The questions based on averages are quite frequent in the quantitative section of the CAT. These questions are generally related to the basic and elementary maths skills and so, most of the candidates can easily solve the given questions. But, as the CAT exam is all about time, being able to quickly solve CAT quantitative aptitude questions is very crucial.

To help the CAT aspirants prepare this topic more effectively, a detailed lessons on averages is given here. Also, some useful shortcut techniques are explained which can help to find the average along with various illustrations.

**What are Averages?**

Averages can be defined as the central value in a set of data. Average can be calculated simply by dividing the sum of all values in a set by the total number of values. In other words, an average value represent the middle value of a data set. The data set can be of anything like age, monety, runs, etc.

**Average = [ Sum of Data (or observations) in a set / Number of data (or observations) in that set]**

**Example: **

*What is the average of first five consecutive odd numbers?*

**Solution:**

The first five consecutive odd numbers are: 1, 3, 5, 7, 9.

Here, the number of data or observations is 5 and the sum of these 5 numbers is 25.

So, average = 25 / 5 = 5.

**Also Check:**

**Shortcut techniques in Averages:**

Questions based on averages can be easily solved using shortcuts. By using shortcuts, any question can be solved quickly and efficiently which can save a lot of time. So, some shortcuts to solve average questions are explained below along with illustrations.

**Shortcut to find average or change in average from a set of values**

**Example 1:**

*The average of a batsman in 16 innings is 36. In the next innings, he is scoring 70 runs. What will be his new average?*

*a) 44**b) 38**c) 40**d) 48*

**Solution:**

**Conventionally solving:**

New average = (old sum+ new score)/(total number of innings) = ((16 ×36)+70)/((16+1)) = 38

**Shortcut technique:**

**Step 1)** Take the difference between the new score and the old average = 70 – 36= 34

**Step 2)** This is 34 extra runs which is spread over 17 innings. So, the innings average will increase by 34/17 = 2

**Step 3)** Hence, the average increases by => 36+2 = 38.

*Here are a few more average questions and their solutions using the same technique.*

**Example 2: **

*The average marks of 19 children in a particular school is 50. When a new student with marks 75 joins the class, what will be the new average of the class?*

**Solution:**

**Step 1) **Take the difference between the old average and the new marks = 75-50=25

**Step 2)** This score of 25 is distributed over 20 students => 25/20 = 1.25

**Step 3)** Hence, the average increases by 1.25=> 50+1.25 = 51.25.

*Here is another question where the average dips.*

**Example 3:**

*The average age of Mr. Mark’s 3 children is 8 years. A new baby is born. Find the average age of all his children?*

**Solution:**

The new age will be 0 years. The difference between the old average and the new age = 0-8= -8

This age of 8 years is spread over 4 children => (-8/4= -2) Hence, the average reduces to 8-2= 6 years.

**Shortcut to find new value when average is given**

*Now here is a technique which will help to compute the new value when the average is given. Take this question for example:*

**Example 1:**

*The average age of 29 students is 18. If the age of the teacher is also included the average age of the class becomes 18.2. Find the age of the teacher?*

*a) 28**b) 32**c) 22**d) 24*

**Solution:**

**Conventionally solving,**

Let the average age of the teacher = x

(29 × 18 + x × 1)/30

Solving for x, we get x = 24.

**Shortcut Technique**

*Using the shortcut, based on the same method used previously:*

*Using the shortcut, based on the same method used previously:*

**Step 1:** Calculate the change in average = 18.2 – 18 = 0.2.

This change in 0.2 is reflected over a sample size of 30.

The new age is increased by 30 × 0.2 = 6 years above the average i.e. 18 + 6 = 24; which is the age of the teacher.

**Thinkquest-** The average age of 26 students in an MBA school is 30. One student among these quits the school in between. Can you find the age of that student if the new average is 29.8?

**Illustrations On Averages**

**Question 1:**

*Average goals scored by 15 selected players in EPL is 16.Maximum goals scored by a player is 20 and minimum is 12.Goals scored by players is between 12 and 20. What can be maximum number of players who scored at least 18 goals ?*

*a) 10**b) 5**c) 9**d) 6**e) None of these*

**Solution: Option (c)**

To maximize the number of players who scored 18 and above number of goals, one should assume that only one person has scored 20. To counter him, there will be one person who will score 12 goals.

i.e. 15 – 2 = 13 players left.

Now to maximize the 18 and above goals for every two players who are scoring 18, there will be one player scoring 12. This is done, to arrive at the average of 16. We will have 8 players with a score of 18 and 4 players with a score of 12.The last player will have a score of 16 Thus, the maximum number of people with 18 and more goals = 9.

**Question 2:**

*The average weight of a group of 8 girls is 50 kg. If 2 girls R and S replace P and Q, the new average weight becomes 48 kg.The weight of P= Weight of Q and the weight of R= Weight of S.Another girl T, is included in the group and the new average weight becomes 48 kg. Weight of T= Weight of R. Find the weight of P?*

*a) 48 kgs**b) 52 kgs**c) 46 kgs**d) 56 kgs*

**Solution: Option (d)**

8 x 50 +R+S-P-Q= 48×8 R+S-P-Q=-16

P+Q-R-S= 16 R=S and P=Q

P-R=8

One more person is included and the weight = 48 kg

Let the weight be a = (48 × 8 + a)9/9 = 48

A = 48 kg= weight of R

=> Weight of P= 48+8= 56 kg.

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