**An average,** or additional correctly the arithmetic mean is, in crude terms, the sum of n different data divided by n.

For example, if a cricket player scores 35, 45 and 37 runs in first,2^{nd} and 3^{rd} innings individually, then his average runs in 3 innings is equal to

35+45 +372/3 = 39 runs.

Therefore, the two most used formula in this chapter are:

Average = Total of data /No. of data

And, Total = Average ×No. of data

**Ex. 1:** The average age of 30 boys in a class is equal to 14 yrs. When the age of the tutorial educator is comprised the average becomes 15 yrs. Find the age of the class teacher.

Soln: Total ages of 30 boys = 14 x 30 = 420 yrs.

Whole ages when class educator is comprised = 15 x 31 = 465 yrs

Age of class educator = 465 – 420 = 45 yrs.

**Direct Formula:**

Age of new entrant=New average + No. of ancient associates ×Upsurge in average = 15 + 30(15 – 14) = 45 yrs

**Ex. 2:** The average weightiness of 4 men is enlarged by 3 kg when one of them who weighs 120 kg is swapped by another man. What is the weight of the new man?

** Soln**: Quicker Approach: If the average is increased by 3 kg, then the sum of weights increases by 3 x 4 = 12 kg.

And this surge in weight is due to the additional weight comprised due to the inclusion of a new person.

:: Weight of new man = 120 + 12 = 132 kg Direct Formula: Weight of new person = weight of removed person +

No. of persons x increase in average = 120 + 4 x 3 = 132 kg

** Ex. 3:** The average of marks obtained by 120 candidates in a certain O examination is 35. If the average marks of conceded applicants are 39 and that of the failed applicants is 15, what is the number of

candidates who passed the examination?

**Soln:** no of passed candidates: no of failed candidates = 20:4= 5:1

No of passed candidates = 120/5+1 = 100

**Ex. 4:** The average of 11 outcomes is 50. If the average of 1^{st} six outcomes is 49 and that of the last six is 52, find the sixth outcome.

** Soln:** The total of 11 outcomes = 11 50 = 550

The total of first 6 outcomes = 6 x 49 = 294

The total of last 6 outcomes = 6 x 52 = 312

The 6th outcome is common to both;

.. Sixth outcome = 294 + 312 – 550 = 56 Direct Formula:

6th outcome = 50 + 6{(52 – 50) + (49 – 50)} = 50 +6 {2-1} = 56

**Ex. 5:** The average age of 8 people in a group is increased by 2 years when two men aged 35 yrs and 45 yrs are replaced by two women. Find the average age of these two women.

**Soln:** By the direct formula used in Ex. 2, the total age of two women

= 2 x 8 +(35 +45) = 16 + 80 = 96 yrs.

average age of two women= = 48 yrs.

**Ex. 6:** The average age of a family of 6 associates is 22 yrs. If the age of the newest associate be 7 yrs, then what was the average age of the family at the birth of the newest associate?

** Soln:** Total ages of all associates = 6 x 22 = 132 yrs.

7 yrs ago, total sum of ages = 132 – (6 x 7) = 90 yrs.

However, at that period there were 5 associates in the family.

Average at that time = 90 + 5 = 18 yrs.

** Ex. 7:** A man purchased 13 shirts of 50 each, 15 pants of $60 each and 12 couples of shoes at 65 a pair. Find the average value of each article.

**Soln:** Direct Method: Average = 13×50+15×60+12×65/ 13+15+12 = -Rs 58.25

**Ex. 8:** The average score of a cricketer in 2 matches is 27 and in three all other matches are 32. Formerly find the average score in all the five matches.

**Soln:** Direct Method:

Average in 5 matches = 2×27+3 x 32/2+3 = 54+96/5-=30

**Ex. 9:** The average of 11 outcomes is 30, that of the first five is 25 and that of the last five is 28. Find the value of the 6th number.

**Soln:** **Direct Formula:**

6th number = Total of 11 outcomes – (Total of first five + Total of last five outcomes)

= 11 × 30 – (5 25 + 5×28)= 330 – 265 = 65

Note: Ex 4 and Ex 9 are different. In Ex 4 the 6th result is common to both the groups but in Ex 9 the 6th result is excluded in both theresults.

** Ex. 10:** In a discussion, there are 30 boys whose average age is reduced by 4 months, when one boy aged 18 years is substituted by a new boy. Find the age of the new boy.

** Soln**: This example is similar to Ex. 2. The lone change is that in Ex 2 the average upsurges afterward auxiliary whereas in this case the average reductions. Thus, you can see the change in direct formula.

Direct Formula:

Age of fresh *people*= Age of removed people -No. of people× Decrease in average age

= 18-30 ×4/ 12 = 18 – 10 = 8 8 months.

** Ex. 11:** A batsman in his 17th turns makes a score of 85, and thereh. increases his average by 3. What is his average after 17 turns?

**Soln**

**Direct Formula:** Average after 16 turns = 85 – 3 x 17 = 34

Average after 17 turns = 85 – 3(17 – 1) = 37