Average : Shortcut & Tricks | Quantitative Techniques for CLAT PDF Download

The Average

The Sum of all the numbers/elements/quantities divided by the total number of numbers/elements/quantities is called the Average.

Average = (Sum of all numbers)/( Total number of numbers)

With the help of the Following Tricks, you can solve the problem of Average easily-

Trick 1:

The average of two or more numbers/quantities is called the mean of these numbers, which is given by
Average(A) = (sum of observation/quantities)/(No. of observation/quantities)
∴ S = A * n

Trick 2:

The average of ‘n’ consecutive natural numbers starting from 1,2,3, ……n.
Average of 1,2,3, …..n = (n+1)/2

Trick 3:

The average of squares of ‘n’ consecutive natural numbers starting from 1.
Average of 1², 2², 3², 4² ….. x² = (n+1)(2n+1)/6

Trick 4:

The average of cubes of first ‘n’ consecutive natural numbers.
Average of 1³, 2³, 3³ ….. n³ = n(n+1)²/4

Trick 5:

The average of first ‘n’ consecutive even natural numbers.
Average of 2, 4, 6, ….. 2n = (n-1)

Trick 6:

The average of first ‘n’ consecutive odd natural numbers.
Average of 1, 3, 5, ….. (2n – 1) = n

Trick 7:

The average of certain consecutive numbers.
Average of a, b, c, ……… n is (a + n)/2

Trick 8:

The average of 1st ‘n’ multiples of certain numbers x.
Average of x = (x(1 + n))/2

Trick 9:

If A goes from P to Q with the speed of x km/h and returns from Q to P with the speed of y km/h, then the average speed of the total journey is
Average speed = 2xy/(x + y)

Trick 10:

If a distance is travelled with three different speeds a km/h, b km/h and c km/h, then
The average speed of total journey = 3abc/(ab + bc + ca) km/h

Examples

Q1. A student calculates the arithmetic mean of the following 5 numbers: 10, 15, 20, 25 and x He found the mean is 15. Find out the value of x?
(a) 3 
(b) 7 
(c) 17 
(d) 5
Ans: d
Sol:
Mean = (10+15+20+25+X)/5=15
70+X = 75
X = 5

Q2: Out of 20 boys, 5 are each of 1 m 10 cm height, 10 are of 1 m 20 cm and rest of 1 m 30 cm. The average height of all of them is-
(a) 1 m 10 cm
(b) 1 m 25 cm
(c) 1 m 20 cm
(d) 1 m 12 cm
Ans: c
Sol:
Average Height
= (5 X 1.10 + 10 X 1.20 + 5 X 1.30)/20
= (5.5 + 12 + 6.5)/20
= 24/20
= 1.20
= 1 meter 20 cm

Q3: In 2021, the average monthly income of an employee was 3,000. For the first nine months of the year, his average monthly income was 3100 and for the last four months, it was 4,500. His income in the ninth month of the year was –
(a) 16000
(b) 5080  
(c) 8200
(d) 9900
Ans: d
Sol: 
Employee’s income in the eighth month
= (3100 × 9 + 4 × 4500 – 12 × 3000)
= (27900 + 18000 – 36000)
= (45900 – 36000)
= 9900

Q4: The average yearly expenditure of a company for the first five years is 25700, for the next two years 24900 and for the last five years 30300. If the company is saved 53200 during 12 years, the average yearly earnings of the company are (approx value, not exact value)-
(a) 30000
(b) 31850
(c) 31917
(d) 35807
Ans: c
Sol:  
Total expenditure of the company
= (5 × 25700 + 2 × 24900 + 5 × 30300)  
= (128500 + 49800 + 151500)
= 329800
Total earning = (329800 + 53200) = 383000
Required average yearly earning = 383000/12 = 31917

Q5: A man goes to the market from his house and returns house from the market at speeds of 10 km/hr and 15 km/hr respectively. Find out his average speed?
(a) 12 km/hr
(b) 15 km/hr
(c) 14 km/hr
(d) 13 km/hr
Ans: a
Sol:
We know,
Average speed = (2×S1×S2)/(S1+S2)
Here, S1 = 10 km/hr & S2 = 15 km/hr
Average speed = 2×15×10/(15+10) km/hr
 = 300/25 km/hr
= 12 km/hr