Basics of geometry including concepts of triangles, similar triangles, quadrilaterals, polygons and circles are oft repeated questions both in the GMAT Problems Solving section and in the GMAT Data Sufficiency section of the GMAT Math Section. A good grasp of the fundamentals of Geometry will help you crack questions from this topic with relative ease. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts:
1. Definition of various terms used in geometry
2. Properties of concepts explained, with illustrations
3. Detailed explanation of the concepts in angles
4. Introduction into triangles and related properties with illustrative examples
5. Different types of triangles
6. Concepts and theorems, such as Apollonius Theorem and Pythagoras Theorem, associated with triangles with illustrative examples
7. Explanation of relationship between area of triangles, inradius and circumradius
8. Introduction of different polygons, meaning of quadrilaterals, squares, rectangles, rhombus, parallelogram, kite and trapezium
9. Circles and various associated concepts and properties, with illustrative examples
10. Different types of circles
11. 22 solved examples with illustrations
12. 13 exercise problems with the answer key and explanatory answers
Here is an example of a typical solved example in this chapter.
Sample Question
Question
In a TriangleABC. AB = 4.5, AC = 6 and BC = 7.5. The points D & E lie on the line segment BC with BD = 1.5 and EC = 3. What is the value of the /_DAE?
Explanatory Answer
Geometry Solved Example - Solution Image The triangle is right-angled (4.5, 6 and 7.5).
Therefore /_A is right-angled.
BD = 1.5, EC = 3 => BE = 4.5 and DC = 6
Hence AB = BE = 4.5 and AC = CD = 6 and
Therefore
/_BAE = /_BEA = /_DEA and /_DAC = /_ADE
In triangle ADE, /_ADE + /_DEA + /_DAE = 180o
=> /_DAC + /_BAE + /_DAE = 180o
=> (90o + /_DAE) + /_DAE = 180o
=> /_DAE = 45o
Sunday, August 26, 2007
Set Theory
Basics of set language theory such as union of sets, intersection of sets get tested in the GMAT problem solving and data sufficiency sections. With a good grasp of the basic formulae governing set language, one should be able to attempt these questions with ease. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts
1. Introduction of the concept of sets and methods to represent sets.
2. Different types of sets explained with illustrative examples.
3. Cardinal number of sets, power sets, number of subsets for a set.
4. The De-Morgan's Law.
5. Using Venn Diagrams to solve set language questions explained with illustrative examples.
6. 7 solved examples, including Venn Diagram representation.
7. 5 Comprehensive exercise problems with answer key and explanatory answers.
8. An objective type speed test with around 35 questions along with explanatory answers and answer key are provided for the speed test.
Here is an example of a typical solved example in this chapter.
Sample Question
Question
In a room of 50 people whose dresses have either red or white color, 30 are wearing red dress, 16 are wearing a combination of red and white. How many are wearing dresses that have only white color?
Explanatory Answer
Venn Diagram Number of people wearing a red dress = 30
i.e., n(R) = 30
Number of people wearing a combination of red and white = 16
i.e., n (R Intersection W) = 16
The total number of people in the room = number of people who are wearing dresses that have either red or white color = n (R Union W) = 50.
We know,
n (R Union W) = n(R) + n(W) - n(R Intersection W)
50 = 30 + n(W) - 16
50 - 14 = n(W) - 16
n(W) = 36
i.e., the number of people who are wearing a white dress = 36.
Therefore, number of people who are wearing white dress only = n(W) - n(R Intersection W) = 36 - 16 = 20
Additional Practice Questions in Set Language
Looking for additional practice questions in Set Theory? A collection of some questions available at
1. Introduction of the concept of sets and methods to represent sets.
2. Different types of sets explained with illustrative examples.
3. Cardinal number of sets, power sets, number of subsets for a set.
4. The De-Morgan's Law.
5. Using Venn Diagrams to solve set language questions explained with illustrative examples.
6. 7 solved examples, including Venn Diagram representation.
7. 5 Comprehensive exercise problems with answer key and explanatory answers.
8. An objective type speed test with around 35 questions along with explanatory answers and answer key are provided for the speed test.
Here is an example of a typical solved example in this chapter.
Sample Question
Question
In a room of 50 people whose dresses have either red or white color, 30 are wearing red dress, 16 are wearing a combination of red and white. How many are wearing dresses that have only white color?
Explanatory Answer
Venn Diagram Number of people wearing a red dress = 30
i.e., n(R) = 30
Number of people wearing a combination of red and white = 16
i.e., n (R Intersection W) = 16
The total number of people in the room = number of people who are wearing dresses that have either red or white color = n (R Union W) = 50.
We know,
n (R Union W) = n(R) + n(W) - n(R Intersection W)
50 = 30 + n(W) - 16
50 - 14 = n(W) - 16
n(W) = 36
i.e., the number of people who are wearing a white dress = 36.
Therefore, number of people who are wearing white dress only = n(W) - n(R Intersection W) = 36 - 16 = 20
Additional Practice Questions in Set Language
Looking for additional practice questions in Set Theory? A collection of some questions available at
Probability
The concepts in this topic are essentially an extension of the fundamentals learnt in Permutation Combination. Many of the probability questions are restricted permutation questions. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts:
1. Introduction to the concept of probability
2. Meaning of sample space and events along with illustrative examples to explain the same
3. Types of experiments in probability
4. Introduction to independent events, mutually exclusive events and probability of complimentary events.
5. Explanation of compound events
6. Method to evaluate the probability of an event with illustrative example and shortcut methods
7. 21 solved examples covering typical questions in tossing of coins, rolling of dice, picking cards from a pack of cards.
8. 15 exercise problems with the answer key and also explanatory answers
9. An objective type speed test with around 40 questions along with explanatory answers and answer key are provided for the speed test
Here is an example of a typical solved example in this chapter.
Sample Question
Question
A bag contains 5 yellow balls and 6 orange balls. When 4 balls are drawn at random simultaneously from the bag, what is the probability that not all of the balls drawn are orange?
Explanatory Answer
Sample Space (Denominator) : Four balls can be drawn from a bag containing 11 balls in 11C4 ways
Event (Numerator) : The number of ways in which all four balls drawn will be orange = 6C4.
Probability: The probability that all four balls drawn are orange
=
Therefore, the probability that not all of the balls drawn are orange = 1 - probability of all four being orange
=>
=>
1. Introduction to the concept of probability
2. Meaning of sample space and events along with illustrative examples to explain the same
3. Types of experiments in probability
4. Introduction to independent events, mutually exclusive events and probability of complimentary events.
5. Explanation of compound events
6. Method to evaluate the probability of an event with illustrative example and shortcut methods
7. 21 solved examples covering typical questions in tossing of coins, rolling of dice, picking cards from a pack of cards.
8. 15 exercise problems with the answer key and also explanatory answers
9. An objective type speed test with around 40 questions along with explanatory answers and answer key are provided for the speed test
Here is an example of a typical solved example in this chapter.
Sample Question
Question
A bag contains 5 yellow balls and 6 orange balls. When 4 balls are drawn at random simultaneously from the bag, what is the probability that not all of the balls drawn are orange?
Explanatory Answer
Sample Space (Denominator) : Four balls can be drawn from a bag containing 11 balls in 11C4 ways
Event (Numerator) : The number of ways in which all four balls drawn will be orange = 6C4.
Probability: The probability that all four balls drawn are orange
=
Therefore, the probability that not all of the balls drawn are orange = 1 - probability of all four being orange
=>
=>
Permutation and Combination
Permutation combination is considered to be a hard topic by many GMAT aspirants. It is widely believed that questions from this topic appear when you score in the higher percentile levels in the GMAT Math section. If you have good understanding of the basics of this topic, it will be quite easy to crack questions that appear from permutation combination. 4GMAT's Math Lesson Book in this chapter covers the following areas
1. Independent Events, Product Rule, Sampling with and without replacement, sampling with and without ordering (arrangement).
2. Introduction to Permutation, combination. Difference between permutation and combination. Formulae associated with permutation and combination.
3. Examples of sampling with replacements, r-sequence and r-multisets.
4. Solved examples involving permutation and combination concepts in listing numbers
5. Solved examples involving re-arranging letters of words and their ranks
6. Concepts and solved examples on tossing of coins
7. Concepts and solved examples on rolling of a die and multiple dice
8. Solved examples on drawing one or more cards from a pack of cards
9. Typical permutation problems such as circular permutation, arranging boys and girls in a line etc.,
10. Typical combination problems such as questions on making musical albums, chess boards etc.,
11. About 5 illustrative examples to explain concepts; over 50 solved examples (with shortcuts wherever applicable to hard and tough questions) to acquaint you with as many different questions as possible; around 25 exercise problems with answer key and explanatory answers to provide you with practice and an objective type speed test with 50 questions. Explanatory answers and answer key are provided for the speed test.
Here is an example of a typical solved example in this chapter.
Sample Question
Question:
Each of the 11 letters A, H, I, M, O, T, U, V, W, X and Z appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many three letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?
Explanatory Answer
There are three possible cases that will satisfy the condition of forming three letter passwords with at least 1 symmteric letter.
Case 1: 1 symmetric and 2 asymmetric or
Case 2: 2 symmetric and 1 asymmetric or
Case 3: all 3 symmetric
= {(11C1 * 15C2) + (11C2 * 15C1) + 11C3} * 3!
= {11 * permutation + permutation * 15 + permutation} x 6
= {1155 + 825 + 165} * 6
= 2145 * 6 = 12870
1. Independent Events, Product Rule, Sampling with and without replacement, sampling with and without ordering (arrangement).
2. Introduction to Permutation, combination. Difference between permutation and combination. Formulae associated with permutation and combination.
3. Examples of sampling with replacements, r-sequence and r-multisets.
4. Solved examples involving permutation and combination concepts in listing numbers
5. Solved examples involving re-arranging letters of words and their ranks
6. Concepts and solved examples on tossing of coins
7. Concepts and solved examples on rolling of a die and multiple dice
8. Solved examples on drawing one or more cards from a pack of cards
9. Typical permutation problems such as circular permutation, arranging boys and girls in a line etc.,
10. Typical combination problems such as questions on making musical albums, chess boards etc.,
11. About 5 illustrative examples to explain concepts; over 50 solved examples (with shortcuts wherever applicable to hard and tough questions) to acquaint you with as many different questions as possible; around 25 exercise problems with answer key and explanatory answers to provide you with practice and an objective type speed test with 50 questions. Explanatory answers and answer key are provided for the speed test.
Here is an example of a typical solved example in this chapter.
Sample Question
Question:
Each of the 11 letters A, H, I, M, O, T, U, V, W, X and Z appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many three letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?
Explanatory Answer
There are three possible cases that will satisfy the condition of forming three letter passwords with at least 1 symmteric letter.
Case 1: 1 symmetric and 2 asymmetric or
Case 2: 2 symmetric and 1 asymmetric or
Case 3: all 3 symmetric
= {(11C1 * 15C2) + (11C2 * 15C1) + 11C3} * 3!
= {11 * permutation + permutation * 15 + permutation} x 6
= {1155 + 825 + 165} * 6
= 2145 * 6 = 12870
Arithmetic and Multiplicative Progressions (AP, GP).
One could expect one or two questions based on the topic sequences and series. The typical series tested included arithmetic progression and geometric or multiplicative progression. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts in the topic sequences and progressions.
1. Introduction to Arithmetic Progression
2. Explanation with formulae to find the nth term of an arithmetic progression and the sum of n terms of an arithmetic progression.
3. Illustrative and solved examples to find the value of the common difference, first term and the number of terms, given the sum of n terms or the nth term and first term of an AP.
4. Property changes when an AP is transformed by addition, multiplication of constant terms.
5. Introduction to Multiplicative Progression (Geometric Progression)
6. Formulae to find the nth term of a multiplicative progression and the sum up to n terms of a multiplicative progression.
7. Introduction to the concept of infinitely decreasing multiplicative progression and the formulae to find the sum of such a sequence.
8. Geometric Mean or the mean of the Multiplicative progression.
9. Transformation witnessed when a Multiplicative Progression is multiplied by a constant.
10. Introduction to Harmonic progression. Formulae to find the nth term of a Harmonic Progression.
11. Relation between Arithmetic mean, Multiplicative mean and Harmonic mean.
12. 2 illustrative examples to explain concepts; 25 solved examples (with shortcuts wherever applicable) to acquaint you with as many different questions as possible; around 20 exercise problems with answer key and explanatory answers to provide you with practice and an objective type speed test with 40+ questions. Explanatory answers and answer key are provided for the speed test.
Here is an example of a typical solved example in this chapter.
Sample Question
Question
There are 4 terms in an A.P. such that the sum of the two means is 21 and the product of the extremes is 54. What are the terms of the A.P?
Explanatory Answer
Let the four terms be a - 3d, a - d, a + d and a + 3d.
The sum of the two means = a - d + a + d = 2a = 21 or a = 10.5
The product of the two extremes = (a - 3d)(a + 3d) = a2 - 9d2 = 54 => 10.52 - 9d2 = 54
=> 9d2 = 110.25 - 54 = 56.25
=> 3d = 7.5 => d = 2.5.
The four terms are a - 3d = 10.5 - 7.5 = 3, a - d = 10.5 - 2.5 = 8; a + d = 10.5 + 2.5 = 13 and a + 3d = 10.5 + 7.5 = 18. ie. 3, 8, 13, 18
Note
In the above expression, the term 'a' is not the first term as is generally assumed and the common difference is not 'd'. There is actually no term as 'a' as part of this progression. The terms are a - 3d, a - d ... and the common difference is 2d.
1. Introduction to Arithmetic Progression
2. Explanation with formulae to find the nth term of an arithmetic progression and the sum of n terms of an arithmetic progression.
3. Illustrative and solved examples to find the value of the common difference, first term and the number of terms, given the sum of n terms or the nth term and first term of an AP.
4. Property changes when an AP is transformed by addition, multiplication of constant terms.
5. Introduction to Multiplicative Progression (Geometric Progression)
6. Formulae to find the nth term of a multiplicative progression and the sum up to n terms of a multiplicative progression.
7. Introduction to the concept of infinitely decreasing multiplicative progression and the formulae to find the sum of such a sequence.
8. Geometric Mean or the mean of the Multiplicative progression.
9. Transformation witnessed when a Multiplicative Progression is multiplied by a constant.
10. Introduction to Harmonic progression. Formulae to find the nth term of a Harmonic Progression.
11. Relation between Arithmetic mean, Multiplicative mean and Harmonic mean.
12. 2 illustrative examples to explain concepts; 25 solved examples (with shortcuts wherever applicable) to acquaint you with as many different questions as possible; around 20 exercise problems with answer key and explanatory answers to provide you with practice and an objective type speed test with 40+ questions. Explanatory answers and answer key are provided for the speed test.
Here is an example of a typical solved example in this chapter.
Sample Question
Question
There are 4 terms in an A.P. such that the sum of the two means is 21 and the product of the extremes is 54. What are the terms of the A.P?
Explanatory Answer
Let the four terms be a - 3d, a - d, a + d and a + 3d.
The sum of the two means = a - d + a + d = 2a = 21 or a = 10.5
The product of the two extremes = (a - 3d)(a + 3d) = a2 - 9d2 = 54 => 10.52 - 9d2 = 54
=> 9d2 = 110.25 - 54 = 56.25
=> 3d = 7.5 => d = 2.5.
The four terms are a - 3d = 10.5 - 7.5 = 3, a - d = 10.5 - 2.5 = 8; a + d = 10.5 + 2.5 = 13 and a + 3d = 10.5 + 7.5 = 18. ie. 3, 8, 13, 18
Note
In the above expression, the term 'a' is not the first term as is generally assumed and the common difference is not 'd'. There is actually no term as 'a' as part of this progression. The terms are a - 3d, a - d ... and the common difference is 2d.
Pipes, Cisterns and Work, Time
Concepts Covered The typical questions that appear from this topic are word problems. The basic concept tested in this chapter is analogous to the concepts tested in Speed Time Distance. So, if you have a good grasp of Speed Time Distance topic, then you will be able to understand and solve questions in this topic with a lot of ease. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts:
1. The chapter begins with the rules governing the questions in pipes and cisterns and it is explained with illustrative examples
2. There are 17 solved examples in questions related to pipes and cisterns. Some of the tough math questions in this topic are featured here.
3. The concepts regarding work and time are explained with examples.
4. There are 13 solved examples for questions related to work and time. Shorcuts to few hard math questions in work time are provided.
5. 24 exercise problems in both pipes and cisterns as well as work and time have been provided with the answer key and explanatory answers to the questions
6. An objective type speed test with around 50 questions with explanatory answers and answer key provided for the speed test
Here is an example of a typical solved example in this chapter.
Sample Question
Question
Pipe A fills a tank in 30 minutes. Pipe B can fill the same tank 5 times as fast as pipe A. If both the pipes were kept open when the tank is empty, how much time will it take for the tank to overflow?
Explanatory Answer
Pipe B fills the tank 5 times as fast as pipe A.
Therefore, pipe B will fill the tank in one-fifth of the time that pipe A takes.
Pipe B will fill the tank in 30 / 5 = 6 minutes.
In 1 minute, pipe A will fill 1 / 30th of the tank and pipe B will fill 1 / 6th of the tank.
Therefore, together, the two pipes will fill (1/30 + 1/6) = (1+5 / 30) = 6/30 = 1/5th of the tank in a minute
Hence, the two pipes working together will take 5 minutes to fill the tank.
1. The chapter begins with the rules governing the questions in pipes and cisterns and it is explained with illustrative examples
2. There are 17 solved examples in questions related to pipes and cisterns. Some of the tough math questions in this topic are featured here.
3. The concepts regarding work and time are explained with examples.
4. There are 13 solved examples for questions related to work and time. Shorcuts to few hard math questions in work time are provided.
5. 24 exercise problems in both pipes and cisterns as well as work and time have been provided with the answer key and explanatory answers to the questions
6. An objective type speed test with around 50 questions with explanatory answers and answer key provided for the speed test
Here is an example of a typical solved example in this chapter.
Sample Question
Question
Pipe A fills a tank in 30 minutes. Pipe B can fill the same tank 5 times as fast as pipe A. If both the pipes were kept open when the tank is empty, how much time will it take for the tank to overflow?
Explanatory Answer
Pipe B fills the tank 5 times as fast as pipe A.
Therefore, pipe B will fill the tank in one-fifth of the time that pipe A takes.
Pipe B will fill the tank in 30 / 5 = 6 minutes.
In 1 minute, pipe A will fill 1 / 30th of the tank and pipe B will fill 1 / 6th of the tank.
Therefore, together, the two pipes will fill (1/30 + 1/6) = (1+5 / 30) = 6/30 = 1/5th of the tank in a minute
Hence, the two pipes working together will take 5 minutes to fill the tank.
Circular and Straight Races
The chapter on Races is essentially an extention to the concepts covered in the Speed Time and Distance Chapter and Ratio Proportion chapter. The lesson book has questions on two types of races - Straight Races and Circular Races. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts:
1. Definition of the term race
2. Explanation of the concept of races along a straight track
3. Illustrative Examples are given to explain the above mentioned concept
4. 9 solved examples are provided with detailed explanation for races along a straight track
5. The concept of race in a circular path is explained along with examples
6. 4 solved examples are given with detailed explanation
7. 13 exercise problem of both types of races with the answer key and explanatory answers
8. An objective type speed test with around 40 questions with explanatory answers and answer key provided for the speed test
Here is an example of a typical solved example in this chapter.
Sample Question
Question:
P and Q run around a circular track of radius 49 meters starting from the same point at the same time and in the same direction. If 'P' runs at a speed of 15m/min and 'Q' runs at a speed of 10m/min, when will they meet again for the first time?
Explanatory Answer
As P and Q run in the same direction, their relative speed = 15 - 10 = 5 m/min
Circumference of the circular track is the length of the circular track = 2 * pi * r = 2 * (22/7) * 49 = 308 m.
Time taken for the two of them to meet for the first time after they have started a race along a circular track, starting simultaneously from the same point is given by
(length of the circular track / relative speed) = 308 / 5 = 61.6 minutes
Therefore, they will meet for the first time after 1 hour, 1 minute and 36 seconds.
1. Definition of the term race
2. Explanation of the concept of races along a straight track
3. Illustrative Examples are given to explain the above mentioned concept
4. 9 solved examples are provided with detailed explanation for races along a straight track
5. The concept of race in a circular path is explained along with examples
6. 4 solved examples are given with detailed explanation
7. 13 exercise problem of both types of races with the answer key and explanatory answers
8. An objective type speed test with around 40 questions with explanatory answers and answer key provided for the speed test
Here is an example of a typical solved example in this chapter.
Sample Question
Question:
P and Q run around a circular track of radius 49 meters starting from the same point at the same time and in the same direction. If 'P' runs at a speed of 15m/min and 'Q' runs at a speed of 10m/min, when will they meet again for the first time?
Explanatory Answer
As P and Q run in the same direction, their relative speed = 15 - 10 = 5 m/min
Circumference of the circular track is the length of the circular track = 2 * pi * r = 2 * (22/7) * 49 = 308 m.
Time taken for the two of them to meet for the first time after they have started a race along a circular track, starting simultaneously from the same point is given by
(length of the circular track / relative speed) = 308 / 5 = 61.6 minutes
Therefore, they will meet for the first time after 1 hour, 1 minute and 36 seconds.
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