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MATH FOR GMAT, SAT & SAT - Integers - Fractions - Decimals - Percents
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CONTENTS
INTEGERS
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
EVEN-ODD NUMBERS
INTEGERS
CONSECUTIVE NUMBERS
Consecutive Positive Integers
are integers that follow each other in
order:1, 2, 3, 4, 5, …
CONSECUTIVE INTEGERS
are even integers that follow each other in
order:2, 4, 6, 8, 10, …
EVEN NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
CONSECUTIVE EVEN INTEGERS
Consecutive Odd Integers
are odd integers that follow each other in
order:1, 3, 5, 7, …
ODD NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
CONSECUTIVE ODD INTEGERS
even eveneven odd oddodd
odd oddodd even eveneven
…, n – 2, n – 1, n , n + 1, n + 2, n + 3, …
EVEN/ODD NUMBERS
even even even
odd odd even
even odd odd
odd even odd
EVEN / ODD NUMBERS
EVEN / ODD NUMBERS
even even even
odd even even
even odd even
odd odd odd
EVEN / ODD NUMBERS
EVEN / ODD NUMBERS
Consecutive Prime Numbers
are prime numbers that follow each other
in order:2, 3, 5, 7, 11, …
CONSECUTIVE
NUMBERS
COUNTING INTEGERS
COUNTING CONSEC. INTEGERS
Counting Consecutive Integers
12, 13, 14, 15, 16, 17, 18, 19, 20
20 – 12 + 1 = 9
COUNTING CONSEC. EVEN/ODD INTEGERS
If the result is an integer number,
that is the answer.
Counting Consecutive
Even/Odd Integers
12, 13, 14, 15, 16, 17, 18, 19
19 – 12 + 1 = 8
8 / 2 = 4
Counting Consecutive
Even/Odd Integers
11, 12, 13, 14, 15, 16, 17, 18
18 – 11 + 1 = 8
8 / 2 = 4
Subtract the smallest number
from the largest number and
add 1, divide by 2.
COUNTING CONSEC. EVEN/ODD INTEGERS
If the result is not an integer number,
see how the series starts and ends.
Counting Consecutive
Even/Odd Integers
11, 12, 13, 14, 15, 16, 17, 18, 19
19 – 11 + 1 = 9
9 / 2 = 4.5
Counting Consecutive
Even/Odd Integers
10, 11, 12, 13, 14, 15, 16, 17, 18
18 – 10 + 1 = 9
9 / 2 = 4.5
CONSECUTIVE
NUMBERS
DIVISIBILITY
FACTOR / DIVISOR
FACTOR / DIVISIOR
a number that can be divided by another number without a
remainder.
MULTIPLE / DIVISIBLE
MULTIPLE / DIVISIBLE
Multiples of 2
Multiples of 3
DIVISIBILITY FACTS
1 is a factor/divisor of every integer.
0 is a multiple of every integer.
The factors of an integer include
positive and negative integers.
Factors of 4
Factors of 12
Prime Numbersare natural numbers
that has no positive divisors other than 1
and itself.
PRIM
E N
UM
BERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 …
If is prime number then, it won’t have any
factor such that .
2 1, 2
3 1, 3
5 1, 5
7 1, 7
11 1, 11
13 1, 13
17 1, 17
19 1, 19
Current Largest Prime
257,885,161 – 117,425,170 digits longJan 25, 2013, University of Central Missouri
Composite Number
a natural number greater than 1 that is not a prime number.
CO
MPO
SIT
E
NU
MBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 …
If k is composite number then, it
will have at least one factor p such
that 1 < p < k.
4 1, 2, 4
6 1, 2, 3,6
8 1, 2, 4, 8
9 1, 3, 9
12 1, 2, 3, 4, 6, 12
14 1, 2, 7, 14
15 1, 3, 5, 15
16 1, 2, 4, 8, 16
1 1 11 1, 11 21 1, 3, 7, 21
2 1, 2 12 1, 2, 3, 4, 6, 12 22 1, 2, 11, 22
3 1, 3 13 1, 13 23 1, 23
4 1, 2, 4 14 1, 2, 7, 14 24 1, 2, 3, 4, 6, 8, 12, 24
5 1, 5 15 1, 3, 5, 15 25 1, 5, 25
6 1, 2, 3, 6 16 1, 2, 4, 8, 16 26 1, 2, 13, 26
7 1, 7 17 1, 17 27 1, 3, 9, 27
8 1, 2, 4, 8 18 1, 2, 3, 6, 9, 18 28 1, 2, 4, 7, 14, 18
9 1, 3, 9 19 1, 19 29 1, 29
10 1, 2, 5, 10 20 1, 2, 4, 5, 10, 20 30 1, 2, 3, 5, 6, 10, 15, 30
Prime Factorization
is the decomposition of a composite number
into prime factors,
which when multiplied together equal the
original integer.
PRIME FACTORIZATION
PRIME FACTORIZATION
Eighter way, the result is
2 2 3 5 = 60 or 22 3 5 = 60
60
6 10
2 3 2 5
60
2 30
3
2 15
5
NUMBER OF DIVISORS
60 = 22 3
1 51
= 60
1 x 60
2 x 30
3 x 20
4 x 15
5 x 12
6 x 103 x 2 x 2 = 12
• Take all the exponents from the
prime factorization and add 1 to
each of them.
• Multiply the modified exponents
together.
of two integers is the largest positive integer that
divides the numbers without a remainder.
GCF
GCF
Prime factors of :18 = 2 × 3 × 3
Prime factors of :24 = 2 × 2 × 2 × 3
There is one 2 and one 3 in common.
The GCF of 18 and 24 is 2 × 3 = 6
GCF (GCD)
36
4 9
2 2 3 3
54
6 9
2 3 3 3
Shared Factors: 2, 3, 3
Multiply (GCF): 2 3 3 = 18
Find the GCF of 36 and 54:
The Least Common Multiple (LCM)
of two integers or more integers, is the smallest positive integer that is
divisible by all the numbers.
LCM
LCM
LCM
Factors Multiples1 2 3 4 6 12 12 24 36 48 60 72 84 96 108 …
1 2 3 6 9 18 18 36 54 72 90 108 126 …
GCF = 6 LCM = 36
GCF and LCM
The remainder
is the amount "left over" after performing
the division of two integers which do not
divide evenly.
REMAINDER
1- 6
3
2 7divisor
remainder
dividend
quotient
7 = 2∙3 + 1
dividend = divisor∙quotient + remainder
The remainder r when n is divided by a
nonzero integer d is zero if and only if n is
a multiple of d.
Dividing by 4
Divisible by
means that when you divide one number by another the result is a
whole number.
DIVISIBILITY BY 2
2, 40, 258, 1020
Last digit is even
DIVISIBILITY BY 3
69 6+9 = 15
504 5+0+4 = 9
1938 1+9+3+8 = 21
Sum of digits is a multiple of 3
DIVISIBILITY BY 4
512, 720, 1424, 1620
Last two digits are multiple of 4
DIVISIBILITY BY 5
25, 50, 560, 1005
Last digit is 5 or 0
DIVISIBILITY BY 6
72 7+2 = 9
1200 1+2+0+0 = 3
1860 1+8+6+0 = 15
Sum of the digits is multiple
of 3 and the last digit is even
DIVISIBILITY BY 7
3101 310 – 2 = 308
308 30 – 16 = 14
Take the last digit off the
number, double it and
subtract the doubled number
from the remaining number
DIVISIBILITY BY 9
729 7+2+9 = 18
810 8+1+0 = 9
9918 9+9+1+8 = 27
Sum of digits is a multiple of 9
DIVISIBILITY BY 10
30, 70, 100, 250, 560
Last digit is 0
Divisibility Rules
A number is divisible by … DivisibleNot
Divisible
2 If the last digit is even 3,728 357
3 If the sum of the digits is a multiple of 3 120 155
4 If the last two digits form a number divisible by 4 144 142
5 If the last digit is 0 or 5 150 123
6 If the number is divisible by both 2 and 3 48 20
9 If the sum of the digits is divisible by 9 729 811
10 If the last digit is 0 50 53
DIVISIBILITY
FRACTIONS
EQUIVALENT FRACTIONS
NAMING FRACTIONS
FRACTIONS
It is useful to think of a fraction
bar as a symbol for division.
The denominator of a fraction
can’t be equal to zero.
SIGNS IN A FRACTION
numeratorfraction
denominator
Any two of the
three signs of a
fraction may be
changed without
altering the value
of the fraction.
SIGNS IN A FRACTION
2 2 2 2
5 5 5 5
2 2 2 2
5 5 5 5
COMPARING FRACTIONS
Same Denominator
COMPARING FRACTIONS
Same Numerator
1
3
1
4
1
5
1
6
5 4?
8 7Cross- multiplication
COMPARING FRACTIONS
5 7?4 8
35 32
5 4
8 7
2 7
15 15
Make sure the denominators are the same.
Add the numerators, put the answer over
the denominator.
Simplify the fraction.
ADDING FRACTIONS
9
15
3
5
Make sure the denominators are the same.
Subtract the numerators. Put the
answer over the same denominator.
Simplify the fraction.
SUBTRACTING FRACTIONS
2 9 1
15 10 5
25
30
4 27 6
30
4 27 6
30 30 30
5
6
5 6
7 4
3 3 7 217
5 5 1 5
Multiply the numerators.
Multiply the denominators.
Simplify the
fraction.
MULTIPLYING FRACTIONS
30
28
15
14
1 3
2 5
Turn the second fraction upside-down
(this is now a reciprocal).
Multiply the first fraction by that
reciprocal.
Simplify the
fraction.
DIVIDING FRACTIONS
1 5
2 3
1 5
2 3
5
6
22
3
Distribute the exponent into the
numerator as well as into the denominator.
Evaluate the numerator
and the denominator.
Simplify the
fraction.
POWER OF FRACTIONS
2
2
2
3
4
9
4
9
Distribute the root into the
numerator as well as into the denominator.
Evaluate the numerator
and the denominator.
Simplify the
fraction.
ROOTS OF FRACTIONS
4
9
2
3
TRICKY OPERATIONS
The reciprocal of a is .1
a
The reciprocal of 2 is .1
2
The reciprocal of is .a b1
a b
The reciprocal of is .3
4
4
3
a
a c a d a dbc b d b c b c
d
COMPLEX FRACTIONS
A fraction with fractions in the
numerator or denominator.
Proper Fractionfraction that is less than one, with the numerator
less than the denominator.
Improper Fraction
a fraction in which the numerator is greater
than the denominator.
Multiply the whole number
part by the denominator
Add the numerator
The result is the new numerator (over the same denominator)
25
7
MIXED NUMBER TO IMPROPER FRACTION
5 7 2
7
37
7
Divide the denominator into the numerator.
The quotient becomes the whole number.
The remainder becomes the new numerator.
7
2
IMPROPER FRACTION TO MIXED NUMBER
13
2
MIXED NUMBERS
Part fraction whole
3 3100 100 75
4 4of
25252525
252525
PART – FRACTION
Part fraction whole
1 1100 100 50
2 2of
50
50
PART – FRACTION
FRACTIONS
DECIMALS
2 decimal
places
1 decimal
place
3 decimal
places
Divide the top
of the fraction
by the bottom.
FRACTION TO DECIMAL
50.625
8
40.571428
7
FRACTION TO DECIMAL
Write down the decimal divided by 1.
Multiply both top and bottom by 10 for every number after the decimal
point.
Simplify (or reduce) the
fraction.
DECIMAL TO FRACTION
0.750.75 100
1 100
75
100
3
4
Terminating Decimals
When the denominator has only factors 2, 5, a
combination of both or of its powers.
TERMINATING DECIMALS
1.5
2
71.4
5
2
4
3 3.06
50 2 5
3 3.1875
16 2
Repeating Decimals
when the denominator has other factors than 2 and 5 or its powers.
REPEATING DECIMALS
10.333
3
120.1212
99
40.571428571428...
7
50.384615384615...
13
repeatingdecimals
repeatingdecimals
10.333
3
120.1212
99
40.571428571428...
7
50.384615384615...
13
repeatingdecimals
repeatingdecimals
LENGTH OF THE CLUSTER
2n
d
4th
6th
3rd
6th
9th
1 20.111... 0.222...
9 9
3 70.333... 0.777...
9 9
COMMON REP. DECIMALS
11 120.1111... 0.1212...
99 99
25 830.2525... 0.8383...
99 99
COMMON REP. DECIMALS
127 2150.127127... 0.215215...
999 999
853 6150.853853... 0.615615...
999 999
COMMON REP. DECIMALS
OPERATIONS
WITH
DECIMALS
ADDING DECIMALS
Line up decimal
points.
132.7
96.543
229.243
SUBTRACTINGDECIMALS
Line up decimal
points.
132.7
96.543
36.157
It is not necessary to align the
decimal points.
Add the number of digits to the right of the decimal points in the
decimals being multiplied.
MULTIPLYING DECIMALS
125.3
1.2
2506
1253
150.36
MULTIPLYING DECIMALS
12.53
1.2
2506
1253
15.036
Move the decimal point in the divisor to
the right until the divisor becomes an
integer.
Move the decimal point in the dividend the same number of
places.
Proceed with the division.
DIVIDING DECIMALS
1.6 128.32
80.2
160 12832
1280
320
320
0
DIVIDING DECIMALS
1.6 12.832
8.02
1600 12832
12800
3200
3200
0
DECIMALS
PERCENTS
Percents:
a percentage is a number or ratio expressed as a fraction of 100.
PERCENTS
Percent means
hundredths or
number out of
100.
1%
2%
20%
PERCENTS
Percent means
hundredths or
number out of
100.
%100
11%
100
22%
100
2020%
100
nn
PERCENT EQUIVALENTS
14
12
34
25%
0.25
50%
0.50
75%
0.75
PERCENT EQUIVALENTS
16
13
23
16.6%
0.1666
33.33%
0.333
66.66%
0.666
PERCENT EQUIVALENTS
110
15
12
10%
0.1
20%
0.20
50%
0.5
100
percentPart whole
PERCENTS FORMULA
PERCENTS FORMULA
1225
100x
45 9100
x
6015
100x
2540 160
100
Percent increaseIs the ratio of the increase of two
numbers divided by the original number multiplyied by 100.
100%increase
Percent increaseoriginal whole
100%
(100 + n)%
n %
100%increase
Percent increaseoriginal whole
PERCENT INCREASE
The price of a tour goes up from
$80 to $100. What is the percent
increase?
20100% 25%
80Percent increase
PERCENT INCREASE
The price of a tour goes up from $80 to
$100. What is the percent increase?
20100% 25%
80Percent increase
100%decrease
Percent decreaseoriginal whole
(100 – n) %
100 %
n %
100%decrease
Percent decreaseoriginal whole
PERCENT DECREASE
The price of a tour goes down from
$100 to $80. What is the percent
decrease?
20100% 20%
100Percent decrease
PERCENT DECREASE
COMBINED PERCENT INCREASE
A price went up 10% one year, and the
new price went up 20% the next year.
What is the combined percent
increase?
32% increase110 120
100 132100 100
COMBINED PERCENT DECREASE
A price went down 10% one year, and
the new price went down 20% the next
year. What is the combined percent
decrease?
28% decrease90 80
100 72100 100
COMBINED PERCENT INC/DEC
A price went down 20% one year, and
the new price went up 10% the next
year. What is the combined percent
decrease?
12% decrease80 110
100 88100 100
COMBINED PERCENT INC/DEC
INICIAL
AMMOUNT
%
INCREASE /
DECREASE
PARTIAL
RESULT
%
INCREASE /
DECREASE
FINAL
RESULT
100 + 10% 110 -10% 99
100 - 10% 90 + 10% 99
100 + 20% 120 - 20% 96
100 - 20% 80 + 20% 96
100 + 50% 150 - 50% 75
100 - 50% 50 + 50% 75
100 100100 100
100 100
n n
100 100100 100
100 100
n n
COMBINED PERCENT INC/DEC
INTEREST
Interestis a fee paid by a
borrower of assets to the owner as a form of
compensation for the use of the assets.
INTEREST
1
I P r n
F P I
F P rn
Simple interest (I) is determined
by multiplying the interest rate (r)
by the principal (P) by the
number of periods (n).
SIMPLE INTEREST
SIMPLE INTEREST
1
I P r n
F P I
F P rn
SIMPLE INTEREST
Carine deposits $ 1,000 into a special bank account
which pays a simple annual interest rate of 5% for 3
years. How much will be in her account at the end of the
investment term?
P = 1,000
r = 5% = 0.05
n = 3
1
1,000 1 0.05 3
1,150
F P rn
F
F
55% 1,000 1,000 50
100of
SIMPLE INTEREST
55% 1,000 1,000 50
100of
55% 1,000 1,000 50
100of
Simple Interest on 1,000.00 after:
SIMPLE INTEREST
Interest (I) calculated on the initial
principal (P) and also on the
accumulated interest of previous
periods of a deposit or loan.
1n
F P r
I F P
COMPOUND INTEREST
COMPOUND INTEREST
1n
F P r
I F P
Annual rate = 12%, compounded:0 12 months6
6% 6%
COMPOUND INTEREST
Principal = $ 100, Annual rate = 12%,
Time = 1 year, compounded:
COMPOUND INTEREST
1
100 1 0.12F
2
100 1 0.06F
4
100 1 0.03F
12
100 1 0.01F
COMPOUND INTEREST
Carine deposits $ 1,000 into a special bank account
which pays a compound annual interest rate of 5% for 3
years. How much will be in her account at the end of the
investment term?
P = 1,000
r = 5% = 0.05
n = 3
3
1
1,000 1 0.05
1,157.625
nF P r
F
F
COMPOUND INTEREST
55% 1,000.00 1,000.00 50.00
100of
55% 1,050.00 1,050.00 52.50
100of
55% 1,102.50 1,102.50 55.13
100of
Compound Interest on 1,000.00 after:
COMPOUND INTEREST
SIMPLE INTEREST vs
COMPOUND INTEREST
Prin
cip
al
Co
mp
ou
nd
Inte
rest
Sim
ple
Inte
rest
Co
mp
ou
nd
Inte
rest
Sim
ple
Inte
rest
Co
mp
ou
nd
Inte
rest
Sim
ple
In
tere
st1,0
50
.00
1,1
00
.00
1,1
50
.00
P = 1,000.00
r = 5% = 0.05
n = 3 years
PERCENTS
SUMMARY
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