Class1 algebra and arithmetic

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A rope 20.6 meters long is cut into two pieces. If the length of one piece of rope is 2.8
meters shorter than the length of the other, what is the length, in meters, of the longer
piece of rope?
A)7.5 B)8.9 C)9.9 D)10.3 E)11.7

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How many cases do you need if you have to pack 112 pairs of
shoes into cases that each hold 28 shoes?
Rpta: 16
SUNDAY MONDAY TUESDAY WEDNESDAY THURSDAY
FRIDAY SATURDAY What day comes three days after the day
which comes two days after the day which comes immediately
after the day which comes two days after Monday?
Rpta: Tuesday
Source: Only A True Nerd Can Ace This IQ Test

60 is a multiple of each of its factors.
60 is divisible by each of its divisors
LEAST COMMON MULTIPLE (of two nonzero integers a and b): The least
positive integer that is a multiple of both a and b.
GREATEST COMMON DIVISOR(of two nonzero integer a and b): the
greatest positive integer that is a divisor of both a and b
DIVISIBILITY, EVEN AND ODD INTEGERS
15 is divisible by 3
15 is multiple of 3
3 is a factor of 15
3 goes into 15
3 divides 15
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PROPERTIES OF INTEGERS

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Samantha has two pieces of cloth. One piece is 72 inches wide and the
other piece is 90 inches wide. She wants to cut both pieces into strips of
equal width that are as wide as possible. How wide should she cut the
strips?
Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel
both exercised today. How many days will it be until they exercise
together again?

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Mrs. Evans has 120 crayons and 30 pieces of paper to give to her
students. What is the largest number of students she can have in her
class so that each student gets equal number of crayons and equal
number of paper
Rosa is making a game board that is 16 inches by 24 inches. She
wants to use square tiles. What is the largest tiles she can use?

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WHOLE NUMBERS
 Whole numbers are simply the numbers
0,1,2,3,4,5, … (and so on)
 But numbers like 1/2 , 1.1 and 3. 5 are not whole
numbers.
COUNTING HUMBERS
 Counting numbers are whole numbers, but
without the zero. Because you can’t “count” zero.
 So they are 1,2,3,4,5,… (and so on)
NATURAL NUMBERS
 Natural numbers can mean either “counting
numbers” or “whole numbers”, depending on the
subject.

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If N is a positive integer, then the least value of N for which N! is
divisible by 1,000 is?
A)1 B)4 C)9 D)15 E)30
www.DominateTheGMAT.com
If n=20!+17, then n is divisible by which of the following
I. 15
II. 17
II. 19
(A) None
(B) I only
(C) II only
(D) I and II
(E) II and III
Source: GMAT 2016 Test Guide

Those rules come in very handy. An integer is divisible by:
2 if the integer is even.
3 if the sum of the integer’s digits is a multiple of 3.
5 if the integer ends in 0 or 5.
9 if the sum of the integer’s digits is multiple of 9.
10 if the integer ends in 0.
FACTORS ARE DIVISORS
2 is a factor of 6 = 6 is a multiple of 2
2 is a divisor of 6 = 6 is divisible by 2
2 divides 6= 2 goes into 6 (evenly) (without a remainder)
Even integer: divided by 2
Odd integer: otherwise
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DIVISIBILITY RULES FOR SMALL INTEGERS

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Terms Used in Division
 The terms used in division are dividend, divisor, quotient
and remainder.
 D=dq + r
 100 divided by 45 is 2 remainder 10
 24 divided by 4 is 6 remainder 0. In general, the remainder
is 0 if and only if:
 The dividend is ……… of the divisor
 The dividend is a ……………. of the divisor
 Prime number: is an integer greater than 1 that has only
two positive ……… 1 and itself
 Prime factorization, prime divisor
 12 = 22
.3

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When positive integer n is divided by 5, the remainder is 1. When n is divided
by 7, the remainder is 3. What is the smallest positive integer k such that k +
n is a multiple of 35?
A)3 B)4 C)12 D)32 E)35
GMAT 2018 Test Guide

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
𝑎
𝑏
 When we invert the fraction, we find its
reciprocal

𝑏
𝑎
 4
3
7
: mixed number
FRACTIONS

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After 4,000 gallons of wáter were added to a large wáter tank that was already
filled to ¾ of its capacity, the tank was then at 4/5 of its capacity. How many
gallons of wáter does the tank hold when filled to capacity?
A)5,000 B)6,200 C)20,000 D)40,000 E)80,000

DECIMALS
The decimal number system is based on representing numbers using powers
of 10. The place value of each digit corresponds to a power of 10.
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Source: Teaching Packs

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STANDARD FORM OF A DECIMAL NUMBER
In Britain this is another name for Scientific Notation, where you write down a
number this way:
STANDARD FORM OF AN EQUATION
The “Standard Form” of an equation is:
(some expression)=0
In other words, “=0” is on the right, and everything else in on the left.
Source: https://www.mathsisfun.com/algebra/standard-form.html

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DIRECTLY PROPORTIONAL AND INVERSELY
PROPORTIONAL
 DIRECTLY PROPORTIONAL: As one amount
increases another amount increases at the same rate
Example. How much you earn is directly proportional to
how many hours you work. Work more hours, get more
pay; in direct proportion.
Earnings ∝ Hours worked
 INVERSELY PROPORTIONAL: When one value
decreases at the same rate that the other increases.
Example. Speed and travel time
Speed and travel time are Inversely proportional
because the faster we go the shorter the time.
As speed goes up, travel time goes down

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A total of 5 liters of gasoline is to be poured into two empty containers with
capacities of 2 liters and 6 liters, respectively, such that both containers will be
filled to the same percent of their respective capacities. What amount of
gasoline, in liters, must be poured into the 6-liter container?
A)4
1
2
B)4 C) 3
3
4
D)3 E) 1
1
4
Five machines at a certain factory operate at the same constant rate. If four of
these machines, operating simultaneously, take 30 hours to fill a certain
production order, how many fewer hours does it take all five machines,
operating simultaneously, to fill the same production order?
A)3 B)5 C)6 D)16 E)24

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•y ∝ x
•y = kx
This equation gives us a straight line. The gradient of the line is k.
Straight line
Source: Bitesize-Math-Graphs and proportion - Higher

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•y ∝ x2
•y = kx2
This equation gives us a curve. The larger the value of k, the steeper the
graph
Quadratic

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•y ∝ x3
•y = kx3
Cubic

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y ∝ 𝑥
y =k 𝑥
You have similar shaped curves for any powers between 0 and 1. Again,
increasing k will make the graph steeper.
Square root

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•y ∝ 1/x ²
•y = k/x ²
Inverse proportion leads to curved graphs.
Inverse proportion

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The acceleration of a particle is inversely proportional to the square of the time since it
was fired. If the acceleration of the particle 20 seconds after it was fired was 5m/s2,
what was its acceleration 5 seconds later? (Source Mathisfun)
A)3.2m/s2 B)4m/s2 C)6.4m/s2 D)20m/s2
The circumference (C cm) of a circle is directly proportional to its diameter (d cm).
The circumference of a circle of diameter 3.5 cm is 11 cm What is the circumference of a
circle of diameter 4.2 cm?
A)9.17cm B)11.7cm C)13.2cm D)14cm
It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they
work at the same rate?
A)2.5hours B)3hours C)3.2hours D)1.5hours

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The total cost of filling up your car with gas varies directly with the
number of gallons of gasolina you are purchasing. If a gallon of gas costs
$2.25, how many gallons could you purchase for $18?
A)9 B)11 C)8 D)14
Arthur is typing a paper that is 390 words long. He can type 30 words in a
minute. How long will it take for him to type the paper?

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The sum of the weekly salaries of 5 employees is $3250. If each of the 5 salaries is to
increase by 10 percent, then the average (arithmetic mean) weekly salary per employee
will increase by
A)$52.50 B)$55.00 C)$57.50 D)$62.50 E)$65.00
A student’s average (arithmetic mean) test score on 4 tests is 78. What must be the
student’s score on a 5th test for the student’s average score on the 5th test to be 80?
A)80 B)82 C)84 D)86 E)88

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SEQUENCES
A sequence is a list of things (usually numbers) that are in order.
3,5,7,9,…
Notación
To make it easier to use rules, we often use this special style
𝑥 𝑛 is the term
n is the term number
Arithmetic sequences: The difference between one term and the next is a
constant.
Geometric sequences: Each term is found by multiplying the prevoious
term by a constant.
Triangular numbers: 1,3,6,10,15,21,28,36,45,…𝑥 𝑛 =
𝑛(𝑛+1)
2
Fibonacci Sequence: 𝑥 𝑛 = 𝑥 𝑛−1 + 𝑥 𝑛−2

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The sequence 𝑎1, 𝑎2, … , 𝑎 𝑛, … is such that 𝑎 𝑛 = 2𝑎 𝑛−1 − 𝑥 for all positive integers
𝑛 ≥ 2 and for certain number x. If 𝑎3 = 27 𝑎𝑛𝑑 𝑎5 = 99, what is the value of x?
A)3 B)9 C)18 D)36 E)45
Source: GMATH 2018 Test Guide
The infinite sequence 𝑎1, 𝑎2, … , 𝑎 𝑛 … is such that 𝑎1 = 2, 𝑎2 = −3, 𝑎3 = 5, 𝑎4 =
− 1, and
𝑎 𝑛 = 𝑎 𝑛−4 for n>4. What is the sum of the first 97 terms of the sequence?
A)72 B)74 C)75 D)78 E)80

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WORD PROBLEMS
RATE PROBLEMS
The distance that an object travels is equal to the product of the average speed
at which it travels and the amount of time it takes that distance, that is,
𝑹𝒂𝒕𝒆𝒙𝑻𝒊𝒎𝒆 = 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆
If a car travels at an average speed of 70 kilometers per hour for 4 hours, how
many kilometers does it travel?
On a 400 mile trip, Car X traveled half the distance at 40 miles per hour
(mph) and the other half at 50 mph. What was the average speed of Car
X?
44
4
9

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During a certain time period, Car X traveled north along a straight road at a constant
rate of 1 mile per minute and used fuel at a constan trate of 5 gallons every 2 hours.
During this time period, if Car X used exactly 3.75 gallons of fuel, how many miles did
Car X travel?
A)36 B)37.5 C)40 D)80 E)90
Abdul, Barb, and Carlos all live on the same straight road, on which their school is
also located. The school is halfway between Abdul’s house and Barb’s house. Barb’s
house is halfway between the school and Carlos’ house. If the school is 4 miles from
Carlos’ house, how many miles is Abdul’s house from Carlos’ house?
A)1 1/3 B)2 C)4 D)6 E)8

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In a recent election, Ms Robbins received 8,000 votes cast by independent
voters, that is, voters not registered with a specific political party. She also
received 10 percent of the votes cast by those voters registered with a political
party. If N is the total number of votes cast in the election and 40 percent of
the votes cast were cast by independent voters, which of the following
represents the number of votes that Ms. Robbins received?
A)0.06N+3,200
B)0.1N+7,200
C)0.4N+7,200
D)0.1N+8,000
E)0.06N+8,000

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Some rate problems can be solved by using ratios
If 5 shirts cost $44, the, at this rate, what is the cost of 8 shirts?
$70,40
A flat patio was built alongside a house as shown in the figure above. If all
angles are right angles, what is the área of the patio in square feet?
A)800 B)875 C)1,000 D)1,100 E)1,125

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WORK PROBLEMS
In a work problema, the rates at which certain persons or machines work alone are
usually given, and it is necessary to compute the rate at which they work together
(or viceversa).
The basic formula for solving work problems is
1
𝑟
+
1
𝑠
=
1
𝑏
, where r and s are, for
example, the number of hours it takes Rae and Sam, respectively, to complete a job
when working alone, and b is the number of hours it takes Rae and Sam to do the
job when working together. The reasoning is that in 1 hour Rae does
1
𝑟
of the job,
Sam does
1
𝑠
of the job, and Rae and Sam together do
1
𝑏
of the job.

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If Machine X can produce 1,000 bolts in 4 hours and Machine Y can produce
1,000 bolts in 5 hours, in how many hours can Machines X and Y, working
together at these constant rates, produce 1,000 bolts?
If Art and Rita can do a job in 4 hours when working together at their
respective constan trates and Art can do the job alone in 6 hours, in how
many hours can Rita do the job alone?

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MIXTURE PROBLEMS
In mixture problems, substances with diferent characteristics are combined,
and it is necessary to determine the characteristics of the resulting mixture.
Examples:
If 6 pounds of nuts that cost $1.20 per pound are mixed with 2 pound of
nuts that cost $1.60 per pound, what is the cost per pound of the mixture?
$1.30
How many liters of a solution that is 15 percent salt must be added to 5
liters of a solution that is 8 percent salt so that the resulting solution is 10
percent salt?
2liters

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The company at which Mark is employed has 80 employees, each of whom
has a different salary. Mark’s salary of $43,700 is the second-highest salary in
the first quartile of the 80 salaries. If the company were to hire 8 new
employees at salaries that are less than the lowest of the 80 salaries, what
would Mark’s salary be with respect to the quartiles of the 88 salaries at the
company, assuming no other changes in the salaries?
A) The fourth-highest salary in the first quartile
B) The highest salary in the first quartile
C) The second-lowest salary in the second quartile
D) The third-lowest salary in the second quartile
E) The fifth-lowest salary in the second quartile

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Jackie has two solutions that are 2 percent sulfuric acid and 12 percent
sulfuric acid by volumen, respectively. If these solutions are mixed in
appropiate quantities to produce 60 liters of a solution that is 5 percent
sulfuric acid, approximately how many liters of the 2 percent solution will be
required?
A)18 B)20 C)24 D)36 E)42

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By weight, liquid A makes up 8 percent of solution R and 18 percent of solution S. If 3
grams of solution R are mixed with 7 grams of solution then liquid A accounts for
what percent of the weight of the resulting solution?
10% 13% 15% 19% 26%
Source: GRE Test Guide
After driving to a riverfront parking lot, Bob plans to run south along the river, turn
around, and return to the parking lot, running north along the same path. After
running 3.25 miles south, he decides to run for only 50 minutes more. If Bob runs at a
constant rate of 8 minutes per mile, how many miles farther south can he run and
still be able to return to the parking lot in 50 minutes?
A)1.5 B)2.25 C)3 D)3.25 E)4.75

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INTEREST PROBLEMS
Interest can be computed in two basic ways. With simple annual interest, the interest is
compute don the principal only and is equal to (principal)x(interest rate)x(time). If
interest if compounded, then interest is compute don the principal as well as on any
interest already earned.
If $8,000 is invested at 6 percent simple annual interest, how much interest will it have
get after 3 months?
If $10 000 is invested at 10 percent annual interest, compounded semiannual balance
after 1 year?
The balance after the first 6 months would be
10,000 + 10,000 0.05 = $10,500
The balance after one year would be 10,000 + 10,500 0.05 = $11,500

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Note that the interest rate for each 6-month period is 5% which is half of the 10% annual
balance after one year can also be expressed as
10,000(1 +
0.10
2
)2
dollars
Alex deposited x dollars into a new account that earned 8 percent annual interest,
compounded annually. One year later Alex deposited an additional x dollars into the
account. If there were no other transactions and if the account contained w dollars at the
end of two years, which of the following expresses x in terms of w?
A)
𝑤
1+1.08
B)
𝑤
1.08+1.16
C)
𝑤
1.16+1.24
D)
𝑤
1.08+(1.08)2 E)
𝑤
(1.08)2+(1.08)3

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DISCOUNT
If a Price is discounted by n percent, then the price becomes (100-n) percent of the
original price
A certain customer paid $24 for a dress. If that price represented a 25 percent discount
of the original price of the dress, what was the original price of the dress?
$32
The price of an ítem is discounted by 20 percent and then this reduced price is
discounted by an additional 30 percent. These two discounts are equal to an overall
discount of what percent?
44%
Profit
Gross profit is equal to revenues minus expenses, or selling price minus cost.
A certain appliance costs a merchant $30. At what price should the merchant sell the
appliance in order to make a gross profit of 50 percent of the cost of the appliance?
$45

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Creryl purchased 5 identical hollow pine doors and 6 identical solid oak doors for the
house she is building. The regular price of each solid oak door was twice the regular
price of each hollow pine door. However, Cheryl was given a discount of 25% off the
regular price of each solid oak door. If the regular price of each hollow pine door was
$40, what was the total price of all 11 doors?
A)$320 B)$540 C)$560 D)$620 E)$680
A manufacturer of a certain product can expect that between 0.3 percent and
0.5 percent of the units manufactured will be defective. If the retail price is
$2,500 per unit and the manufacturer offers a full refund for defective units, how
much money can the manufacturer expect to need to cover the refunds on
20,000 units?
A)Between $15,000 and $25,000
B) Between $30,000 and $50,000
C) Between $60,000 and $100,000
D) Between $150,000 and $250,000
E) Between $300,000 and $500,000

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Last month a certain music club offered a discount to preferred
customers. After the first compact disc purchased, preferred customers
paid $3.99 for each additional compact disc purchased. If a preferred
customer purchased a total of 6 compact discs and paid $15.95 for the
first compact disc, then the dollar amount that the customer paid for the 6
compact discs is equivalent to which of the following?
A)5(4.00)+15.9
B)5(4.00)+15.95
C)5(4.00)+16.00
D)5(4.00-0.01)+15.9
E)5(4.00-0.05)+15.95

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Source: Only A True Nerd Can Ace This IQ Test
If Milly gives tilly $60 the money they have is in the ratio 2:1;
however, if Tilly gives Milly $10 the ratio is 1:3. How much money
did Milly and Tilly have before they exchanged any money?

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ALGEBRA
Algebra is based on the operations of arithmetic and on the concept of
an unknown quantity, or variable. Letters such as x or n are used to
represent unknown quantities.
The expression 19𝑥2
− 6𝑥 + 3 consists of the terms 19x2, −6x, and 3,
where 19 is the coefficient of 𝑥2, −6 is the coefficient of x, and 3 is a
constant term (or coefficient of ). Such an
expression is called a second degree (or quadratic) polynomial in x
since the highest power of x is 2.

EXPONENTS AND ROOTS
𝐵 𝐸 where. B:base and E:Exponent
74: seven to the fourth power
Five to the third power is ….
When the exponent is 2, we call the process squaring. Thus, 6
squared is 36, 8 squared is 64.
A negative number raised to an ….. power is always positive and a
negative number raised to an …. power is always negative.
-42
: “the negative of 4 squared” the exponent is applied before the
negative sign.
For all nonzero numbers a, 𝑎0
=1, the expression 00
is
………………………………
If the exponent is negative:
𝑎−𝑛=1/𝑎 𝑛
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46
When you multiply exponential terms that have the same base, add the exponents.
𝑎5
𝑥𝑎3
= 𝑎8
a to the fifth times a to the third equals a to the eighth
When you divide exponential terms that have the same base, subtract the exponents.
𝑥 𝑦
𝑥2
= 𝑥 𝑦−2
PRETTY MUCH ANYTHING TO THE ZEROTH POWER: ONE
But 𝟎 𝟎
= 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
A square root: exponent = ½
It is a root of order two. Higher order roots of a positive number n are defined similarly:
For orders 3 and 4:
the cube root 3
𝑛
The fourth root 4
𝑛
NEGATIVE POWER: ONE ------ A POSITIVE POWER
𝑎. 𝑎. 𝑎
𝑎. 𝑎. 𝑎. 𝑎. 𝑎.
=
1
𝑎. 𝑎
=
1
𝑎2
=
𝑎3
𝑎5
= 𝑎3−5
= 𝑎−2
APPLY TWO EXPONENTS: MULTIPLY THE EXPONENTS
𝑎2 4
= 𝑎8
Taller 2018
EXPONENTS AND ROOTS
23/10/2018

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SQUARE ROOT: POWER OF ONE HALF
 Consider this equation:
 9 𝑥 2 = 9
 2x =1 -> x = ½
 Now we know that 9
1
2
2
= 9 = 9
2
 So we can conclude that
 9 = 9
1
2
 A square root is equivalent to an exponent
of ½

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 One way to indicate the repeating part of a
decimal that repeats without end is to use
a bar over the digits that repeat.

15
14
= 1.0714285
 Every ………… number can be expressed as
a terminating or repeating decimal. The
converse is also true; that it, every
terminating decimal represents a …………..
Number.

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 Not all decimals are terminating or
repeating; for instance:
 Such numbers are called ……………..
numbers
𝜋 ∶ 𝑃𝑖
𝑒: 𝐸𝑝𝑠𝑖𝑙𝑜𝑛
2 =1.4142135….

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REAL NUMBERS
 𝑹𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 = 𝑰𝒓𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝒏. +𝑹𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝒏.
 The absolute value of x:
 Triangle inequality:
𝑎 + 𝑏 ≤ 𝑎 + 𝑏
 Less than
 Greater than
 Equal to

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COMBINING LIKE TERMS
AND PULLING OUT TERMS
 3𝑥2
+ 2𝑥2
= (3 + 2)𝑥2
Three x squared
plus two x squared equals the quantity
three plus two, times x squared
 The word quantity indicates parentheses.
 PULLING OUT THE COMMON FACTOR
 3𝑥2
+ 7𝑥 + 2𝑥2
− 𝑥 = 5𝑥2
+ 6𝑥 = 5𝑥 + 6 𝑥
 2 + 2 𝜋 = 2(1 + 𝜋)

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RATIOS AND PERCENT
 Ratios can be reduced to lowest terms. For example, if
there are 8 apples and 12 oranges, then the ratio of
the number of apples to oranges is still 2 to 3, the
ratio 9 to 12 is equivalent to the ratio 3 to 4.
 Consider r, s and t, then their relative sizes can also be
expressed as a ratio with the notation “r to s to t”
 If there are 5 apples, 30 pears, and 20 oranges in a
basketm then the ratio of the numbers of apples to
pears to oranges is 5 to 30 to 20, but it can be
reduced to 1 to 6 to 4 dividing each number by the
……………………………………. Of 5, 30 and 20 which is 5

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Percent change
 When a quantity changes from an initial
positive amount to another positive
amount, for instance, an employee’s salary
that is raised, you can compute the
amount of change as a percent of the
initial amount.
 If a quantity increases from 600 to 750

𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒
𝑏𝑎𝑠𝑒
=
750−600
600
= 25%

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 An investment in a mutual fund increased
by 12% in a single day. If the value of the
investment before the increase was $1300,
what was the value after the increase?

5523/10/2018 Taller 2018
 The monthly enrollment at a preschool
decreased by 8% during one month and
increased by 6% during next month. What
was the cumulative percent change for the
two months

23/10/2018 Taller 2018 56
Operations with Algebraic
Expressions
An algebraic expression has one or more
variables and can be written:
2𝑥 ; 𝑦 −
1
3
;
8
𝑛 + 𝑝
; 5𝑧2
− 𝑤3
𝑧
Identities
 𝑎 + 𝑏 2
= 𝑎2
+2𝑎𝑏 + 𝑏2
(The square of binomial)
 𝑎 − 𝑏 3 = 𝑎3 −3𝑎𝑏(𝑎 − 𝑏) + 𝑏3
 𝑎2−𝑏2= (𝑎 + 𝑏)(𝑎 − 𝑏)

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PASCAL’S TRIANGLE

23/10/2018 Taller 2018 58
Solving quadratic equations
 A quadratic equation in the variable x:
 𝑎𝑥2 + 𝑏𝑥 + 𝑐 =0
 Where a y b are real numbers and 𝑎 ≠ 0
Quadratic formula:
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
What the dismininant tells you

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Solving Linear Inequalities
 𝑥 ≤ 𝑦 x is less than or equal to y
 𝑥 ≥ 𝑦 x is greater than or equal to y
 𝑥 < 𝑦 x is less than y
 𝑥 > 𝑦 x is greater than y
 −3 ≤ 𝑦 < 5 -3 is less than or equal to y,
and y is less than 5
 y is greather or equal to -3 and less than 5

23/10/2018 Taller 2018 60
Which of the following could be the graph of all values of x that satisfy
the equality ?
2 − 5𝑥 ≤ −
6𝑥−5
3
Source: GRE 2012

23/10/2018 Taller 2018 61
A garden center sells a certain grass seed in 5-pound bags at $13.85 per bag, 10-
pound bags at $20.43 per bag, and 25 pound bags at $32.25 per bag. If a
customer is to buy at least 65 pounds of the grass seed, but no more than 80
pounds, what is the least possible cost of the grass seed that the customer will
buy?
A)$94.03 B)$96.75 C)$98.78 D)$102.07 E)$105.36

23/10/2018 Taller 2018 62
To FLIP the sign when dividing by a
negative. If:
−2𝑥 > 4
𝑥 < −2
In inequalities, you can’t multiply or
divide by a variable unless you know
its sign.

23/10/2018 Taller 2018 63
COORDINATE GEOMETRY
The figure above shows the (rectangular) coordinate plane. The horizontal
line is called the x-axis and the perpendicular vertical lines is called the y-
axis. The point at which these two axes intersect with one another is called
the origin.
The axes divide the plane into four quadrants, I, II, III and IV

23/10/2018 Taller 2018 64
Each point in the plane has an x-coordinate and a y coordinate. A point
is identified by an ordered pair (x,y) of numbers in which the x-
coordinate is the first number and the y-coordinate is the second
number.
In the graph above, the (x,y) coordinates of point P are (2,3) since P is 2
units to the right of the y-axis (that is, x=2) and 3 units above the x-axis

23/10/2018 Taller 2018 65
One way to find the distance between two points in the coordinate plane is to
use the Pythagorean theorem
To find the distance between points R and S using the Pythagorean theorem, draw the
trinagle as shown. Note that Z has (x,y) coordinates (-2,-3), RZ=7, and ZS=5. Therefore,
the distance between R and S is equal to
72 + 52 = 74
For a line in the coordinate plane, the coordinates of each point on the line satisfy a
linear equation of the form 𝑦 = 𝑚𝑥 + 𝑏 (or the form x=a if the line is vertical). For
example, each point on the line on the next page satisfies the equation 𝑦 = −
1
2
𝑥 + 1.
One can verify this for the points (-2,2),(2,0), and (0,1) by substituting the respective
coordinates for x and y in the equation

23/10/2018 Taller 2018 66
The graph of a quadratic function is called a parábola and always has the
shape of the curve above, although it may be upside down or have a greater
or lesser width. Note that the roots of the equation 𝑓 𝑥 = 𝑥2 − 1 = 0 and
x=1 and x=-1; these coincide with the x-intercepts since x-intercepts are
found by setting y=0 and solving for x. Also, the y-intercept is f(0)=-1 because
this is the value of y corresponding to x=0. For any function f, the x-intercepts
are the solutions of the equation f(x)=0 and y-intercept is the value f(0)
If all the points were graphed for −2 ≤ 𝑥 ≤ 2, then the graph would appear as follows

23/10/2018 Taller 2018 67
In the xy-plane, the point with coordinates (-6,-7) is the center
of circle C. The point with coordinates (-6,5) lies inside C, and
the point with coordinates (8,-7) lies outside C. If m is the
radius of C and m is an integer, what is the value of m?
GRE Test Guide

23/10/2018 Taller 2018 68
Functions
 ℎ 𝑥 = 𝑥2
+ 5 for −2 ≤ 𝑥 ≤ 2
 Where f(x) is called the value of f at x and
is obtained by substituting the value of x in
the expression above.
 The domain of a function is the set of all
permissible inputs.
 𝑓 𝑥 =
2𝑥
𝑥−6
; 𝑔 𝑥 = 𝑥3
+ 𝑥 + 2 − 10

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Graphs of functions
 Consider the functions defined by
 𝑔 𝑥 = 𝑥 + 1 2 and 𝑓 𝑥 = 𝑥 + 2
 The graph of f is the graph of 𝑥 shifted
upward by 2 units.
 The graph of g is the graph of 𝑥2 shifted to
the left by 1 unit.

23/10/2018 Taller 2018 71
In general
 The graph of h(x)+c is the graph of h(x)
shifted upward by c units.
 The graph of h(x)-c is the graph of h(x)
shifted downward by c units.
 The graph of h(x+c) is the graph of h(x)
shifted to the left by c units.
 The graph of h(x-c) is the graph of h(x)
shifted to the right by c units.

23/10/2018 Taller 2018 72
 Consider functions defined by
𝑓 𝑥 = 2 𝑥 − 1 and 𝑔 𝑥 = −
𝑥2
4
 The graph of f is the graph of 𝑥 shifted to
the right by 1 unit and then stretched
vertically away from the x-axis by a factor
of 2.
 The graph of g is the graph of 𝑥2
shrunk
vertically by a factor of ¼ an then
reflected in the x-axis.

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In general, for any function h(x) and any positive number c, the
following are true
The graph of ch(x) is the graph of h(x) stretched vertically by a factor
of c if c>1.
The graph of ch(x) is the graph of h(x) shrunk vertically by a factor of
c if 0<c<1.

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In general
 The graph of ch(x) is the graph of h(x)
stretched vertically by a factor of c if c>1.
 The graph of ch(x) is the graph of h(x)
shrunk vertically by a factor of c if 0<c<1

23/10/2018 Taller 2018 75
Find an algebraic expression to represent each of the following.
(a) The square of y is subtracted from 5, and the result is
multiplied by 37.
(b) Three times x is squared, and the result is divided by 7
(c) The product of (x+4) and y is added to 18

23/10/2018 Taller 2018 76
The figure above shows the graph of the function f in the xy-plane,
what is the value of f(f(-1))?
A)-2 B)-1 C)0 D)1 E)2
GRE Test guide 2018

78
[1] Mastering Inequality Questions on the GMAT-Veritas Prep
https://www.youtube.com/watch?v=qkPUHuwJkOE
[2] GCSE Bitesize Math Graph and proportion
http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/proportionhirev3.shtml
[3] Math is fun Directly Proportional and Inversely Proportional
https://www.mathsisfun.com/algebra/directly-inversely-proportional.html
[4]The official Guide for GMAT review 2016
Graduate management admission council mba.com
[5]GMAT Official Guide 2018
Graduate Management Admission Council mba.com
[6]Manhattan GMAT Foundations of GMAT Math 5th Edition-GMAT Strategy Guide Arithmetic and Algebra
Dan Gonzales
Create an account with Manhattan GMAT at the website:
www.manhattanprep.com/register
[7] The oficial Cambridge Guide to IELTS for academic and general training
-Pauline Cullen
-Amanda French
-Vanessa Jakeman
[8] The oficial Guide to GRE revised general test 2nd Edition
Mc Graw-Hill Education product www.mhprofessional.com
23/10/2018 Taller 2018

79
THANKS FOR YOUR ATTENTION!
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Class1 algebra and arithmetic

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