### AVERAGE SHORTCUT : Calculate Average with a Trick

CALCULATE AVERAGE SHORTCUT TRICKS

You know that the quantitative aptitude section is most important in bank exams in PO and Clerk and for other competitive exams because if you want a good score in the bank exam then you have to score well in maths. In competitive exams the most important thing is time management, if you know how to manage your time then you can do well in Bank Exams. That’s where maths shortcut tricks and formulas are comes into action. So continuously we are providing shortcut tricks on different maths topics.
One of the most important topics in maths is average. You should know how to calculate the Average in a very short time. For this here we are providing shortcut tricks to calculate averages in maths.

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

HOW TO CALCULATE THE AVERAGE:
For example, if a batsman scores 30, 50, and 25 runs in the first, second, and third innings respectively, then his average runs in 3 innings is equal to
30+50+25/3; =39 runs

Therefore, the two most used formulas in this question are:
Average= total of data/no. of data
And total = average × no. of data

Important Formulae Related to Calculate Average of numbers:
1. Average of first n natural number = (n+1)/2
2. Average of the first n even number = (n+1)
3. Average of first n odd number = n
4. Average of consecutive number = (First number +Last number)/2
5. Average of 1 to n odd numbers = (Last odd number+1)/2
6. Average of 1 to n even numbers = (Last even number+2)/2
7. Average of squares of first n natural numbers = [(n+1) (2n+1)]/6
8. Average of the cubes of first n natural number = [n (n+1) ^2]/4
9. Average of n multiples of any number = [Number*(n+1)]/2

Examples with shortcut tricks on how to calculate Average are given below:
1. Ex: the average age of 30 boys in a class is equal to 14 yrs. When the age of the class teacher is included the average becomes 15 yrs. Find the age of the class teacher?
Solution: total ages of 30 boys = 14 × 30 = 420yrs.
Total ages when class teacher is included = 15 × 31 = 465yrs.
So the age of the class teacher = 465 – 420 = 45 yrs.
Shortcut trick: age of new entrant = new average + no. of old members × Increase in average = 15 + 30 (15-14) = 45 yrs.

2. Ex: the average weight of 4 men is increased by 3 kg when one of them who weighs 120kg is replaced by another man. What is the weight of the new man?
Solution: by shortcut trick: if the average is increased by 3 kg, then the sum of weight increases by 3 × 4 = 12 kg.
And this increase in weight is due to the extra weight included due to the inclusion of new person.
So weight of new man = 120 + 12 = 132 kg.
We can also solve this question by Direct formula: weight of new person = weight of removed person+ no. of persons × increase in average = 120 + 4 * 3= 132 kg.

3. Ex: the average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed the examination?
Solution: Let the number of passed candidates be x.
The total marks = 120 × 35 = 39x + (120-x) * 15
Or, 4200 = 39x + 1800 – 15x
Or, 24x = 2400
X = 100
So the number of passed candidates = 100.

We can also solve this question by shortcut trick direct formula:
Number of passed candidates =
total candidates ( total average – failed average) / passed the average-failed average
And the number of failed candidates=
Total candidates (passed average – total average) / passed average – failed average
by this, we can solve the number of passed candidates = 120 (35 – 15) / 39 – 15
= 100

4. Ex: A batsman in his 17th innings makes a score of 85, and thereby increases his average by 3. What is his average after 17 innings?
Solution: 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
Shortcut tricks or direct formula:
Average after 16 innings = 85 – 3 * 17 = 34
Average after 17 innings = 85 – 3 (17 – 1) = 37.

5. Ex: 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?
Solution: 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 / 11 =24
Or x, = 264 – 215 = 49
Shortcut tricks or direct formula:
Required score = 11 × 24 – 21.5 × 10 = 49
Note: the above formula is based on the theory that the difference is counted due to the score in the last innings.

6. Ex: in a class, there are 20 boys whose average age is decreased by 2 months, when one boy aged 18 years is replaced by a new boy. Find the age of the new boy.
Solution: Shortcut trick:
Age of new person = age of removed person – no. of persons × decrease in average age
= 18 – 20 × 2/12
= 18 – 10/3
= 44/3
= 14 yrs 8 months.

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