I.Introduction

Basic Math Review

Before we proceed for detailed discussion of the question in quantitative section, it is necessary to review the basic concepts used in Math. This chapter is intended to give you an intensive review of some of the basic concepts in Math, to familiarize you with basic Math terminology, formulae and general mathematical information. We will be covering the basic review on Arithmetic, Algebra, Geometry, Data analysis, and Data Interpretation.

After familiarizing yourself with the basic concepts, terminology, formulae in Arithmetic, you should take an Arithmetic Diagnostic Test, to identify your weak areas and use the Arithmetic Review that follows to strengthen these areas. You should follow the same procedure for other subjects such as Algebra, Geometry, Statistics, etc.

Even if you are strong in Math, you may like to brush up on the topics and to refresh your memory of important concepts. If you are weak in Math, you should read through the complete review.

Basic Arithmetic Review Common Symbols and Terms

= is equal to

≠ is not equal to

> is Greater than

< is less than

≥ is Greater than or equal to

≤ is less than or equal to

Square the result when a number is multiplied by itself, E.g. 3 x 3 = 9

Cube the result when a number is multiplied by itself twice E.g.3 x 3 x 3 = 27

English System Measurements

Length:

12 inches (in) = 1 foot (ft)

3 feet = 1 yard (yd)

36 inches = 1 yard

1760 yards = 1 mile (mi)

5280 feet = 1 mile

Weight:

16 ounces (oz) = 1 pound (lb)

2000 pounds = 1 ton (T)

Capacity:

2 cups = 1 pint (pt)

2 pints = 1 quart (qt)

4 quarts = 1 gallon (gal)

4 pecks = 1 bushel

Time:

365 days = 1 year

52 weeks = 1 year

10 years = 1 decade

100 years = 1 century

Metric System

Length:

Kilometer (km or KM) = 1000 meters (m)

Hectometer (hm) = 100 meters

Decameter (dam) = 10 meters

10 decimeter (dm) = 1 meter

100 centimeters (cm) = 1 meter

Volume:

1000 milliliters (ml or ML) = 1 liter (l or L)

1000 liters = 1 kiloliter (kl or KL)

Mass:

1000 milligram (mg) = 1 gram (g)

1000 grams = 1 kilogram (kg)

1000 kilogram = 1 metric ton (t)

Some approximations:

One meter is slightly more than a yard

One kilometer is about 0.6 mile

One kilogram is about 2.2 pounds

One liter is slightly more than a quart. - 3 -

About Numbers

Types and Nature of Numbers

Natural Number: These are also called counting numbers as these numbers are the ones which we use for counting purpose. It is represented by ‘N’.

For e.g. N = {1, 2, 3, …}

Whole Number: It includes all natural numbers and Zero. It is represented by ‘W’

For e.g. W = {0, 1, 2, 3, …}

Integers: It includes all whole numbers along with negative numbers. It is represented by ‘I’.

For e.g. ’I’ = {-,…..-2,-1,0,1,2,…+ }

Zero (0) is neither positive nor negative.

Natural numbers consists of only positive integers.

Whole numbers include natural numbers and zero.

Even Numbers: A number, which is completely divisible by 2, is called an even number. In other words, such numbers have 2 as factor when they are written as product of different numbers.

For e.g. : 30 = 2 x 3 x 5

42 = 2 x 3 x 7

A number is said to be a factor or sub multiple of another when it divides the other number exactly, without leaving the reminder.

For e.g. 5 and 3 are factors of 15.

Odd Numbers: These numbers are not completely divisible by 2. In other words, a number, which is not even, is an odd number.

For e.g. 1, 3, 5, 7, 9…..

It may be noted that zero is an exception to this even – odd classification.

0 is considered as even integer.

Real Numbers : Any measurement carried out in the physical world leads to some meaningful figure or value or number. This number is called a real number.

It consists of two groups:-

(i) Rational Numbers and

(ii) Irrational Numbers.

(i) Rational Numbers: - A rational number can always be represented by a fraction of the form p/q, where ‘p’ and ‘q’ are integers ‘q’ is not equal to zero. All integers and fractions are rational numbers.

(ii) Irrational Numbers: - An irrational number cannot be expressed in the form of p/q where ‘q’ is not equal to. For e.g.: √3, √2. It gives only an approximate answer in the form of a fractional or decimal number. The digits after the decimal point are non- ending. The same holds true for pi = 3.14 ….. which is again irrational. Alternatively, we can say that it is an infinite nonrecurring decimal number.

Prime Numbers: A Prime Number is a number, which has no factors besides itself and unity i.e., it is divisible only by itself and 1 and not divisible by any other numbers.

For E.G. :- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

Note:

(i) 2 is the only even number which is prime

(ii) All Prime Numbers other than 2 are odd numbers but all odd numbers are not prime numbers

For e.g.:- 9 is an odd number but not a prime number as it is divisible by 3.

(iii) 0 and 1 are not prime numbers

(iv) Prime numbers are always positive numbers. There are no negative prime numbers.

Composite Number: - A composite number is one, which has other factors besides itself and unity. Thus, it is a non prime number.

For e.g. : 4, 6, 9, 14, 15 etc

Tests Of Divisibility

1. Divisibility by 2: A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.

Example: 84932 is divisible by 2, while 65935 is not.

2. Divisibility by 3: A number is divisible by 3, if the sum of its digits is divisible by 3,

Example: 592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4+8 + 2) = 30, which is divisible by 3. But, 864329 is not divisible by 3, since sum of its digits = (8 + 6 + 4 + 3 + 2 + 9) = 32, which is not divisible by 3.

3. Divisibility by 4: A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

Example: 892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4. But, 749282 is not divisible by 4, since the number formed by the last two digits is 82, which is not divisible by 4.

4. Divisibility By 5: A number is divisible by 5, if its unit's digit is either 0 or 5. Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.

5. Divisibility by 6: A number is divisible by 6, if it is divisible by both 2 and 3. Ex. The number 35256 is clearly divisible by 2. Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3, Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.

6. Divisibility by 8: A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.

Example: 953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8. But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8.

7. Divisibility by 9: A number is divisible by 9, if the sum of its digits is divisible by 9.

Example: 60732 is divisible by 9, since sum of digits = (6 + 0 + 7 + 3 + 2) = 18, which is divisible by 9.

But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is not divisible by

9.

8. Divisibility by 10: A number is divisible by 10, if it ends with 0. e.g. 96410, 10480 are divisible by 10, while 96375 is not.

9. Divisibility by 11: A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.

Example: The number 4832718 is divisible by 11, since:

(Sum of digits at odd places) - (Sum of digits at even places) = (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11.

1. Divisibility by 12: A number is divisible by 2, if it is divisible by both 4 and 3.

Example: Consider the number 34632. The number formed by last two digits is 32, which is divisible by 4. (Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.

2. Divisibility by 14: A number is divisible by 14, if it is divisible by 2 as well as 7.

3. Divisibility by 15: A number is divisible by 15, if it is divisible by both 3 and 5.

4. Divisibility by 16: A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16.

Example: 7957536 is divisible by 16, since the number formed by the last four digits is 7536, which is divisible by 16.

Operations on Numbers:

Methods of BODMAS:

B stands for brackets, ‘0’ for ‘of ‘, ‘D’ for division, ’M’ for multiplication, ‘A’ for addition, ‘S’ for subtraction.

Illustrative Examples

1. When a certain number is divided by 12, a remainder of 1 is left. However, when the same number is divided by 14, a remainder of 5 is left. Give one such number.

Solution:-

We have to find out the number, which when divided by 12, leaves a remainder of 1. Thus, the general expression for such number would be (12n + 1). Further, when

the same number is divided by 14, a remainder of 5 is left. Thus, any number of type (12 n + 1), which fits into the next criterion also, is the required number. Hence, 145 is such a number.

1. Which least number should be added to 11148 to make it exactly divisible by seven?

Solution:-

When you divide 11148 by 7, we get 1592 as the quotient and 4 as remainder. Least number that should be added to it to make it exactly divisible by 7 is 3. Hence, 11148 + 3 = 11151, this is the number completely divisible by 7.

2. Which is the smallest number which, when subtracted from the sum of squares of 11 and 13, gives a perfect square?

Solution:-

Sum of square of 11 and 13 = 121 + 169 = 290. The perfect square nearer to 290 = 289 (172).

Therefore, 1 should be deducted from 290.