MATH Workbook - GMAT classroom training, GMAT Verbal, CAT/ XAT

44 Pages·2009·0.25 MB·English

MATH Workbook

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Foreword

One of the prime objectives of education is to develop thinking skill in learners.
Thinking skills is essential to success in education, career and life in general.
Mathematical reasoning is one of the essential skills to be an effective critical thinker.
An individual who is adept at mathematical reasoning is able
• to read between the numbers to deduce patterns
• to relate various parameters and arrive at a relationship specific to the
problem
• to construct alternate scenarios with the same parameters thereby resulting
in multiple solutions
• to correlate a known or an analogous formula/theorem to the given problem
• to converge at a solution using different approaches
• to convert a problem from the given form into another less complex form
We are convinced that the preparation towards the complex area of mathematical
reasoning should comprise more than referring to discussion of sample questions.
Therefore this module attempts to present before you learning experiences which
will empower you with essential mathematical reasoning and problem solving tools.

This module is presented in a format which aims to create a virtual teacher who
would hand hold you while you explore the realm of math. The concepts are
explained in a lucid manner with minimalist words and maximum transfer learning.

It takes more than one reading to fully assimilate and appreciate the concepts.
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While solving problems, initially use the pen and paper extensively. After you learn
computational techniques try to ‘visual think’ the computation.

Remember math learning is not about remembering formula or tables or theorems, it
is all about learning the processes involved in successful problem solving.
Enjoy learning

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Number System and averages
1.

There are different terms in number system. Natural integers, fractions,
mixed fractions, prime numbers, composite numbers, imaginary numbers,
real numbers, rational numbers, irrational numbers, decimals, odd
numbers, even numbers, co-prime numbers are some of the commonly
used terms in the number system.

1.1 Operations on a fractions

Fractions can be multiplied, added, divided and subtracted
To add fractions or to subtract them there are two methods
Method 1:
Make sure the denominators are the same number
4 6
To add and
5 50
4
Convert to a fraction with denominator 50
5
This can be done by multiplying 4 and 5 by 10
4 4x 1040
=
Hence =
5 5x 1050
4 6 406 6 4
+ =
Therefore + =
5 50 505050

Method 2:
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Find LCM of the denominators
5116
Subtract -
8 6

LCM of the denominators(8,6) is 24

Multiply each fraction with a number such that the denominator = 24
5151 x 3153
==
88 x 324
1616x 484
= =
66x 424
5116 1538469
- = - =
8 6 242424

Practice Exercise 1.1
Fill in the space with the correct answer
5 7
1. + = ---------------------------------------------
2 3

9 7
2. + = ---------------------------------------------
4 5

1 2
3. 3 + 4 = ---------------------------------------------
3 5

5 1
4. 4 – 2 = ---------------------------------------------
16 8

3 3
5. 2 – 1 = ---------------------------------------------
16 8

5116
6. + = ---------------------------------------------
8 6

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Fractions can be multiplied by multiplying the denominators and
numerators separately
Practice Exercise 1.2
Fill in the space with the correct answer
5 7
1. x = ---------------------------------------------
2 3

9 7
2. x = ---------------------------------------------
4 5

1 2
3. 3 x 4 = ---------------------------------------------
3 5

5 1
4. 4 x 2 = ---------------------------------------------
16 8

3 3
5. 2 x 1 = ---------------------------------------------
16 8

Fractions can be divided by multiplying the reciprocal of one fraction with
the other fraction
575315
‚= x =
232714

Invert 7/3 to 3/7 and multiply with 5/2

Practice Exercise 1.3
Fill in the space with the correct answer
9 7
1. ‚ = ---------------------------------------------
4 5

1 2
2. 3 4‚ = ---------------------------------------------
3 5

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5 1
3. 4 2 ‚ = ---------------------------------------------
16 8

3 3
4. 2 1 ‚ = ---------------------------------------------
16 8

5116
5. ‚ = ---------------------------------------------
8 6

1.2 Method to determine the square root of a given number
Observe the method to find the square roots of 23456

1.Start from Left, Segregate two numbers at a time.
i.e 2,34,56
2. Find the square of a number which is lesser than
the left most number(i.e. 2) . Place the number (1)
to the left of the line. Place the square below the left
(cid:129) x most number.
2,34,56
(cid:129)
1
3. Subtract the two numbers.
2x1 34
× x
4. Bring the next two numbers down.
5. Double the number which is placed above the line
(1) and place on the left of the number which is
brought down.
15
2,34,56
1 6. Find the value of x such that (2 x) × (x) gives a
1
number equal to or less than 134. (i.e. 23 x 3 or 24 x
25 1 34
1 25 4 or 25 x 5 or 26 x 6....)
30y 9 56
. y
7. Since 25 x 5is nearest to 134. Replace x as 5 &
subtract.
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(cid:129) 5 9 8.Double the number above the top line (15) and
2,34,56 write it on the left region (30y)
(cid:129)
1 fl
251 34 fl
9. Determine the value of y (30y) × (y) which gives a
1 25
value lesser than or equal to 956.
30 3 9 56
× 3 9 09
10. Replace y as 9.
7 4
Nearest Square Root of 23,456 = 159.
Practise Exercise 1.4
Enter the nearest square root of the given number in the space provided
1. 2345 -------------------------
2. 5684 -------------------------
3. 430 -------------------------

1.3 Method to test whether given number is prime
A prime number is a number which can be divided by itself and one only.
Eg. 2,3,5,7,11,13...
1 is not a prime number, 2is the only even prime number.
To determine whether the given number is a prime number or not divide
the given number by all the prime numbers starting from 2 till the square
root of the given number. If it is divisible by any of the numbers then the
given number is not a prime number.

State whether the given number is prime or not prime in the space provided

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Practice Exercise 1.5
1. 567 --------------------------------
2. 787 --------------------------------
3. 431 --------------------------------

1.4 Method to find LCM of given numbers
LCM is the least common multiple of given set of numbers
Find LCM of 48, 24, 36
Divide successively by numbers which divides atleast one of the numbers
given
2 divides 48,24 and 36 2 48, 24,36
2 24, 12,18
Now 3 divides 12,6,and 9
12, 6,09
3
4, 2, 03
3
Now 2 divides 4 and 2 and 3 divides 3
2 4, 2, 1
2 2, 1, 1
Divide the numbers by suitable numbers
1, 1, 1
Until they get simplified to 1 each
Multiply all the numbers which divided 48,24,36
LCM = 2 x 2 x 3 x 3 x 2 x 2 = 144

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Practice Exercise 1.6
Find LCM of the following numbers
1. 25, 10 and 8 = ----------------------
2. 34, 7, 9 = ----------------------
3. 18, 24 , 48 = ----------------------
4. 25 , 30 , 50 = ----------------------

1.5 Compare fractions
Fractions can be compared by
1. Making the numerator similar
2. Making the denominator similar
By making numerator similar
5 11
Is greater than
7 13
To compare these fractions
Method 1: Make numerator similar
Determine LCM of 5 and 11
LCM = 55
5
Multiply by 11 to make numerator 55
7
55x 1155
= =
. Similarly
77x1177
1111x555
= =

1313x 565
The numerators of both fractions are same(55).
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