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 -
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.
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