GMAT Exam  >  GMAT Notes  >  Quantitative for GMAT  >  Detailed Notes: Averages

Detailed Notes: Averages | Quantitative for GMAT PDF Download

Introduction

  • This chapter forms the backbone concept of most questions in the Quantitative Aptitude & Data Interpretation sections. This is a crucial chapter & quick solving methods in this concept will help you save time - which is an essential factor for your success.
    Detailed Notes: Averages | Quantitative for GMAT

What is an Average

  • Average (or Mean) is defined as the ratio of the sum of the quantities to the number of quantities.Detailed Notes: Averages | Quantitative for GMAT
  • Following are some examples on average.

Example 1: Find the average of first five natural numbers. 

  • Sol: First five natural numbers:

    1, 2, 3, 4, 5

    Step 1: Calculate the sum

    Sum = 1 + 2 + 3 + 4 + 5

    Sum = 1 + 2 + 3 + 4 + 5= 15

    Step 2: Count the total number of values

    Total numbers = 5

    Step 3: Calculate the average

    Average = 155 = 3

    Therefore, The average of the first five natural numbers is 3.

Example 2: If a person with age 45 joins a group of 5 persons with an average age of 39 years. What will be the new average age of the group?

  • Sol: Total age will be 45 + 5× 39 = 240. And there will be 6 persons now.
    So, the average will be 240/6 = 40.
    (or)
    Since, 45 is 6 more than 39, by joining the new person, the total will increase by 6 and so the average will increase by 1.
    So, the average is 39 + 1 = 40.

Example 3: Two students with marks 50 and 54 leave class VIII A and move to class VIII B. As a result, the average marks of class VIII A fall from 48 to 46. How many students were there initially in class VIII A?

  • Sol: The average of all the students of class VIII A is 46, excluding these two students.
    They have 4 and 8 marks more than 46. So, with the addition of these two students, 12 marks are adding more, and hence the average is increasing 2.
    There should be 6 students in that class, including these two. This is the initial number of students.

Weighted Mean

  • Let N1, N2, N3, …. Nn be the weights attached to variable values X1, X2, X3, …….. Xn respectively. Then the weighted arithmetic mean, usually denoted by:
    Detailed Notes: Averages | Quantitative for GMAT
  • For any two different quantities taken in different ratios. The weighted average is just like a see-saw. 
    The more the ratio of a quantity,  more will be the inclination of the average from mid-value towards the value with more ratios.

Example: The average marks of 30 students in a section of class X are 20 while that of 20 students of the second section is 30. Find the average marks for the entire class X.

Sol: Given:

  • Section A: Number of students = 30, Average marks = 20
  • Section B: Number of students = 20, Average marks = 30

Step 1: Calculate Total Marks for Each Section

Total marks for Section A:

Total MarksA = 301 × 20 = 600

Total marks for Section B:

Total MarksB = 201 × 30 = 600

Step 2: Calculate Total Marks for the Entire Class

Total Marksclass = Total MarksA + Total MarksB = 600 + 600 = 1200

Step 3: Calculate Total Number of Students

Total Students = 30 + 20 = 50

Step 4: Calculate the Average Marks for the Entire Class

Average Marksclass = 120050 = 24

Therefore , average marks of the entire class is 24.

Real Facts About Average

Fact 1: If each number is increased/decreased by a certain quantity n, then the mean also increases or decreases by the same quantity.

Fact 2: If each number is multiplied/ divided by a certain quantity n, then the mean also gets multiplied or divided by the same quantity.

Fact 3: If the same value is added to half of the quantities and the same value is subtracted from the other half quantities, then there will not be any change in the final value of the average.

Average Speed

  • It is the total distance traveled divided by the time it takes to travel the distance.
  • Detailed Notes: Averages | Quantitative for GMATIf d1 & d2 are the distances covered at speeds v1 & v2 respectively and the time taken are t1 & t2 respectively, then the average speed over the entire distance (x+ x2) is given by:
    Detailed Notes: Averages | Quantitative for GMAT
  • If both the distances are equal i.e. d1 = d2 = d then, Detailed Notes: Averages | Quantitative for GMAT  (i.e. Harmonic mean of two velocities)
  • But if both the time taken are equal i.e. t1 = t2 = t then,
    Average speedDetailed Notes: Averages | Quantitative for GMAT (i.e. Algebraic mean of two velocities)

Example: The average of 10 consecutive numbers starting from 21 is:

Sol: The average is simply the middle number, which is the average of 5th & 6th no. i.e, 25 & 26 i.e. 25.5

Question for Detailed Notes: Averages
Try yourself:The average of 5 numbers is 20. If one number is removed from the group, the average becomes 18. What is the number that was removed?
View Solution

Examples

Example 1: A man travels 120 km at 60 km per hour and returns the same distance at 40 km per hour. What is his average speed for the entire journey?

  • Sol: The formula for average speed in a round trip is:
  • 2 × Speed1 × Speed2Speed1 + Speed2
  • Here, Speed1 = 60 km per hour and Speed2 = 40 km per hour.
  • Using the formula:
  • 2 × 60 × 4060 + 40 = 4800100 = 48 km per hour
  • Therefore , the answer to this question is 48 km per hour

Example 2: A grocer mixes two types of sugar. Type A costs ₹30 per kg, and Type B costs ₹50 per kg. How many kilograms of Type B sugar must he mix with 10 kg of Type A to get a mixture worth ₹40 per kg?

  • Sol: Let x be the quantity of Type B sugar in kg.
  • Using the weighted average formula:
  • 30 × 10 + 50 × x10 + x = 40
  • Now, simplify the equation:
  • 300 + 50x = 40 × (10 + x)
  • 300 + 50x = 400 + 40x
  • Rearrange terms:
  • 50x – 40x = 400 – 300
  • 10x = 100
  • x = 10

Example 3: A company has three departments:

  1. Department A has 10 employees with an average salary of ₹60,000.
  2. Department B has 15 employees with an average salary of ₹80,000.
  3. Department C has 5 employees with an average salary of ₹50,000.
    What is the overall average salary of all employees in the company?

Sol: First, calculate the total salary for each department:

  • Department A: 10 × 60,000 = 600,000
  • Department B: 15 × 80,000 = 1,200,000
  • Department C: 5 × 50,000 = 250,000

Now, sum all the salaries and divide by the total number of employees:

600,000 + 1,200,000 + 250,00010 + 15 + 5 = 2,050,00030 = 68,333.33

Answer: ₹68,333.33

Example 4: A jar contains 10 liters of a solution with 30% salt. Two liters are removed and replaced with pure water. What is the new concentration of salt?

  • Sol: Step 1: Initial amount of salt in the solution =  30 × 10100 = 3 liters of salt
  • Step 2: Salt removed = 30 × 2100 = 0.6 liters of salt
  • Step 3: Remaining salt after removal: 3 − 0.6 = 2.4 liters.
  • The total volume of the solution is still 10 liters after replacing the removed 2 liters with pure water.
  • Step 4: New concentration of salt =  2.410 = 0.24 or 24%

Example 5: The average marks of 3 classes are as follows:

  • Class A: 50 students with an average of 75.
  • Class B: 30 students with an average of 80.
  • Class C: 20 students with an average of 85.
    What is the overall average of all the students?

Sol: First, calculate the total marks for each class:

  • Class A: 50 × 75 = 3,750
  • Class B: 30 × 80 = 2,400
  • Class C: 20 × 85 = 1,700

Now, sum the total marks and the total number of students:

Total marks:

3,750 + 2,400 + 1,700 = 7,850

Total number of students:

50 + 30 + 20 = 100

The overall average is:

7,850100 = 78.5

Question for Detailed Notes: Averages
Try yourself:
The average weight of a group of 6 people is 70 kg. If one person with a weight of 80 kg leaves the group, what will be the new average weight of the group?
View Solution

The document Detailed Notes: Averages | Quantitative for GMAT is a part of the GMAT Course Quantitative for GMAT.
All you need of GMAT at this link: GMAT
110 videos|110 docs|120 tests

Top Courses for GMAT

FAQs on Detailed Notes: Averages - Quantitative for GMAT

1. What is the difference between average and weighted mean?
Ans.The average, or arithmetic mean, is calculated by adding all the values together and dividing by the number of values. The weighted mean, on the other hand, takes into account the importance or frequency of each value by assigning weights, thus providing a more accurate representation when the values have different levels of significance.
2. How is the average speed calculated?
Ans.Average speed is calculated by dividing the total distance traveled by the total time taken. The formula is: Average Speed = Total Distance / Total Time. This calculation is essential for understanding overall travel efficiency.
3. Why is it important to understand averages in the GMAT exam?
Ans.Understanding averages is crucial for the GMAT exam because many quantitative and data interpretation questions involve calculating or analyzing averages. Mastery of this concept can help test-takers efficiently solve problems related to data sets, statistics, and real-world scenarios.
4. Can the average be affected by outliers?
Ans.Yes, the average can be significantly affected by outliers, which are values that are much higher or lower than the other values in a data set. Outliers can skew the average, making it unrepresentative of the overall data. This is why it's often useful to also consider the median or mode in such cases.
5. What are some real-life applications of averages?
Ans.Averages have numerous real-life applications, such as calculating grades in schools, determining average income in economics, assessing performance metrics in businesses, and analyzing statistics in sports. Understanding averages helps in making informed decisions based on data.
110 videos|110 docs|120 tests
Download as PDF
Explore Courses for GMAT exam

Top Courses for GMAT

Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

Detailed Notes: Averages | Quantitative for GMAT

,

Exam

,

ppt

,

Summary

,

Viva Questions

,

MCQs

,

video lectures

,

practice quizzes

,

Detailed Notes: Averages | Quantitative for GMAT

,

past year papers

,

Semester Notes

,

Free

,

Sample Paper

,

shortcuts and tricks

,

Previous Year Questions with Solutions

,

study material

,

pdf

,

Important questions

,

Objective type Questions

,

mock tests for examination

,

Extra Questions

,

Detailed Notes: Averages | Quantitative for GMAT

;