Averages

In this post, i am going to tell about the shortcuts in averages.

Points to remember in Averages:

Average =  ( Sum of observations ) . Numbers of observations

• An average, or more accurately an arithmetic mean is, in general terms, the sum of n different data divided by n.

Average =  Total of data No. of data

Total = Average × No. of data

•  If the value of each item in a group is increased/decreased by the same value x, then the average of the group also increases/decreases by x. For instance, if the income of each person in a group increases by 15, the average income of the group also increases by Rs. 15.

This is valid only when the value of each item increases/decreases by the same amount.

•  If the average age of group of people is x years, then their average age after n years will be (x + n) and their average age n years ago would have been (x – n) years.
This is because with each passing year, each person’s age increases by 1 and vice versa.

•  If the value of each item in a group is multiplied/divided by the same value x (where x ≠ 0 in the case of division), then the average of the group also gets multiplied/divided by x.

•  The average of a group always lies between the smallest value and the longest value in that group.

• If the terms in Arithmetic progression and the number of terms is odd,then the average will be equal to middle term.
• EX: 

the average of 4 ,6,8 is    6.

• If the terms are in Arithmetic progression and the number of term are even ,the average is equal to average of  middle term.
• Ex:

the average of 4,6,8,10 is (6+8)/2     =7

• the average of  N odd numbers is N.
• the average of N even numbers is N+1.

exercise problems:

1)Ex. 1:  The average of 11 results is 50. If the average of first six results is 49 and that of last six is 52, find the sixth result.

Sol. :  The total of 11 results = 11 × 50 = 550
The total of first 6 results = 6 × 49 = 294
The total of last 6 results = 6 × 52 = 312
The 6th result is common on both;
∴ Sixth result = 294 + 312 – 500 = 56

Ex. 2 :  The average age of a family of 6 members is 22 years. If the age of the youngest member be 7 years, then what was the average age of the family at the birth of the youngest member?
 

Sol. :  Total ages of all members = 6 × 22 = 132 years.
7 years ago, total sum of ages = 132 – (6 × 7) = 90 years.
But at that time there were 5 members in the family.
∴ Average at that time = 90 ÷ 5 = 18 years.

Ex. 3:  A man bought 13 shirts of Rs. 50 each, 15 pants of Rs. 60 each and 12 pairs of shoes at Rs. 65 a pair. Find the average value of each article.
Sol. : By the use of averge formula,  

Average =	13 × 50 + 15 × 60 + 12 × 65	= Rs. 58.25
	13 + 15 + 12

Ex. 4 :  The average score of a cricketer in two matches is 27 and in three other matches is 32. Then find the average score in all the five matches.

Sol. : By the use of average formula,

Average in 5 matches =	2 × 27 + 3 × 32	=	54 + 96	= 30.
	2 + 3		5

Ex. 5 :  The average of 11 results is 30, that of the first five is 25 and that of the last five is 28. Find the value of the 6th number.

Sol. : By the use of average formula,

6th number = Total of 11 results – (Total of first five results + Total of last five results)
= 11 × 30 – (5 × 25 + 5 × 28)
= 330 – 265 = 65

Ex. 6: A batsman in his 17th innings makes a score of 85, and thereby increases his average by 3. What is this average after 17 innings?
Sol. :  Let the average after 16th innings be x, then

16x + 85 = 17 (x + 3) = Total score after 17th innings.

∴     x = 85 – 51 = 34

∴    average after 17 innings = x + 3 = 34 + 3 = 37.

Ex. 7: A cricketer has completed 10 innings and his average is 21.5 runs. How many runs must he make in his next innings so as to raise his average to 24?
Sol. :  Total of 10 innings = 21.5 × 10 = 215

Suppose he needs a score of x in 11th innings; then

average in 11 innings =	215 + x	= 24
	11

or, x = 264 – 215 = 49

Ex8: the average of 9 consecutive even numbers is 68.what is the product of the smallest and the largest of number

Sol:

                n=9

             the numbers are 

              60   62  64   66  68  70   72  74 76 

             product of smallest and largest =60*76       =4560.

EX 9:the  average of set of 24 numbers is 32.if each number is tripled then what will be the resultant age?

from the above points to remember 

                    

                                      =32*3

                                      =96

EX:In a examination students average marks were 71 per parer.if they had obtain 15 more marks in English ,9 more marks in science ,its average marks per paper will be 74.find the number of papers in the exam.

sol:

          let the number of papers is n

             

          the  number of paper in exam=(71n+15+9)/n=74

                                                       n  =     8.