In this article on

**Average Concepts and Tricks with Solved Examples Chapter 1**, we are sharing some important properties of Average with important questions which are usually asked in Quantitative Aptitude section of various competitive examinations. Once you have understood the basics of Average, you can solve any problem from this topic in no time.Here we will discuss various different types of questions on Average and the easiest, quickest and simplest way to deal with them. Let us first start with some basic examples followed by some important properties of Average and then we will move towards some advance level questions.

As we all know that the term ‘

**Average**‘ is defined as the sum of all the observations divided by the total number of observations. Let’s take some examples to understand it more clearly.**Q.) Chetan Bhagat bought 65 books for ₹ 1050 from one shopkeeper and 50 books for ₹ 1020 from another shopkeeper. What’s is the average price he paid per book ?**

**:**

__Solution__**Average = Total Price of Books / Number of Books**

So, Average = (1050 + 1020)/ (65 + 50)

= 2070 / 115 = 18

= 2070 / 115 = 18

*[ANS]***Q.) The average of 5 positive integers is 436. The average of first two numbers is 344 and the average of last two numbers is 554. What is the third number ?**

**: As we all know that when average is given then we can find the sum of elements which is equal to,**

__Solution__

Sum = Average × Number of Elements

Now let 5 integers be a,b,c,d and e. So,

a + b + c + d + e = 436 × 5

Sum of first two numbers (a+b) = 344 × 2

Sum of last two numbers (d+e)= 554 × 2

Sum of last two numbers (d+e)= 554 × 2

Then,

(344 × 2) + c + (554 ×2) = 436 × 5

688 + c + 1108 = 2180

688 + c + 1108 = 2180

[ Tip – To multiply 436 by 5, multiply 436 by 10 and divide it by 2]

c + 1796 = 2180

c = 2180 – 1796

c = 384

c = 2180 – 1796

c = 384

**[ANS]**So the required third number is 384.

**Q.) The average temperature of first 3 days is 27 degrees and the average temperature of the next 3 days is 29 degrees. If the average of the whole week is 28.6 degrees, then what is the temperature of last day ?**

**: This question is based on the example we discussed above. Here we will use the same formula,**

__Solution__Sum = Average × Number of Elements

Sum of temperature of first 3 days = 27 × 3

Sum of temperature of next 3 days = 29 × 3

Sum of temperature of whole week = 28.6 × 7

Sum of temperature of next 3 days = 29 × 3

Sum of temperature of whole week = 28.6 × 7

Let the last day temperature be ‘x’. Hence,

(27×3) + (29×3) + x = 28.6 × 7

81 + 87 + x = 200.2

168 + x = 200.2

x = 200.2 – 168 = 32.2

81 + 87 + x = 200.2

168 + x = 200.2

x = 200.2 – 168 = 32.2

*[ANS]*By all the above examples you might have got an idea about the basic solution of Average questions. Now let us learn some properties which are very important to solve the average problems at a jiffy.

**Property 1**: Suppose there are ‘n’ numbers – a1, a2, a3, a4 …………. a’n’, then it’s average will be

A = (a1 + a2 + a3 + a4 ………….. a’n’) / n

Now if each number is increased by ‘x’ , then the new average will be

A’ = { (a1+x) + (a2+x) + (a3+x) + ……………. + (a’n’+x) } / nIt can also be written as,

A’ = { (a1 + a2 + …….. + a’n’) + ‘n’x } / n

A’ = {(a1 + a2 + …….. + a’n’) / n} + ‘n’x / n

A’ = {(a1 + a2 + …….. + a’n’) / n} + x

A’ = {(a1 + a2 + …….. + a’n’) / n} + ‘n’x / n

A’ = {(a1 + a2 + …….. + a’n’) / n} + x

Or A’ = A + x

In short,

**if we increase every quantity by ‘x’, then the average will also increase by ‘x’. The same condition applies for subtraction as well as multiplication.****Property 2**:

**Average of consecutive numbers**

Whenever the numbers are increasing constantly in a definite same pattern then these numbers are known as consecutive numbers. For example : 4, 6, 8, 10, 12 and 14 are consecutive numbers as the difference between all the numbers are same.

So,

**Average of consecutive numbers = (a+h)/2**here, a = first term of consecutive numbers, and

h = last term of consecutive numbers

h = last term of consecutive numbers

Now the question arises as how to calculate the sum of consecutive numbers. So to understand this, let say there are 6 consecutive numbers.

Then,

**Sum of consecutive numbers = {(a+h) / 2} × 6**Now what if there are 3, 5 or 7 consecutive integers, or if there are odd number of consecutive numbers.

**So when there are odd number of consecutive integers**(for example : 2, 4, 6, 8 and 10)**then****the average will be the middle term**. In this case 6 will be our average.So always remember this, if there are even number of consecutive integers then average will be equal to sum of first term and last term divided by 2 and if there are odd number of consecutive integers then average will be the middle term.

Now, ket us discuss an example based on these properties.

**Q.) In a school, nine students have 10, 20, 30, 40, 50, 60, 70, 80 and 90 chocolates respectively. If each one of them is given 5 chocolates in addition, then find the new average.**

**: Here the given number of chocolates are consecutive numbers, and since there are (nine) odd number of consecutive integers, the average will be the middle term.**

__Solution__Hence

**(using property 2)**the average = 50and now since 5 chocolates are given to every student, thus

**using property 1,**The new Average = 50 + 5 = 55

*[ANS]*As each quantity is increasing by 5, so the average will also increase by 5. Therefore our answer will be 55.

That’s all in this article, hopefully you find it useful and if you does then don’t forget to share it with your near and dear ones. We will be publishing more problems in the chapter 2 of Average Concepts and Tricks with Solved Examples, so stay connected with our website.

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