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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

The document Average : Shortcut & Tricks | Quantitative Techniques for CLAT is a part of the CLAT Course Quantitative Techniques for CLAT.
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FAQs on Average : Shortcut & Tricks - Quantitative Techniques for CLAT

1. What is the concept of the Average Average?
Ans. The concept of the Average Average refers to a shortcut or trick used to quickly calculate the average of a set of numbers. It involves finding the average of the average values of subsets within the original set.
2. How can the Average Average be calculated?
Ans. To calculate the Average Average, follow these steps: 1. Divide the original set of numbers into subsets of equal size. 2. Calculate the average of each subset. 3. Find the average of the calculated subset averages.
3. Why is the Average Average a useful shortcut?
Ans. The Average Average is a useful shortcut because it allows for a quicker calculation of the average of a large set of numbers. By dividing the set into subsets, the calculation becomes more manageable and less time-consuming.
4. Can the Average Average be applied to any set of numbers?
Ans. Yes, the Average Average can be applied to any set of numbers. However, it is most effective and accurate when the subsets are of equal size. If the subsets are uneven, the result may not represent the true average of the entire set.
5. Are there any limitations or drawbacks to using the Average Average method?
Ans. While the Average Average method provides a convenient shortcut, it is important to note that it may not always yield the most precise result. If the subsets are not representative of the entire set or if there are outliers within the subsets, the calculated average may be skewed. Therefore, it is advisable to use this method with caution and consider the characteristics of the data set before applying it.
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