Average

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,2nd  and 3rd 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 1st  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

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