =1.2/.8

=12:8

=3:2

**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*30 = 1200

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

41*31 = 200+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

- 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*3 = 105

Total price of apples =480-105 = 375

Average price of apples = 375/75

=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 x1,x2,x3,x4,x5,x6,x7

sum of seven numbers = 26*7 = 182

Sum of first 4 numbers i.e. x1+x2+x3+x4 = 27*4 = 108

Sum of last 4 numbers i.e. x4+x5+x6+x7 = 25*4 = 100

X1+x2+x3+x4+x4+x5+x6+x7 =108+100 = 208

Fourth number i.e. x4 = 208-182 =2

**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 13*2 = 26

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

Now, sum of the age of A,B and C is 17*3 = 51

Now, the age of C is 51-36 = 15

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

**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*20 = 400

Average age of students is 2 less than that of whole group . it means that the age of whole group decrease 20*x = 20x

so,400= 2x

X = 400/2

= 200

**Find the average of 5,12,19,26,………………..,,705.**

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

Average = (5+705)/2

= 710/2

= 355