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Murtaza Khan Khajjak The Quantitative secti on measures your basic mathemati cal skill s, understanding of el ementarymathemati cal concepts, and the ability to reason quantitatively and solve problems i n a quanti tative setting. There i s a balance of questi ons requiri ng basi c knowl edge of ari thmetic, al gebra, geometry, and data analysis. These are essential content areas usually studied at the hi gh school level.

The questions i n the quantitative section can al so be from

• Discrete Quantitative Question

• Quantitative Comparison Question

• Data Interpretation Question etc.

The di stributi on in thi s gui de i s only to facilitate the candi dates. This di stri bution is not a part of test templ ate, so, a test may contain all the questi ons of one format or may have a random number of questi ons of different formats.

This chapter i s divi ded i nto 4 major secti ons. The fi rst di scusses the syll abus/contents in each secti on of the test respecti vely and the remai ning three secti ons address the questi on format, guide lines to attempt the questions i n each format and some exampl e questi ons.

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The foll owing are some key points, whi ch are phrased here to refresh your knowl edge of basic arithmeti c princi pl es.

o ais negative

o ais zero

o ais positive

• The only number that is equal to its opposite is 0 (e.g. 0 a a only if a =−=)

• If 0 is multiplied to any other number, it will make it zero ( 00 a×=).

• Product or quotient of two numbers of the same sign are always positive and of a different sign are always negative. E.g. if a positive number is multiplied to a negative number the result will be negative and if a negative number is divided by another negative number the result will be positive.

See the following tables for all combinations.

• The sum of two positive numbers is always positive.

• The sum of two negative numbers is always negative.

• Subtracting a number from another is the same as adding its opposite

a −b = a + (−b)

• The reciprocal of a number a is a

1

• The product of a number and its reciprocal is always one

× 1 = 1

a

a

• Dividing by a number is the same as multiplying by its reciprocal

b

a ÷ b = a × 1

• Every integer has a finite set of factors (divisors) and an infinite set of

multipliers.

• If a and b are two integers,the following four terms are synonyms

o a is a divisor of b

o a is a factor of b

o b is a divisible b y a

o b is a multiple of a

They all mean that when a is divided by b there is no remainder.

• Positive integers, other than 1, have at least two positive factors.

• Positive integers, other than 1, which have exactly two factors, are known

as prime numbers.

• Every integer greater than 1 that is not a prime can be written as a product

of primes.

To find the prime factorization of an integer, find any two factors of that

number, if both are primes, you are done; if not, continue factorization

until each factor is a prime.

E.g. to find the prime factorization of 48, two factors are 8 and 6. Both of

them are not prime numbers, so continue to factor them.

Factors of 8 are 4 and 2, and of 4 are 2 and 2 (2 × 2 × 2).

Factors of 6 are 3 and 2 (3 × 2).

So the number 48 can be written as 2 × 2 × 2 × 2 × 3.

• The Least Common Multiple (LCM) of two integers a and b is the smallest

integer which is divisible by both a and b, e.g. the LCM of 6 and 9 is 18.

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• The Greatest Common Divisor (GCD) of two integers a and b is the largest

integer which divides both a and b, e.g. the GCD of 6 and 9 is 3.

• The product of GCD and LCM of two integers is equal to the products of

numbers itself. E.g. 6 × 9 = 54

3 × 18 = 54 (where 3 is GCD and 18 is LCM of 6 and 9).

• Even numbers are all the multiples of 2 e.g. (… , −4, −2, 0, 2, 4, …)

• Odd numbers are all integers not divisible by 2 ( … , −5, −3, −1, 1, 3, 5, … )

• If two integers are both even or both odd, their sum and difference are

even.

• If one integer is even and the other is odd, their sum and difference are

odd.

• The product of two integers is even unless both of them are odd.

• When an equation involves more than one operation, it is important to

carry them out in the correct order. The correct order is Parentheses,

Exponents, Multiplication and Division, Addition and Subtraction, or just

the first letters PEMDAS to remember the proper order.

Exponents and Roots

• Repeated addition of the same number is indicated by multiplication:

17 + 17 + 17 + 17 + 17 = 5 × 17

• Repeated multiplication of the same number is indicated by an exponent:

17 × 17 × 17 × 17 × 17 = 175

In the expression 175, 17 is called base and 5 is the exponent.

• For any number b: b1 = b and bn = b × b × … × b, where b is used n times

as factor.

• For any numbers b and c and positive integers m and n:

o bmbn = bm+n

o m n

n

m

b

b

b = −

o (bm )n = bmn

o bmcm = (bc)m

• If a is negative, an is positive if n is even, and negative if n is odd.

• There are two numbers that satisfy the equation 9 x2 = : x = 3 and x = −3 .

The positive one, 3, is called the (principal) square root of 9 and is denoted

by symbol 9 . Clearly, each perfect square has a square root:

0 = 0 , 9 = 3, 36 = 6 , 169 = 13 , 225 = 25 etc.

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• For any positive number a there is a positive number b that satisfies the

equation a = b .

• For any positive integer, ( a)2 = a × a = a .

• For any positive numbers a and b:

o

ab = a × b

and b

a

b

a =

o 5 = 25 = 9 +16 ≠ 9 + 16 = 3 + 4 = 7

+ ≠ +

as

a b a b

• Although it is always true that ( a)2 = a , a = a 2

is true only if a is

positive as (−5)2 = 25 = 5 ≠ −5

• For any number a, 2

n

an = a .

• For any number a, b and c:

• a(b + c) = ab + ac a(b − c) = ab − ac

and i f a ≠ 0

• a

c

a

b

a

b c = +

( + )

a

c

a

b

a

b c = −

( − )

Inequalities

• For any number a and b, exactly one of the following is true:

a > b or a = b or a < b .

• For any number a and b, a > b means that a − b is positive.

• For any number a and b, a < bmeans that a − b is negative.

• For any number a and b, a = bmeans that a − b is zero.

• The symbol ≥ means greater than or equal to and the symbol ≤ means less

than or equal to. E.g. the statement x ≥ 5means that x can be 5 or any

number greater than 5.

The statement 2 < x < 5is an abbreviation of 2 < x and x < 5 .

• Adding or subtracting a number to an inequality preserves it.

• If a < b , then a + c < b + c and a − c < b − c .

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e.g. 5 < 6⇒5 +10 < 6 +10 and 5 −10 < 6 −10

• Adding inequalities in same direction preserves it:

If a < b and c < d , then a + c < b + d .

• Multiplying or dividing an inequality by a positive number preserves it. If

a < b and c is a posi tive number, then a×c < b×c and c

b

c

a <

.

• Multiplying or dividing an inequality by a negative number reverses it. If

a < b and c is a negative number, then a×c > b×c and c

b

c

a >

.

• If sides of an inequality are both positive and both negative, taking the

reciprocal reverses the inequality.

• If 0 < x <1and a is posi tive, then xa < a .

• If 0 < x <1and m and nare integers wi th m > n , then

xm < xn < x .

• If 0 < x <1, then x > x .

•

If 0 < x <1, then

1 x

x

>

and

1 1

x

>

Properties of Zero

• 0 is the only number that is neither negative nor positive.

• 0 is smaller than every positive number and greater than every negative

number.

• 0 is an even integer.

• 0 is a multiple of every integer.

• For every number a : a + 0 = a and a − 0 = a .

• For every number a : a×0 = 0 .

• For every posi tive integer n : 0 0 n = .

• For every number a (including 0): 0

0

a ÷ and a

are undefined symbols.

• For every number a (other than 0):

0 a 0 0

a

÷ = = .

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• 0 is the only number that is equal to i ts opposi te: 0 = −0 .

• If the product of two or more numbers is 0, at least one of them is 0.

Properties of One

For any number a : a×1= a and 1

a = a .

• For any number n : 1 1 n = .

• 1 is the divisor of every integer.

• 1 is the smallest positive integer.

• 1 is an odd integer.

• 1 is not a prime.

Fractions and Decimals

• When a whole is divided into n equal parts, each part is called one nth of

the whole, written

1

n . For example, if a pizza is cut (divided) into 8 equal

slices, each slice is one eighth (

1

8 ) of the pizza; a day is divided into 24

equal hours, so an hour is one twenty-fourth

( 1 )

24

of a day and an inch is

one twelfth (

1

12 ) of a foot. If one works for 8 hours a day, he works eight

twenty-fourth (

8

24 ) of a day. If a hockey stick is 40 inches long, it

measures forty twelfths

(40)

12 of a foot.

• The numbers such as

1

8 ,

1

24 ,

8

24 and

40

12 , in which one integer is written

over the second integer, are called fractions. The center line is called the

fraction bar. The number above the bar is called the numerator, and the

number below the bar is called denominator.

• The denominator of a fraction can never be 0.

• A fraction, such as

1

24 , in which the denominator is greater than

numerator, is known as a proper fraction. Its value is less than one.

• A fraction, such as

40

12 , in which the denominator is less than numerator, is

known as an improper fraction. Its value is greater than one.

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• A fraction, such as,

12

12 in which the denominator is equal to the numerator,

is also known as an improper fraction. But, Its value is one.

• Every fraction can be expressed in decimal form (or as a whole number) by

dividing the number by the denominator.

3 0.3, 3 0.75, 8 1, 48 3, 100 12.5

10 4 8 16 8

= = = = =

• Unlike the examples above, when most fractions are converted to decimals,

the division does not terminate, after 2 or 3 or 4 decimal places; rather it

goes on forever with some set of digits repeating it.

2 0.66666..., 3 0.272727..., 5 0.416666..., 17 1.133333...

3 11 12 15

= = = =

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• To compare two decimals, follow these rules:

o Whichever number has the greater number to the left of the decimal

point is greater: since 11 > 9, 11.0001 > 9.8965 and since 1 > 0,

1.234 > .8. (Recall that if a decimal is written without a number on

left of decimal point, you may assume that a 0 is there, so, .8 =

0.8).

o If the numbers to the left of the decimal point are equal, proceed as

follows:

• If the numbers do not have the same number of digits to the right

of the decimal point, add zeroes to the end of the shorter one to

make them equal in length.

• Now compare the numbers ignoring the decimal point.

• For example, to compare 1.83 and 1.823, add a 0 to the end of

1.83 forming 1.830. Now compare them, thinking of them as whole

numbers without decimal point: since 1830 > 1823, then 1.830

>1.823.

• There are two ways to compare fractions:

o Convert them to decimals by dividing, and use the method already

described to compare these decimals. For example to compare

2

5

and

1

4 , convert them to decimals.

2 0.4

5

= and

1 0.25

4

= . Now, as 0.4

> 0.25,

2

5 >

1

4 .

o Cross multiply the fractions. For example to compare

2

5 and

1

4 ,

cross multiply:

2

5

1

4

Since

2×4>1×5,

then

2

5 >

1

4 .

• While comparing the fractions, if they have same the denominators, the

fraction with the larger numerator is greater. For example

3 2

5 5

> .

• If the fractions have the same numerator, the fraction with the

smaller denominator is greater. For example

3 3

5 10

> .

• Two fractions are called equivalent fractions if both of them have same

decimal value.

• For example,

1 5

2 10

= as both of these are equal to 0.5.

• Another way to check the equivalence of two fractions is to cross-multiply.

If both of the products are same, the fractions are equivalent. For Example,

to compare

2

5 with

6

15 , cross-mul tiply. Since 2×15 = 6×5 , both of the

fractions are equivalent.

• Every fraction can be reduced to lowest terms by dividing the numerator

and denominator by their greatest common divisor (GCD). If the GCD is 1,

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the fraction is already in lowest terms. For example to reduce

10

15 , divide

both numerator and denominator by 5 (which is GCD of 10 and 15). This

will reduce the fraction to

2

3 .

• To multiply two fractions, multiply their numerators and multiply their

denominators. For example

3 4 3 4 12

5 7 5 7 35

×

× = =

× .

• To multiply a number to a fraction, write that number as a fraction whose

denominator is 1. For example

3 7 3 7 3 7 21

5 5 1 51 5

×

× = × = =

× .

• When a problem requires you to find the fraction of a number, multiply that

fraction with the number. For example, to find two fifth (

2

5 ) of 200,

multiply:

2 200 2 200 400

5 5 1

× = × =

80

80

5

=

/ .

• The reciprocal of a fraction

a

b is another fraction

b

a since 1 a b

b a

× =

• To divide one fraction by the other fraction, multiply the reciprocal of

divisor with the dividend. For example,

22 11 22

7 7

÷ =

2 7

7 11

× 2 2

1

= = .

• To add or subtract the fractions with same denominator, add or subtract

numerators and keep the denominator. For

example

4 1 5 4 1 3

9 9 9 9 9 9

+ = and − = .

Percents

• The word percent means hundredth. We use the symbol % to express the

word percent. For example “15 percent” means “15 hundredths” and can be

written with a % symbol, as a fraction, or as a decimal.

20% 20 0.20

100

= = .

• To convert a decimal to a percent, move the decimal point two places to the

right, adding 0s is necessary, and add the percent symbol (%).

For example, 0.375 = 37.5% 0.3 = 30% 1.25 = 125% 10=1000%

• To convert a fraction to a percent, first convert that fraction to decimal,

than use the method stated above to convert it to a percent.

• To convert a percent to a decimal, move the decimal point two places to the

left and remove the % symbol. Add 0s if necessary.

For example, 25% = 0.25 1% =0.01 100% = 1

• You should be familiar with the following basic conversions:

1 5 0.50 50%

2 10

= = =

1 2 0.20 20%

5 10

= = =

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1 0.25 25%

4

= =

3 0.75 75%

4

= =

• For any positive integer a : a%of 100 is a .

• For any positive numbers a and b : a% of b = b% of a

• The percent change in the quantity is actual change 100%

original amount

× . For example:

If the price of a lamp goes from Rs.80 to Rs.100, the actual increase is

Rs.20, and the percent increase is

20 100% 1 100% 25%

80 4

× = × = .

• If a < b , the percent increase in going from a to b is always greater than

percent decrease in going from b to a .

• To increase a number by k%, mul tiply i t by 1+ k%, and to decrease a

number by k%, mul tiply it by 1− k%. For example, the value of an

investment of Rs. 20,000 after 25% increase is

20,000×(1+ 25%) = 20,000×(1.25) = 25,000 .

• If a number is the resul t of increasing another number by k%, to find the

original number divide i t by 1+ k%, and i f a number is the resul t of

decreasing another number by k%, to find the original number, divide i t by

1− k%.

For example, The government announced an 20% increase in salaries. If

after the increment, The salary of a particular employee is Rs. 18, 000,

what was the original salary?

Original salary (in Rs.) =

18,000 18,000 15,000

1 1 20% 1.20

current salary

percent increase

= = =

+ +

Ratios and Proportions

• A ratio is a fraction that compares two quantities that are measured in the

same units. The first quantity is the numerator and the second quantity is

denominator. For example, if there are 16 boys and 4 girls, we say that the

ratio of the number of boys to the number of girls on the team is 16 to 4,

or

16

4 . This is often written as 16:4. Since a ratio is just a fraction, it can

be reduced or converted to a decimal or a percent. The Following are

different ways to express the same ratio:

16 to 4 , 16 : 4 ,

16

4 ,

4

1 , 0.25 , 25%

• If a set of objects is divided into two groups in the ration a : b , then the

first group contains

a

a + b of the total objects and simi larly the second group

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contains

b

a + b of the total number of objects. This rule applies to extended

ratios, as well . If a set is divided into three groups in the ratio a : b : c , then

the first group contains

a

a + b + c of the total objects, and so on.

• A proportion is an equation that states that two ratios are equivalent. Since

ratios are just fractions, any equation such as

4 10

6 15

= in which each side is

a single fraction is proportion. This proportion states that 4 relates to 6 in

same ratio as 10 relates to 15.

• For each proportion of the form

a c

b d

= , ad = bc . This property can be used

to solve proportions for unknowns (variables). For example: “If 3 oranges

cost Rs.5, how many oranges can you buy for Rs.100”. To solve this

problem we have to set up a proportion. If the number of oranges for

Rs.100 is x , then:

3 3 100 5 3 100 60

5 100 5

x x x x ×

= ⇒ × = × ⇒ = ⇒ =

Averages

• The average of a set of n numbers is the sum of those numbers divided by

n.

average sum of n numbers

n

=

or simply

A Sum

n

=

the technical name for these kind of averages is Arithmetic Mean.

• If you know the average of n numbers, multiply that average with n to get

the sum of numbers.

• If all the numbers in a set are the same, then that number is the average.

• Assume that the average of a set of numbers is A. If a new number x is

added to that set, the new average will be;

o Greater if x is greater than the existing average

o Smaller if x is smaller than the existing average

o Unchanged if x is equal to the existing average

• Arithmetic sequence is an ordered set of numbers, such that, the difference

between two consecutive numbers is the same.

• If there is an arithmetic sequence of n terms, then the average calculation

can be made simple by using these rules.

o The average of the terms in that sequence will be the middle term,

if n is odd.

o If n is even, the average will be the average of two middle terms.

2.1.2 Algebra

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Polynomials

• A monomial is any number or variable or product of numbers and variables.

For example

3, − 4, x, y, 3x, − 2xyz, 5x3 ,1.5xy2 , a3b2 are all monomials.

• The number that appears in front of a variable in a monomial is called the

coefficient. The coefficient of

5x3 is 5. If there is no number, the

coefficient is ei ther 1 or –1, because x means 1x and −x means −1x .

• A polynomial is a monomial or the sum of two or more monomials. Each

monomial that makes up the polynomial is called a term of that polynomial.

• If a polynomial has only one term it is a simple monomial, if it has two

terms, it is known as binomial and if it has three terms, it is called

trinomial.

• Two terms are called like terms if they differ only in their coefficients.

5x3

and

−2x3 are like terms, whereas,

5x3 and

5x2 are not.

• If like terms are involved in a polynomial, they can be combined, by adding

their coefficients, to make that polynomial simpler. The polynomial

3x2 + 4x + 5x − 2x2 − 7 is equivalent to the polynomial

x2 + 9x − 7 .

• All laws of arithmetic are also applicable to polynomials. Most important of

them is PEMDAS.

• Polynomials can be added, subtracted, multiplied or divided.

• To add two polynomials, put a plus sign between them, erase the

parentheses, and combine like terms.

Example:

What is the sum of

5x2 +10x − 7 and

3x2 − 4x + 2 ?

Solution:

2 2

2 2

2

(5 10 7) (3 4 2)

5 10 7 3 4 2

8 6 5

x x x x

x x x x

x x

+ − + − +

= + − + − +

= + −

• To subtract two polynomials, reverse the signs of subtrahend, and add two

polynomials as done before.

Example:

Subtract

3x2 − 4x + 2 from

5x2 +10x − 7

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Solution:

2 2

2 2

2 2

2

(5 10 7) (3 4 2)

(5 10 7) ( 3 4 2)

5 10 7 3 4 2

2 14 9

x x x x

x x x x

x x x x

x x

+ − − − +

= + − + − + −

= + − − + −

= + −

• To multiply monomials, first multiply their coefficients, and then multiply

their variables by adding the exponents.

Example:

What is the product of

3x2 yz from

−2x2 y2 ?

Solution:

2 2 2

2 2 2

4 3

(3 )( 2 )

(3 2)( )( )( )

6

x yz x y

x x y y z

x y z

−

= ×− × ×

= −

• To multiply a monomial by a polynomial, just multiply each term of the

polynomial by the monomial.

Example:

What is the product of 3x from

3x2 − 6xy2 + 2?

Solution:

2 2

2 2

3 2 2

(3 )(3 6 2)

(3 3 ) (3 6 ) (3 2)

9 18 6

x x xy

x x x xy x

x xy x

− +

= × − × + ×

= − +

• To multiply two binomials, multiply each term of first binomial by each term

of second binomial, then add the results.

Example:

What is the product of 3x + y from

3x2 − 6xy2 ?

Solution:

2 2

2 2 2 2

2 2 2 2 3

2 2 2 2 3

(3 )(3 6 )

(3 3 ) (3 ( 6 )) ( 3 ) ( ( 6 ))

(9 ) ( 18 ) (3 ) ( 6 )

9 18 3 6

x y x xy

x x x xy y x y xy

x xy xy xy

x xy xy xy

+ −

= × + × − + × + × −

= + − + + −

= − + −

• The three most important binomial products are:

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o

(x − y)(x + y) = x2 + xy − xy + y2 = x2 + y2

o

(x + y)(x + y) = x2 + xy + xy + y2 = x2 + 2xy + y2

o

(x − y)(x − y) = x2 − xy − xy + y2 = x2 − 2xy + y2

Memorizing these can save a lot of calculation time during the test.

• To divide a polynomial by a monomial, divide each term of the polynomial

by the monomial.

Example:

What is the quotient if

32x2 y +12xy3z is divided by 8xy ?

Solution:

2 3 2 3

32 12 32 12 4 3 2 (by reducing the terms)

8 8 8 2

x y xy z x y xy z x yz

xy xy xy

+

= + = +

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Solving Equations and Inequalities

• The basic principle in solving equations and inequalities is that you can

manipulate them in any way as long as you do the same thing to both

sides. For example you may add a number to both sides, or you may

divide or multiply both sides with same number etc.

• By using the following six-step method, you can solve most of the

equations and inequalities. The method is explained with the help of an

example.

Example:

if

1 3( 2) 2( 1) 1

2

x + x − = x + + , what is the value of x ?

Solution:

Step What to do… Example

1 Get rid of fractions and

decimals by multiplying

both sides by the LCD.

Multiply each side by 2 to get:

x + 6(x − 2) = 4(x +1) + 2

2 Get rid of all parentheses

by solving them.

x + 6x −12 = 4x + 4 + 2

3 Combine like terms on

each side.

7x −12 = 4x + 6

4 By adding and subtracting

get all the variables on

one side (mostly left).

Subtract 4x from each side to get:

3x −12 = 6

5 By adding or subtracting

get all plain numbers on

the other side.

Add 12 to each side to get:

3x =18

6 Divide both sides by the

coefficient of the variable.

(If you are dealing with

an inequality and you

divide with a negative

number, remember to

reverse the inequality.)

Divide both sides by 3 to get:

x = 6

• When you have to solve one variable and the equation/inequality involve

more than one variable, treat all other variables as plain numbers and

apply the six-step method.

Example:

i f a = 3b − c , what is the value of b in terms of a and c ?

Solution:

Step What to do… Example

1 There are no fractions and

decimals.

2 There are no parentheses.

3 There are no like terms.

4 By adding and subtracting get

all the variables on one side.

Remember there is only one

variable b , which is on one

side only.

5 By adding or subtracting get

all plain numbers on the

other side.

Remember we are

considering a and c as

plain number. Add c to

each side to get:

a + c = 3b

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6 Divide both sides by the

coefficient of the variable.

Divide both sides by 3 to

get:

3

a c b

+

=

• It is not necessary to follow these steps in the order specified. Some times

it makes the problem much easier, if you change the order of these steps.

Example:

If x − 4 =11, what is the value of x -8?

Solution:

Going immediately to step 5, add 4 to each side to get: x =15. Now

subtract 8 from both sides to get: x −8 = 7 .

• Doing the same thing on each side of an equation does not mean doing the

same thing to each term of the equation. This is very important if you are

doing divisions, or dealing with exponents and roots.

Example:

If a > 0 and

a2 + b2 = c2 , what is the value of a in terms of b and c .

Solution:

a2 + b2 = c2 ⇒ a2 = c2 − b2 . Now you can’t take a square root of each term

to get a = c − b . You must take the square root of each side:

a2 = c2 − b2 ⇒ a = c2 − b2

• Another type of equation is that in which the variable appears in exponent.

These equations are basically solved by inception.

Example:

If

2x+3 = 32 , what is the value of

3x+2

?

Solution:

2x+3 = 32 ⇒ 2x+3 = 25 ⇒ x + 3 = 5 ⇒ x = 2 .

Now as x = 2 , you can get

x = 2 ⇒ x + 2 = 4 ⇒ 3x+2 = 34 = 81

• A system of equations is a set of two or more equations having two or more

variables. To solve such equations, you must find the value of each

variable that will make each equation true.

• To solve a system of equations, add or subtract them to get a third

equation. If there are more than two equations you can just add them.

Example:

If x + y =10 and x − y =10 what is the value of y ?

Solution:

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Add two equations:

10

2

2 12 6

x y

x y

x x

+ =

− =

= ⇒ =

Now replacing x wi th 6 in the first equation: 6 + y =10 ⇒ y = 4

• If you know the value of one variable in a system of two equations, you can

use this value to get the value of the other variable. As it is done in the

previous question.

Word problems

• To solve word problems, first translate the problem from English to

Algebra. While translating, use variables to represent unknowns. Once

translated, it is easy to solve them using the techniques you have learned

in previous sections.

• Following English to Algebra dictionary will be helpful in translating word

problems to algebraic expressions.

English words

Mathematical

meaning

Symbol

Is, was, will be, had, has, will

have, is equal to, is the same as

Equals =

Plus, more than, sum, increased

by, added to, exceeds, received,

got, older than, farther than,

greater than

Addition +

Minus, fewer, less than,

difference, decreased by,

subtracted from, younger than,

gave, lost

Subtraction −

Times, of, product, multiplied by Multiplication ×

Divided by, quotient, per, for Division

or a

b

÷

More than, greater than Inequality >

At least Inequality ≥

Fewer than, less than Inequality <

At most Inequality ≤

What, how many, etc. Unknown quantity x

(Some variable)

Examples:

o The sum of 5 and some number is 13. 5 + x =13

o Javed was two years younger than Saleem. J = S − 2

o Bilal has at most Rs.10,000. B ≤10000

o The product of 2 and a number exceeds that

number by 5 (is 5 more than that number). 2N = N + 5

• In word problems, you must be sure about what you are answering. Do not

answer the wrong question.

• In problems involving ages, remember that “years ago” means you need to

subtract, and “years from now” means you need to add.

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• Distance problems all depend on three variations of the same formula:

o distance = speed ×time

o

speed distance

time

=

o

time distance

speed

=

Example:

How much longer, in seconds, is required to drive 1 mile at 40 miles per

hour than at 60 miles per hour?

Solution:

The time to drive at 40 miles per hour can be calculated as

time1 =

1

40 hours =

1

40 2

× 60 3

minutes =

3

2

× 60 30

seconds =

90 seconds

The time to drive at 60 miles per hour can be calculated as

time2 =

1

60 hours =

1

60

× 60 minutes = 1×60 seconds = 60 seconds

difference = time1− time2 = 90 − 60 = 30 seconds.

2.1.3 Geometry

Lines and Angles

• An angle is formed at the intersection of two line segments, rays or lines.

The point of intersection is called the vertex. Angles are measured in

degrees.

• Angles are classified according to their degree measures.

• An acute angle measures less than 90°

• A right angle measures 90°

• An obtuse angle measures more than 90° but less than 180°

• A straight angle measures 180°

• If two or more angles combine together to form a straight angle, the sum

of their measures is 180° .

a° + b° + c° + d° =180°

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• The sum of all the measures of all the angles around a point is 360°

a° + b° + c° + d°+ e° = 360°

• When two lines intersect, four angles are formed, two angles in each pair of

opposite angles are called vertical angles. Vertical angles, formed by the

intersection of two lines, have equal measures.

a = c and b = d

• If one of the angles formed by the intersection of two lines is a right angle,

then all four angles are right angles. Such lines are called perpendicular

lines

a = b = c = 90

• In the figure below a line l divides the angle in two equal parts. This line is

said to bisect the angle. The other line k bisects another line into two

equal parts. This line is said to bisect a line.

• Two lines are said to be parallel, if they never intersect each other.

However, if a third line, called a transversal, intersects a pair of parallel

lines, eight angles are formed. And the relationship among theses angles is

shown in the following diagram.

•

• - All four acute angles are equal

a = c = e = g

• - All four obtuse angles are equal

b = d = f = h

• - The sum of any pair of acute and

obtuse angle is 180°, e.g.

a + d =180°,d + e =180°,b + g =180° etc.

Triangles

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• In any triangle, the sum of the measures of the three angles is 180° .

x + y + z =180

• In any triangle:

o The longest side of triangle is opposite the largest angle.

o The shortest side is opposite the smallest angle.

o Sides with the same length are opposite the angles with the same

measure.

• Triangles are classified into three different kinds with respect to the

lengths of sides.

o Scalene: in which all three sides are of different lengths.

o Isosceles: in which two of the sides of triangle are equal in

length, the third is different.

o Equilateral: in which all three sides are equal in length.

• Triangles are also classified with respect to the angles.

o Acute triangle: in which all three angles are acute.

o Obtuse triangle: in which one angle is obtuse and two are acute.

o Right triangle: This has one right and two acute angles.

• In a right triangle, the opposite to the right angle is known as hypotenuse

and is the longest side. The other two sides are called legs.

• In any right triangle, the sum of the measures of the two acute angles is

90° .

• By Pythagorean Theorem, the sum of squares of the lengths of legs of a

right triangle is always equal to the square of length of hypotenuse.

a2 + b2 = c2

• In any triangle, the sum of any two sides is always greater than the third

one. And the difference of any two sides is always less than the third one.

a + b > c and a − b < c

• The perimeter of a triangle is calculated by adding the lengths of al l the

sides of that triangle.

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perimeter = a + b + c

• The area of a triangle is calculated by the formula:

1

2

area = bh

where b is

the base of the triangle and h is the height of the triangle.

o Any side of triangle can be taken as the base.

o Height is the altitude (perpendicular) drawn to the base from its

opposite vertex.

o In a right triangle any leg could be taken as the base, the other

will be the altitude.

Quadrilateral and other Polygons

• A polygon is a closed geometric figure, made up of line segments. The

line segments are called sides and the end points of lines are called

vertices (plural of vertex). Lines, inside the polygon, drawn from one

vertex to the other, are called diagonals.

• The sum of the measures of the n angles in a polygon with n sides is

always (n − 2)×180° .

• In any quadrilateral, the sum of the measures of the four angles is 360° .

• A regular polygon is a polygon in which all of the sides are of the same

length. In any regular polygon, the measure of each interior angle is

(n 2) 180

n

− × °

and the measure of each exterior angle is

360

n

°

.

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• A parallelogram is a special quadrilateral, in which both pairs of opposite

sides are parallel. The Following are some properties of parallelogram.

o Lengths of opposi te sides are equal . AB = CD and AD = BC

o Measures of opposi te angles are equal . a = c and b = d

o Consecutive angles add up to 180° . a + b =180° , b + c =180° etc.

o The two diagonals bisect each other. AE = EC and BE = ED

o A diagonal divides the parallelogram into two triangles that are

congruent.

• A rectangle is a parallelogram in which all four angles are right angles. It

has all the properties of a parallelogram. In addition it has the following

properties:

o The measure of each angle in a rectangle is 90° .

o The diagonals of a rectangle are equal in length.

• A square is a rectangle that has the following additional properties:

o A square has all its sides equal in length.

o In a square, diagonals are perpendicular to each other.

• To calculate the area, the following formulas are required:

o For a parallelogram, Area = bh , where b is the base and h is the

height.

o For a rectangle, Area = lw, where l is the length and w is the

width.

o For a square,

Area = s2 , where s is the side of the square.

• Perimeter for any polygon is the sum of lengths, of all its sides.

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Circles

• A circle consists of all the points that are the same distance from one fixed

point called the center. That distance is called the radius of a circle. The

word radius is also used to represent any of the line segments joining the

center and a point on the circle. The plural of radius is radii.

• Any triangle, such as CED in the figure, formed by connecting the end

points of two radii, is an isosceles.

• A line segment, such as ED in the diagram above, both of whose end

points are on a circle is called a chord.

• A chord that passes through the center of the circle is called the diameter

of the circle. The length of the diameter is always double the radius of the

circle. The diameter is the longest cord that can be drawn in a circle.

• The total length around a circle is known as the circumference of the

circle.

• The ratio of the circumference to the diameter is always the same for any

circle. This ratio is denoted by the symbol π (pronounced as pi).

• 2 C C d C r

d

π = ⇒ =π ⇒ = π where C is the circumference, d is the diameter

and r is the radius of the circle.

• Value of π is approximately 3.14

• An arc consists of two points in a circle and all the points between them.

E.g. PQis an arc in the diagram.

• An angle whose vertex is at the center of the circle is called the central

angle. ∠PCQ in the diagram above is a central angle.

• The degree measure of a complete circle is 360° .

• The degree measure of an arc is the measure of the central angle that

intercepts it. E.g. the degree measure of PQ is equal to the measure of

∠PCQ in the diagram above.

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• If x is the degree measure of an arc, its length can be calculated as 360

x C ,

where C is the circumference.

• The area of a circle can be calculated as

π r2 .

• The area of a sector formed by the arc and two radii can be

calculated as A x

360 , where A is the area of a circle.

2.2 Discrete Quantitative Questions

These are standard multiple-choice questions. Most of such questions require you to do some

computations and you have to choose exactly one of the available choices based upon those

computations. This section will teach you the basic tactics to attempt such questions.

2.2.1 Question format

Each question will consist of a question statement and the choices labeled from A to E. The

number of choices may vary from 2 to 5, but exactly one choice will be correct for each

question.

2.2.2 How to attempt?

Following are some tactics, which will lead you to the correct answer.

• Whenever you know how to answer a question directly, just do it. The

tactics should be used only when you do not know the exact solution, and

you just want to eliminate the choices.

• Remember that no problem requires lengthy or difficult computations. If

you find yourself doing a lot of complex arithmetic, think again. You may

be going in the wrong direction.

• Whenever there is a question with some unknowns (variables), replace

them with the appropriate numeric values for ease of calculation.

• When you need to replace variables with values, choose easy-to-use

numbers, e.g. the number 100 is appropriate in most percent-related

problems and the LCD (least common denominator) is best suited in

questions that involve fractions.

• Apply “back-solving” whenever you know what to do to answer the question

but you want to avoid doing algebra. To understand this tactic read the

following example:

On Monday, a storeowner received a shipment of books. On Tuesday, she

sold half of them. On Wednesday after two more were sold, she had

exactly 2/5 of the books left. How many were in the shipment?

(A) 10 (B) 20 (C) 30 (D) 40 (E) 50

now by this tactic:

Assume that (A) is the correct answer, if so; she must have 3 books on

Wednesday. But 2/5 of 10 are 4, so, (A) is incorrect;

Assume that (B) is the correct answer, if so; she must have 8 books on

Wednesday. 2/5 of 20 are 8, so, (B) is the correct choice, and as there may

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be only one correct choice, there is no need to check for remaining choices.

This tactic is very helpful when a normal algebraic solution for the problem

involves complex or lengthy calculations.

• If you are not sure how to answer the question, do not leave it

unanswered. Try to eliminate absurd choices and guess from the remaining

ones. Most of the times four of the choices are absurd and your answer is

no longer a guess.

Many things may help you to realize that a particular choice is absurd.

Some of them are listed below.

o The answer must be positive but some of the choices are negative

so eliminate all the negative ones.

o The answer must be even but some of the choices are odd so

eliminate all the odd choices.

o The answer must be less then 100, but some of the choices are

greater than 100 (or any other value) so eliminate all choices that

are out of range.

o The answer must be a whole number, but some of the choices are

fractions so eliminate all fractions.

o These are some examples. There may be numerous situations

where you can apply this tactic and find the correct answer even if

you do not know the right way to solve the problem.

Example questions with solutions

The following are some examples, which will help you to master these types of

questions.

Example

If 25% of 220 equals 5.5% of X, what is X?

(A) 10 (B) 55 (C) 100 (D) 110 (E) 1000

Solution:

Since 5.5% of X equals 25% of 220, X is much greater than 220. So, choices

A, B, C, and D are immediately eliminated because these are not larger than

220. And the correct answer is choice E.

(Note: An important point here is that, even if you know how to solve a problem, if

you immediately see that four of the five choices are absurd, just pick the remaining

choice and move on.)

Example

Science students choose exactly one of three fields (i.e. medical sciences,

engineering sciences and computer sciences). If, in a college, three-fifths of

the students choose medical sciences, one-forth of the remaining students take

computer sciences, what percent of the students take engineering sciences?

(A) 10 (B) 15 (C) 20 (D) 25 (E) 30

Solution:

The least common denominator of 3/5 and 1/4 is 20, so assume that there are

20 students in that college. Then the number of students choosing medical

sciences is 12 (3/4 of 20). Of the remaining 8 students, 2 (1/4 of 8) choose

computer sciences. The remaining 6 choose engineering sciences. As 6 is

30% of 20, the answer is E.

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Example

If a school cafeteria needs C cans of soup each week for each student and

there are S students, for how many weeks will X cans of soup last?

(A) CX/S (B) XS/C (C) S/CX (D) X/CS (E) CSX

Solution:

Replace C, S and X with three easy to use numbers. Let C=2, S=5 and X=20.

Now each student will need 2 cans per week and there are 5 students, so 10

cans are needed per week and 20 cans will last for 2 weeks. Now put these

values in choices to find the correct one.

The choices A, B, C, D and E become 8, 50, 1/8, 2 and 200 respectively. So

the choice D represents the correct answer.

2.3 Quantitative Comparison Questions

Some of the questions in the Quantitative section of the test may be

quantitative comparison questions. The Following text will explain you the

format and techniques u need to attempt the questions of this format.

2.3.1 Question format

Such questions consist of two quantities, one in column A and the other in

column B. You have to compare the two quantities. The information

concerning one or both quantities is presented before them. Only the following

four choices will be given:

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

And as it is clear from the choices, only one will be correct at one time. Your job is

to choose one of them after careful comparison. The following text explains some

simple tactics to attempt such questions.

2.3.2 How to attempt

Whenever you encounter a quantitative comparison question, the following

guidelines will help you to find the correct answer quickly.

• If the question involves some variables, replace them with appropriate

numbers. Here are some guidelines in choosing an appropriate number:

o The very best numbers to use are –1, 0 and 1.

o Often fractions between 0 and 1 are useful (e.g. 1/2, 3/4 etc.).

o Occasionally, “large” numbers such as 10 or 100 can be used.

o If there is more than one variable, it is permissible to replace each

with the same number.

o Do not impose any un-specified conditions on numbers. Choose

them randomly.

• Eliminate the choices and choose from the remaining ones. For example If

you found the quantities ever equal, the correct choice could never be A or

B, so, eliminate A and B.

• A quantitative comparison question can be treated as an equation or

inequality. Either:

Column A < Column B, or

Column A = Column B, or

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Column A > Column B

So, you can perform similar operation on both columns to simplify the

problem just as in equations (or inequalities).

Example:

m > 0 and m ≠ 1

• In this example divide both the quantities by m2. This will change

column A to 1 and column B to m. Now the comparison is very simple,

as we know that m is greater than 0 and cannot be 1. So the

relationship is not determinable using the current information. m can

be both greater than 1 or between 0 and less than 1.

2.3.3 Example questions with Answers and

Explanations

Example 1:

A student earned a 75 on each of her first three

math tests and an 80 on her fourth and fifth

tests.

A B

Average after 4 tests Average after 5 tests

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

Remember you want to know which average is higher, not what the averages

are. After 4 tests, the average is clearly less than 80, so an 80 on the fifth

test had to raise the average. So the answer is choice (B).

Example 2:

A B

The time it takes to

drive 40 miles at 35

mph

The time it takes to

drive 35 miles at 40

mph

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

Once again there is no need for calculation, as the speed in column B is higher

than that in column A. It is obvious that it will take less time to travel shorter

distance at a greater speed. So the value in column A is larger. The answer is

option (A).

Example 3:

A B

m2 m3

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20

2

5

5

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

Square each column:

2

20 20 5

2 4

= =

and

2 5 255

5 5

= =

. So both columns are

equal and the answer is choice (C).

Example 4:

A B

13y 15y

To solve this question, subtract 13y from both columns to get 13y −13y = 0 for

column A and 15y −13y = 2y for column B. As there are no restrictions, 2y can

be greater than, less than or equal to 0. So the correct choice is (D).

2.4 Data Interpretation Questions

These questions are based on the information that is presented in the form of

a graph, chart or table. Most of the data is presented graphically. The most

common types of graphs are line graphs, bar graphs and circle graphs. The

objective of such questions is to test your ability to understand and analyze

statistical data.

2.4.1 Question Format

Data interpretation questions always appear in sets, you are presented with

some data in any format (chart, graph or table), and you will then be asked

with some questions about that data.

The following example explains the format of such questions.

Example:

Question 1:

What is the average sale, in million Rs., for the period 1994-2000?

(A) 5.5 (B) 6.0 (C) 7.0 (D) 8.0

(E) 8.5

Question 2:

For which year, the percentage increase in sales from the previous year is the greatest.

(A) 1995 (B) 1996 (C) 1999 (D) 2000

(E) 2001

2.4.2 How to attempt

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• Do not try to answer such questions immediately, first of all read the

presented data carefully. You must be very clear about the data and its

meanings even before reading the first question.

• Do not confuse numbers with percents. This confusion is most likely to

occur when data is presented in pie graphs. For example in the

following graph

0

2

4

6

8

10

12

1994 1995 1996 1997 1998 1999 2000 2001

Years

Sales in million Rs.

Now it would be a great mistake here to think that sales of “TVs & VCRs” is

15% more than the sales of Computers in 2001 by XYZ Corporation. To

know this you have to calculate it as

15 100 60%

25

× =

.

• Try to avoid un-necessary calculations. Most of the questions could

easily be solved by observation and estimation. Use estimation to

eliminate the choices, if you are not able to find the correct answer

without calculation. For example to solve “Question 1” presented in the

example at the start of this section, if you are not sure of the correct

answer, you can then try to cut down the number of possible choices by

observation. You are being asked to tell the percentage increase.

Where as, in year 2000, the sale is decreasing instead of increasing, so,

you can immediately eliminate choice (D) in that question.

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• Your answers must be based upon the information presented in the

given charts and graphs. If your knowledge contradicts any of the data

presented, ignore what you know and stick to the presented data. The

presented data should be the only base for your calculations and

estimations.

• Always use the proper units, there may be some questions that ask you

to compare different data items possibly from different data sets. Be

careful about the units used to represent the data.

• Because graphs and charts present data in a form that enables you to

readily see the relationships among values and to make quick

comparisons, you should always try to visualize you answer in the same

format as the original data was presented.

• Be sure that your answer is reasonable. For example, the profit could

never increase the actual sales, or the expenses could never be negative

etc. While answering the question, first of al l eliminate such unreasonable

choices, and then choose from the remaining ones.

2.5 Practice exercise

1 What is the average of positive integers from 1 to 100 inclusive?

(A) 49

(B) 49.5

(C) 50

(D) 50.5

(E) 51

2 If x + y = 6 , y + z = 7 , and x + z = 9 , what is the average of x , y and z ?

(A)

11

3 (B)

11

2 (C)

22

3

(D) 11 (E) 22

3 In the diagram below, lines l and m are not parallel.

If A represents the average measure of all the eight angles, what is the value

of A?

(A) A = 45

(B) 45 < A < 90

(C) A = 90

(D) 90 < A <180

(E) A =180

4 Aslam has 4 times as many books as Salman and 5 times as many as Javed.

If Javed has more than 40 books, what is the least number of books that

Aslam could have?

(A) 200 (B) 205 (C) 210 (D) 220 (E) 24

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5 Aslam is now 3 times as old as Javed, but 5 years ago, he was 5 times as

Javed was. How old is Aslam now?

(A) 10 (B) 12 (C) 24 (D) 30 (E) 36

6 If x%of y is 10, what is y?

(A)

x

10

Murtaza Khan Khajjak The Quantitative secti on measures your basic mathemati cal skill s, understanding of el ementarymathemati cal concepts, and the ability to reason quantitatively and solve problems i n a quanti tative setting. There i s a balance of questi ons requiri ng basi c knowl edge of ari thmetic, al gebra, geometry, and data analysis. These are essential content areas usually studied at the hi gh school level.

The questions i n the quantitative section can al so be from

• Discrete Quantitative Question

• Quantitative Comparison Question

• Data Interpretation Question etc.

The di stributi on in thi s gui de i s only to facilitate the candi dates. This di stri bution is not a part of test templ ate, so, a test may contain all the questi ons of one format or may have a random number of questi ons of different formats.

This chapter i s divi ded i nto 4 major secti ons. The fi rst di scusses the syll abus/contents in each secti on of the test respecti vely and the remai ning three secti ons address the questi on format, guide lines to attempt the questions i n each format and some exampl e questi ons.

###
**2.1 General Mathematics Review **

**2 . 1. 1 Arithmetic**The foll owing are some key points, whi ch are phrased here to refresh your knowl edge of basic arithmeti c princi pl es.

#### Basic arithmetic

• For any number a, exactly one of the following is true:o ais negative

o ais zero

o ais positive

• The only number that is equal to its opposite is 0 (e.g. 0 a a only if a =−=)

• If 0 is multiplied to any other number, it will make it zero ( 00 a×=).

• Product or quotient of two numbers of the same sign are always positive and of a different sign are always negative. E.g. if a positive number is multiplied to a negative number the result will be negative and if a negative number is divided by another negative number the result will be positive.

See the following tables for all combinations.

• The sum of two positive numbers is always positive.

• The sum of two negative numbers is always negative.

• Subtracting a number from another is the same as adding its opposite

a −b = a + (−b)

• The reciprocal of a number a is a

1

• The product of a number and its reciprocal is always one

× 1 = 1

a

a

• Dividing by a number is the same as multiplying by its reciprocal

b

a ÷ b = a × 1

• Every integer has a finite set of factors (divisors) and an infinite set of

multipliers.

• If a and b are two integers,the following four terms are synonyms

o a is a divisor of b

o a is a factor of b

o b is a divisible b y a

o b is a multiple of a

They all mean that when a is divided by b there is no remainder.

• Positive integers, other than 1, have at least two positive factors.

• Positive integers, other than 1, which have exactly two factors, are known

as prime numbers.

• Every integer greater than 1 that is not a prime can be written as a product

of primes.

To find the prime factorization of an integer, find any two factors of that

number, if both are primes, you are done; if not, continue factorization

until each factor is a prime.

E.g. to find the prime factorization of 48, two factors are 8 and 6. Both of

them are not prime numbers, so continue to factor them.

Factors of 8 are 4 and 2, and of 4 are 2 and 2 (2 × 2 × 2).

Factors of 6 are 3 and 2 (3 × 2).

So the number 48 can be written as 2 × 2 × 2 × 2 × 3.

• The Least Common Multiple (LCM) of two integers a and b is the smallest

integer which is divisible by both a and b, e.g. the LCM of 6 and 9 is 18.

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• The Greatest Common Divisor (GCD) of two integers a and b is the largest

integer which divides both a and b, e.g. the GCD of 6 and 9 is 3.

• The product of GCD and LCM of two integers is equal to the products of

numbers itself. E.g. 6 × 9 = 54

3 × 18 = 54 (where 3 is GCD and 18 is LCM of 6 and 9).

• Even numbers are all the multiples of 2 e.g. (… , −4, −2, 0, 2, 4, …)

• Odd numbers are all integers not divisible by 2 ( … , −5, −3, −1, 1, 3, 5, … )

• If two integers are both even or both odd, their sum and difference are

even.

• If one integer is even and the other is odd, their sum and difference are

odd.

• The product of two integers is even unless both of them are odd.

• When an equation involves more than one operation, it is important to

carry them out in the correct order. The correct order is Parentheses,

Exponents, Multiplication and Division, Addition and Subtraction, or just

the first letters PEMDAS to remember the proper order.

Exponents and Roots

• Repeated addition of the same number is indicated by multiplication:

17 + 17 + 17 + 17 + 17 = 5 × 17

• Repeated multiplication of the same number is indicated by an exponent:

17 × 17 × 17 × 17 × 17 = 175

In the expression 175, 17 is called base and 5 is the exponent.

• For any number b: b1 = b and bn = b × b × … × b, where b is used n times

as factor.

• For any numbers b and c and positive integers m and n:

o bmbn = bm+n

o m n

n

m

b

b

b = −

o (bm )n = bmn

o bmcm = (bc)m

• If a is negative, an is positive if n is even, and negative if n is odd.

• There are two numbers that satisfy the equation 9 x2 = : x = 3 and x = −3 .

The positive one, 3, is called the (principal) square root of 9 and is denoted

by symbol 9 . Clearly, each perfect square has a square root:

0 = 0 , 9 = 3, 36 = 6 , 169 = 13 , 225 = 25 etc.

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• For any positive number a there is a positive number b that satisfies the

equation a = b .

• For any positive integer, ( a)2 = a × a = a .

• For any positive numbers a and b:

o

ab = a × b

and b

a

b

a =

o 5 = 25 = 9 +16 ≠ 9 + 16 = 3 + 4 = 7

+ ≠ +

as

a b a b

• Although it is always true that ( a)2 = a , a = a 2

is true only if a is

positive as (−5)2 = 25 = 5 ≠ −5

• For any number a, 2

n

an = a .

• For any number a, b and c:

• a(b + c) = ab + ac a(b − c) = ab − ac

and i f a ≠ 0

• a

c

a

b

a

b c = +

( + )

a

c

a

b

a

b c = −

( − )

Inequalities

• For any number a and b, exactly one of the following is true:

a > b or a = b or a < b .

• For any number a and b, a > b means that a − b is positive.

• For any number a and b, a < bmeans that a − b is negative.

• For any number a and b, a = bmeans that a − b is zero.

• The symbol ≥ means greater than or equal to and the symbol ≤ means less

than or equal to. E.g. the statement x ≥ 5means that x can be 5 or any

number greater than 5.

The statement 2 < x < 5is an abbreviation of 2 < x and x < 5 .

• Adding or subtracting a number to an inequality preserves it.

• If a < b , then a + c < b + c and a − c < b − c .

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e.g. 5 < 6⇒5 +10 < 6 +10 and 5 −10 < 6 −10

• Adding inequalities in same direction preserves it:

If a < b and c < d , then a + c < b + d .

• Multiplying or dividing an inequality by a positive number preserves it. If

a < b and c is a posi tive number, then a×c < b×c and c

b

c

a <

.

• Multiplying or dividing an inequality by a negative number reverses it. If

a < b and c is a negative number, then a×c > b×c and c

b

c

a >

.

• If sides of an inequality are both positive and both negative, taking the

reciprocal reverses the inequality.

• If 0 < x <1and a is posi tive, then xa < a .

• If 0 < x <1and m and nare integers wi th m > n , then

xm < xn < x .

• If 0 < x <1, then x > x .

•

If 0 < x <1, then

1 x

x

>

and

1 1

x

>

Properties of Zero

• 0 is the only number that is neither negative nor positive.

• 0 is smaller than every positive number and greater than every negative

number.

• 0 is an even integer.

• 0 is a multiple of every integer.

• For every number a : a + 0 = a and a − 0 = a .

• For every number a : a×0 = 0 .

• For every posi tive integer n : 0 0 n = .

• For every number a (including 0): 0

0

a ÷ and a

are undefined symbols.

• For every number a (other than 0):

0 a 0 0

a

÷ = = .

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• 0 is the only number that is equal to i ts opposi te: 0 = −0 .

• If the product of two or more numbers is 0, at least one of them is 0.

Properties of One

For any number a : a×1= a and 1

a = a .

• For any number n : 1 1 n = .

• 1 is the divisor of every integer.

• 1 is the smallest positive integer.

• 1 is an odd integer.

• 1 is not a prime.

Fractions and Decimals

• When a whole is divided into n equal parts, each part is called one nth of

the whole, written

1

n . For example, if a pizza is cut (divided) into 8 equal

slices, each slice is one eighth (

1

8 ) of the pizza; a day is divided into 24

equal hours, so an hour is one twenty-fourth

( 1 )

24

of a day and an inch is

one twelfth (

1

12 ) of a foot. If one works for 8 hours a day, he works eight

twenty-fourth (

8

24 ) of a day. If a hockey stick is 40 inches long, it

measures forty twelfths

(40)

12 of a foot.

• The numbers such as

1

8 ,

1

24 ,

8

24 and

40

12 , in which one integer is written

over the second integer, are called fractions. The center line is called the

fraction bar. The number above the bar is called the numerator, and the

number below the bar is called denominator.

• The denominator of a fraction can never be 0.

• A fraction, such as

1

24 , in which the denominator is greater than

numerator, is known as a proper fraction. Its value is less than one.

• A fraction, such as

40

12 , in which the denominator is less than numerator, is

known as an improper fraction. Its value is greater than one.

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• A fraction, such as,

12

12 in which the denominator is equal to the numerator,

is also known as an improper fraction. But, Its value is one.

• Every fraction can be expressed in decimal form (or as a whole number) by

dividing the number by the denominator.

3 0.3, 3 0.75, 8 1, 48 3, 100 12.5

10 4 8 16 8

= = = = =

• Unlike the examples above, when most fractions are converted to decimals,

the division does not terminate, after 2 or 3 or 4 decimal places; rather it

goes on forever with some set of digits repeating it.

2 0.66666..., 3 0.272727..., 5 0.416666..., 17 1.133333...

3 11 12 15

= = = =

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• To compare two decimals, follow these rules:

o Whichever number has the greater number to the left of the decimal

point is greater: since 11 > 9, 11.0001 > 9.8965 and since 1 > 0,

1.234 > .8. (Recall that if a decimal is written without a number on

left of decimal point, you may assume that a 0 is there, so, .8 =

0.8).

o If the numbers to the left of the decimal point are equal, proceed as

follows:

• If the numbers do not have the same number of digits to the right

of the decimal point, add zeroes to the end of the shorter one to

make them equal in length.

• Now compare the numbers ignoring the decimal point.

• For example, to compare 1.83 and 1.823, add a 0 to the end of

1.83 forming 1.830. Now compare them, thinking of them as whole

numbers without decimal point: since 1830 > 1823, then 1.830

>1.823.

• There are two ways to compare fractions:

o Convert them to decimals by dividing, and use the method already

described to compare these decimals. For example to compare

2

5

and

1

4 , convert them to decimals.

2 0.4

5

= and

1 0.25

4

= . Now, as 0.4

> 0.25,

2

5 >

1

4 .

o Cross multiply the fractions. For example to compare

2

5 and

1

4 ,

cross multiply:

2

5

1

4

Since

2×4>1×5,

then

2

5 >

1

4 .

• While comparing the fractions, if they have same the denominators, the

fraction with the larger numerator is greater. For example

3 2

5 5

> .

• If the fractions have the same numerator, the fraction with the

smaller denominator is greater. For example

3 3

5 10

> .

• Two fractions are called equivalent fractions if both of them have same

decimal value.

• For example,

1 5

2 10

= as both of these are equal to 0.5.

• Another way to check the equivalence of two fractions is to cross-multiply.

If both of the products are same, the fractions are equivalent. For Example,

to compare

2

5 with

6

15 , cross-mul tiply. Since 2×15 = 6×5 , both of the

fractions are equivalent.

• Every fraction can be reduced to lowest terms by dividing the numerator

and denominator by their greatest common divisor (GCD). If the GCD is 1,

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the fraction is already in lowest terms. For example to reduce

10

15 , divide

both numerator and denominator by 5 (which is GCD of 10 and 15). This

will reduce the fraction to

2

3 .

• To multiply two fractions, multiply their numerators and multiply their

denominators. For example

3 4 3 4 12

5 7 5 7 35

×

× = =

× .

• To multiply a number to a fraction, write that number as a fraction whose

denominator is 1. For example

3 7 3 7 3 7 21

5 5 1 51 5

×

× = × = =

× .

• When a problem requires you to find the fraction of a number, multiply that

fraction with the number. For example, to find two fifth (

2

5 ) of 200,

multiply:

2 200 2 200 400

5 5 1

× = × =

80

80

5

=

/ .

• The reciprocal of a fraction

a

b is another fraction

b

a since 1 a b

b a

× =

• To divide one fraction by the other fraction, multiply the reciprocal of

divisor with the dividend. For example,

22 11 22

7 7

÷ =

2 7

7 11

× 2 2

1

= = .

• To add or subtract the fractions with same denominator, add or subtract

numerators and keep the denominator. For

example

4 1 5 4 1 3

9 9 9 9 9 9

+ = and − = .

Percents

• The word percent means hundredth. We use the symbol % to express the

word percent. For example “15 percent” means “15 hundredths” and can be

written with a % symbol, as a fraction, or as a decimal.

20% 20 0.20

100

= = .

• To convert a decimal to a percent, move the decimal point two places to the

right, adding 0s is necessary, and add the percent symbol (%).

For example, 0.375 = 37.5% 0.3 = 30% 1.25 = 125% 10=1000%

• To convert a fraction to a percent, first convert that fraction to decimal,

than use the method stated above to convert it to a percent.

• To convert a percent to a decimal, move the decimal point two places to the

left and remove the % symbol. Add 0s if necessary.

For example, 25% = 0.25 1% =0.01 100% = 1

• You should be familiar with the following basic conversions:

1 5 0.50 50%

2 10

= = =

1 2 0.20 20%

5 10

= = =

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1 0.25 25%

4

= =

3 0.75 75%

4

= =

• For any positive integer a : a%of 100 is a .

• For any positive numbers a and b : a% of b = b% of a

• The percent change in the quantity is actual change 100%

original amount

× . For example:

If the price of a lamp goes from Rs.80 to Rs.100, the actual increase is

Rs.20, and the percent increase is

20 100% 1 100% 25%

80 4

× = × = .

• If a < b , the percent increase in going from a to b is always greater than

percent decrease in going from b to a .

• To increase a number by k%, mul tiply i t by 1+ k%, and to decrease a

number by k%, mul tiply it by 1− k%. For example, the value of an

investment of Rs. 20,000 after 25% increase is

20,000×(1+ 25%) = 20,000×(1.25) = 25,000 .

• If a number is the resul t of increasing another number by k%, to find the

original number divide i t by 1+ k%, and i f a number is the resul t of

decreasing another number by k%, to find the original number, divide i t by

1− k%.

For example, The government announced an 20% increase in salaries. If

after the increment, The salary of a particular employee is Rs. 18, 000,

what was the original salary?

Original salary (in Rs.) =

18,000 18,000 15,000

1 1 20% 1.20

current salary

percent increase

= = =

+ +

Ratios and Proportions

• A ratio is a fraction that compares two quantities that are measured in the

same units. The first quantity is the numerator and the second quantity is

denominator. For example, if there are 16 boys and 4 girls, we say that the

ratio of the number of boys to the number of girls on the team is 16 to 4,

or

16

4 . This is often written as 16:4. Since a ratio is just a fraction, it can

be reduced or converted to a decimal or a percent. The Following are

different ways to express the same ratio:

16 to 4 , 16 : 4 ,

16

4 ,

4

1 , 0.25 , 25%

• If a set of objects is divided into two groups in the ration a : b , then the

first group contains

a

a + b of the total objects and simi larly the second group

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contains

b

a + b of the total number of objects. This rule applies to extended

ratios, as well . If a set is divided into three groups in the ratio a : b : c , then

the first group contains

a

a + b + c of the total objects, and so on.

• A proportion is an equation that states that two ratios are equivalent. Since

ratios are just fractions, any equation such as

4 10

6 15

= in which each side is

a single fraction is proportion. This proportion states that 4 relates to 6 in

same ratio as 10 relates to 15.

• For each proportion of the form

a c

b d

= , ad = bc . This property can be used

to solve proportions for unknowns (variables). For example: “If 3 oranges

cost Rs.5, how many oranges can you buy for Rs.100”. To solve this

problem we have to set up a proportion. If the number of oranges for

Rs.100 is x , then:

3 3 100 5 3 100 60

5 100 5

x x x x ×

= ⇒ × = × ⇒ = ⇒ =

Averages

• The average of a set of n numbers is the sum of those numbers divided by

n.

average sum of n numbers

n

=

or simply

A Sum

n

=

the technical name for these kind of averages is Arithmetic Mean.

• If you know the average of n numbers, multiply that average with n to get

the sum of numbers.

• If all the numbers in a set are the same, then that number is the average.

• Assume that the average of a set of numbers is A. If a new number x is

added to that set, the new average will be;

o Greater if x is greater than the existing average

o Smaller if x is smaller than the existing average

o Unchanged if x is equal to the existing average

• Arithmetic sequence is an ordered set of numbers, such that, the difference

between two consecutive numbers is the same.

• If there is an arithmetic sequence of n terms, then the average calculation

can be made simple by using these rules.

o The average of the terms in that sequence will be the middle term,

if n is odd.

o If n is even, the average will be the average of two middle terms.

2.1.2 Algebra

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Polynomials

• A monomial is any number or variable or product of numbers and variables.

For example

3, − 4, x, y, 3x, − 2xyz, 5x3 ,1.5xy2 , a3b2 are all monomials.

• The number that appears in front of a variable in a monomial is called the

coefficient. The coefficient of

5x3 is 5. If there is no number, the

coefficient is ei ther 1 or –1, because x means 1x and −x means −1x .

• A polynomial is a monomial or the sum of two or more monomials. Each

monomial that makes up the polynomial is called a term of that polynomial.

• If a polynomial has only one term it is a simple monomial, if it has two

terms, it is known as binomial and if it has three terms, it is called

trinomial.

• Two terms are called like terms if they differ only in their coefficients.

5x3

and

−2x3 are like terms, whereas,

5x3 and

5x2 are not.

• If like terms are involved in a polynomial, they can be combined, by adding

their coefficients, to make that polynomial simpler. The polynomial

3x2 + 4x + 5x − 2x2 − 7 is equivalent to the polynomial

x2 + 9x − 7 .

• All laws of arithmetic are also applicable to polynomials. Most important of

them is PEMDAS.

• Polynomials can be added, subtracted, multiplied or divided.

• To add two polynomials, put a plus sign between them, erase the

parentheses, and combine like terms.

Example:

What is the sum of

5x2 +10x − 7 and

3x2 − 4x + 2 ?

Solution:

2 2

2 2

2

(5 10 7) (3 4 2)

5 10 7 3 4 2

8 6 5

x x x x

x x x x

x x

+ − + − +

= + − + − +

= + −

• To subtract two polynomials, reverse the signs of subtrahend, and add two

polynomials as done before.

Example:

Subtract

3x2 − 4x + 2 from

5x2 +10x − 7

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Solution:

2 2

2 2

2 2

2

(5 10 7) (3 4 2)

(5 10 7) ( 3 4 2)

5 10 7 3 4 2

2 14 9

x x x x

x x x x

x x x x

x x

+ − − − +

= + − + − + −

= + − − + −

= + −

• To multiply monomials, first multiply their coefficients, and then multiply

their variables by adding the exponents.

Example:

What is the product of

3x2 yz from

−2x2 y2 ?

Solution:

2 2 2

2 2 2

4 3

(3 )( 2 )

(3 2)( )( )( )

6

x yz x y

x x y y z

x y z

−

= ×− × ×

= −

• To multiply a monomial by a polynomial, just multiply each term of the

polynomial by the monomial.

Example:

What is the product of 3x from

3x2 − 6xy2 + 2?

Solution:

2 2

2 2

3 2 2

(3 )(3 6 2)

(3 3 ) (3 6 ) (3 2)

9 18 6

x x xy

x x x xy x

x xy x

− +

= × − × + ×

= − +

• To multiply two binomials, multiply each term of first binomial by each term

of second binomial, then add the results.

Example:

What is the product of 3x + y from

3x2 − 6xy2 ?

Solution:

2 2

2 2 2 2

2 2 2 2 3

2 2 2 2 3

(3 )(3 6 )

(3 3 ) (3 ( 6 )) ( 3 ) ( ( 6 ))

(9 ) ( 18 ) (3 ) ( 6 )

9 18 3 6

x y x xy

x x x xy y x y xy

x xy xy xy

x xy xy xy

+ −

= × + × − + × + × −

= + − + + −

= − + −

• The three most important binomial products are:

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o

(x − y)(x + y) = x2 + xy − xy + y2 = x2 + y2

o

(x + y)(x + y) = x2 + xy + xy + y2 = x2 + 2xy + y2

o

(x − y)(x − y) = x2 − xy − xy + y2 = x2 − 2xy + y2

Memorizing these can save a lot of calculation time during the test.

• To divide a polynomial by a monomial, divide each term of the polynomial

by the monomial.

Example:

What is the quotient if

32x2 y +12xy3z is divided by 8xy ?

Solution:

2 3 2 3

32 12 32 12 4 3 2 (by reducing the terms)

8 8 8 2

x y xy z x y xy z x yz

xy xy xy

+

= + = +

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Solving Equations and Inequalities

• The basic principle in solving equations and inequalities is that you can

manipulate them in any way as long as you do the same thing to both

sides. For example you may add a number to both sides, or you may

divide or multiply both sides with same number etc.

• By using the following six-step method, you can solve most of the

equations and inequalities. The method is explained with the help of an

example.

Example:

if

1 3( 2) 2( 1) 1

2

x + x − = x + + , what is the value of x ?

Solution:

Step What to do… Example

1 Get rid of fractions and

decimals by multiplying

both sides by the LCD.

Multiply each side by 2 to get:

x + 6(x − 2) = 4(x +1) + 2

2 Get rid of all parentheses

by solving them.

x + 6x −12 = 4x + 4 + 2

3 Combine like terms on

each side.

7x −12 = 4x + 6

4 By adding and subtracting

get all the variables on

one side (mostly left).

Subtract 4x from each side to get:

3x −12 = 6

5 By adding or subtracting

get all plain numbers on

the other side.

Add 12 to each side to get:

3x =18

6 Divide both sides by the

coefficient of the variable.

(If you are dealing with

an inequality and you

divide with a negative

number, remember to

reverse the inequality.)

Divide both sides by 3 to get:

x = 6

• When you have to solve one variable and the equation/inequality involve

more than one variable, treat all other variables as plain numbers and

apply the six-step method.

Example:

i f a = 3b − c , what is the value of b in terms of a and c ?

Solution:

Step What to do… Example

1 There are no fractions and

decimals.

2 There are no parentheses.

3 There are no like terms.

4 By adding and subtracting get

all the variables on one side.

Remember there is only one

variable b , which is on one

side only.

5 By adding or subtracting get

all plain numbers on the

other side.

Remember we are

considering a and c as

plain number. Add c to

each side to get:

a + c = 3b

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6 Divide both sides by the

coefficient of the variable.

Divide both sides by 3 to

get:

3

a c b

+

=

• It is not necessary to follow these steps in the order specified. Some times

it makes the problem much easier, if you change the order of these steps.

Example:

If x − 4 =11, what is the value of x -8?

Solution:

Going immediately to step 5, add 4 to each side to get: x =15. Now

subtract 8 from both sides to get: x −8 = 7 .

• Doing the same thing on each side of an equation does not mean doing the

same thing to each term of the equation. This is very important if you are

doing divisions, or dealing with exponents and roots.

Example:

If a > 0 and

a2 + b2 = c2 , what is the value of a in terms of b and c .

Solution:

a2 + b2 = c2 ⇒ a2 = c2 − b2 . Now you can’t take a square root of each term

to get a = c − b . You must take the square root of each side:

a2 = c2 − b2 ⇒ a = c2 − b2

• Another type of equation is that in which the variable appears in exponent.

These equations are basically solved by inception.

Example:

If

2x+3 = 32 , what is the value of

3x+2

?

Solution:

2x+3 = 32 ⇒ 2x+3 = 25 ⇒ x + 3 = 5 ⇒ x = 2 .

Now as x = 2 , you can get

x = 2 ⇒ x + 2 = 4 ⇒ 3x+2 = 34 = 81

• A system of equations is a set of two or more equations having two or more

variables. To solve such equations, you must find the value of each

variable that will make each equation true.

• To solve a system of equations, add or subtract them to get a third

equation. If there are more than two equations you can just add them.

Example:

If x + y =10 and x − y =10 what is the value of y ?

Solution:

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Add two equations:

10

2

2 12 6

x y

x y

x x

+ =

− =

= ⇒ =

Now replacing x wi th 6 in the first equation: 6 + y =10 ⇒ y = 4

• If you know the value of one variable in a system of two equations, you can

use this value to get the value of the other variable. As it is done in the

previous question.

Word problems

• To solve word problems, first translate the problem from English to

Algebra. While translating, use variables to represent unknowns. Once

translated, it is easy to solve them using the techniques you have learned

in previous sections.

• Following English to Algebra dictionary will be helpful in translating word

problems to algebraic expressions.

English words

Mathematical

meaning

Symbol

Is, was, will be, had, has, will

have, is equal to, is the same as

Equals =

Plus, more than, sum, increased

by, added to, exceeds, received,

got, older than, farther than,

greater than

Addition +

Minus, fewer, less than,

difference, decreased by,

subtracted from, younger than,

gave, lost

Subtraction −

Times, of, product, multiplied by Multiplication ×

Divided by, quotient, per, for Division

or a

b

÷

More than, greater than Inequality >

At least Inequality ≥

Fewer than, less than Inequality <

At most Inequality ≤

What, how many, etc. Unknown quantity x

(Some variable)

Examples:

o The sum of 5 and some number is 13. 5 + x =13

o Javed was two years younger than Saleem. J = S − 2

o Bilal has at most Rs.10,000. B ≤10000

o The product of 2 and a number exceeds that

number by 5 (is 5 more than that number). 2N = N + 5

• In word problems, you must be sure about what you are answering. Do not

answer the wrong question.

• In problems involving ages, remember that “years ago” means you need to

subtract, and “years from now” means you need to add.

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• Distance problems all depend on three variations of the same formula:

o distance = speed ×time

o

speed distance

time

=

o

time distance

speed

=

Example:

How much longer, in seconds, is required to drive 1 mile at 40 miles per

hour than at 60 miles per hour?

Solution:

The time to drive at 40 miles per hour can be calculated as

time1 =

1

40 hours =

1

40 2

× 60 3

minutes =

3

2

× 60 30

seconds =

90 seconds

The time to drive at 60 miles per hour can be calculated as

time2 =

1

60 hours =

1

60

× 60 minutes = 1×60 seconds = 60 seconds

difference = time1− time2 = 90 − 60 = 30 seconds.

2.1.3 Geometry

Lines and Angles

• An angle is formed at the intersection of two line segments, rays or lines.

The point of intersection is called the vertex. Angles are measured in

degrees.

• Angles are classified according to their degree measures.

• An acute angle measures less than 90°

• A right angle measures 90°

• An obtuse angle measures more than 90° but less than 180°

• A straight angle measures 180°

• If two or more angles combine together to form a straight angle, the sum

of their measures is 180° .

a° + b° + c° + d° =180°

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• The sum of all the measures of all the angles around a point is 360°

a° + b° + c° + d°+ e° = 360°

• When two lines intersect, four angles are formed, two angles in each pair of

opposite angles are called vertical angles. Vertical angles, formed by the

intersection of two lines, have equal measures.

a = c and b = d

• If one of the angles formed by the intersection of two lines is a right angle,

then all four angles are right angles. Such lines are called perpendicular

lines

a = b = c = 90

• In the figure below a line l divides the angle in two equal parts. This line is

said to bisect the angle. The other line k bisects another line into two

equal parts. This line is said to bisect a line.

• Two lines are said to be parallel, if they never intersect each other.

However, if a third line, called a transversal, intersects a pair of parallel

lines, eight angles are formed. And the relationship among theses angles is

shown in the following diagram.

•

• - All four acute angles are equal

a = c = e = g

• - All four obtuse angles are equal

b = d = f = h

• - The sum of any pair of acute and

obtuse angle is 180°, e.g.

a + d =180°,d + e =180°,b + g =180° etc.

Triangles

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• In any triangle, the sum of the measures of the three angles is 180° .

x + y + z =180

• In any triangle:

o The longest side of triangle is opposite the largest angle.

o The shortest side is opposite the smallest angle.

o Sides with the same length are opposite the angles with the same

measure.

• Triangles are classified into three different kinds with respect to the

lengths of sides.

o Scalene: in which all three sides are of different lengths.

o Isosceles: in which two of the sides of triangle are equal in

length, the third is different.

o Equilateral: in which all three sides are equal in length.

• Triangles are also classified with respect to the angles.

o Acute triangle: in which all three angles are acute.

o Obtuse triangle: in which one angle is obtuse and two are acute.

o Right triangle: This has one right and two acute angles.

• In a right triangle, the opposite to the right angle is known as hypotenuse

and is the longest side. The other two sides are called legs.

• In any right triangle, the sum of the measures of the two acute angles is

90° .

• By Pythagorean Theorem, the sum of squares of the lengths of legs of a

right triangle is always equal to the square of length of hypotenuse.

a2 + b2 = c2

• In any triangle, the sum of any two sides is always greater than the third

one. And the difference of any two sides is always less than the third one.

a + b > c and a − b < c

• The perimeter of a triangle is calculated by adding the lengths of al l the

sides of that triangle.

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perimeter = a + b + c

• The area of a triangle is calculated by the formula:

1

2

area = bh

where b is

the base of the triangle and h is the height of the triangle.

o Any side of triangle can be taken as the base.

o Height is the altitude (perpendicular) drawn to the base from its

opposite vertex.

o In a right triangle any leg could be taken as the base, the other

will be the altitude.

Quadrilateral and other Polygons

• A polygon is a closed geometric figure, made up of line segments. The

line segments are called sides and the end points of lines are called

vertices (plural of vertex). Lines, inside the polygon, drawn from one

vertex to the other, are called diagonals.

• The sum of the measures of the n angles in a polygon with n sides is

always (n − 2)×180° .

• In any quadrilateral, the sum of the measures of the four angles is 360° .

• A regular polygon is a polygon in which all of the sides are of the same

length. In any regular polygon, the measure of each interior angle is

(n 2) 180

n

− × °

and the measure of each exterior angle is

360

n

°

.

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• A parallelogram is a special quadrilateral, in which both pairs of opposite

sides are parallel. The Following are some properties of parallelogram.

o Lengths of opposi te sides are equal . AB = CD and AD = BC

o Measures of opposi te angles are equal . a = c and b = d

o Consecutive angles add up to 180° . a + b =180° , b + c =180° etc.

o The two diagonals bisect each other. AE = EC and BE = ED

o A diagonal divides the parallelogram into two triangles that are

congruent.

• A rectangle is a parallelogram in which all four angles are right angles. It

has all the properties of a parallelogram. In addition it has the following

properties:

o The measure of each angle in a rectangle is 90° .

o The diagonals of a rectangle are equal in length.

• A square is a rectangle that has the following additional properties:

o A square has all its sides equal in length.

o In a square, diagonals are perpendicular to each other.

• To calculate the area, the following formulas are required:

o For a parallelogram, Area = bh , where b is the base and h is the

height.

o For a rectangle, Area = lw, where l is the length and w is the

width.

o For a square,

Area = s2 , where s is the side of the square.

• Perimeter for any polygon is the sum of lengths, of all its sides.

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Circles

• A circle consists of all the points that are the same distance from one fixed

point called the center. That distance is called the radius of a circle. The

word radius is also used to represent any of the line segments joining the

center and a point on the circle. The plural of radius is radii.

• Any triangle, such as CED in the figure, formed by connecting the end

points of two radii, is an isosceles.

• A line segment, such as ED in the diagram above, both of whose end

points are on a circle is called a chord.

• A chord that passes through the center of the circle is called the diameter

of the circle. The length of the diameter is always double the radius of the

circle. The diameter is the longest cord that can be drawn in a circle.

• The total length around a circle is known as the circumference of the

circle.

• The ratio of the circumference to the diameter is always the same for any

circle. This ratio is denoted by the symbol π (pronounced as pi).

• 2 C C d C r

d

π = ⇒ =π ⇒ = π where C is the circumference, d is the diameter

and r is the radius of the circle.

• Value of π is approximately 3.14

• An arc consists of two points in a circle and all the points between them.

E.g. PQis an arc in the diagram.

• An angle whose vertex is at the center of the circle is called the central

angle. ∠PCQ in the diagram above is a central angle.

• The degree measure of a complete circle is 360° .

• The degree measure of an arc is the measure of the central angle that

intercepts it. E.g. the degree measure of PQ is equal to the measure of

∠PCQ in the diagram above.

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• If x is the degree measure of an arc, its length can be calculated as 360

x C ,

where C is the circumference.

• The area of a circle can be calculated as

π r2 .

• The area of a sector formed by the arc and two radii can be

calculated as A x

360 , where A is the area of a circle.

2.2 Discrete Quantitative Questions

These are standard multiple-choice questions. Most of such questions require you to do some

computations and you have to choose exactly one of the available choices based upon those

computations. This section will teach you the basic tactics to attempt such questions.

2.2.1 Question format

Each question will consist of a question statement and the choices labeled from A to E. The

number of choices may vary from 2 to 5, but exactly one choice will be correct for each

question.

2.2.2 How to attempt?

Following are some tactics, which will lead you to the correct answer.

• Whenever you know how to answer a question directly, just do it. The

tactics should be used only when you do not know the exact solution, and

you just want to eliminate the choices.

• Remember that no problem requires lengthy or difficult computations. If

you find yourself doing a lot of complex arithmetic, think again. You may

be going in the wrong direction.

• Whenever there is a question with some unknowns (variables), replace

them with the appropriate numeric values for ease of calculation.

• When you need to replace variables with values, choose easy-to-use

numbers, e.g. the number 100 is appropriate in most percent-related

problems and the LCD (least common denominator) is best suited in

questions that involve fractions.

• Apply “back-solving” whenever you know what to do to answer the question

but you want to avoid doing algebra. To understand this tactic read the

following example:

On Monday, a storeowner received a shipment of books. On Tuesday, she

sold half of them. On Wednesday after two more were sold, she had

exactly 2/5 of the books left. How many were in the shipment?

(A) 10 (B) 20 (C) 30 (D) 40 (E) 50

now by this tactic:

Assume that (A) is the correct answer, if so; she must have 3 books on

Wednesday. But 2/5 of 10 are 4, so, (A) is incorrect;

Assume that (B) is the correct answer, if so; she must have 8 books on

Wednesday. 2/5 of 20 are 8, so, (B) is the correct choice, and as there may

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be only one correct choice, there is no need to check for remaining choices.

This tactic is very helpful when a normal algebraic solution for the problem

involves complex or lengthy calculations.

• If you are not sure how to answer the question, do not leave it

unanswered. Try to eliminate absurd choices and guess from the remaining

ones. Most of the times four of the choices are absurd and your answer is

no longer a guess.

Many things may help you to realize that a particular choice is absurd.

Some of them are listed below.

o The answer must be positive but some of the choices are negative

so eliminate all the negative ones.

o The answer must be even but some of the choices are odd so

eliminate all the odd choices.

o The answer must be less then 100, but some of the choices are

greater than 100 (or any other value) so eliminate all choices that

are out of range.

o The answer must be a whole number, but some of the choices are

fractions so eliminate all fractions.

o These are some examples. There may be numerous situations

where you can apply this tactic and find the correct answer even if

you do not know the right way to solve the problem.

Example questions with solutions

The following are some examples, which will help you to master these types of

questions.

Example

If 25% of 220 equals 5.5% of X, what is X?

(A) 10 (B) 55 (C) 100 (D) 110 (E) 1000

Solution:

Since 5.5% of X equals 25% of 220, X is much greater than 220. So, choices

A, B, C, and D are immediately eliminated because these are not larger than

220. And the correct answer is choice E.

(Note: An important point here is that, even if you know how to solve a problem, if

you immediately see that four of the five choices are absurd, just pick the remaining

choice and move on.)

Example

Science students choose exactly one of three fields (i.e. medical sciences,

engineering sciences and computer sciences). If, in a college, three-fifths of

the students choose medical sciences, one-forth of the remaining students take

computer sciences, what percent of the students take engineering sciences?

(A) 10 (B) 15 (C) 20 (D) 25 (E) 30

Solution:

The least common denominator of 3/5 and 1/4 is 20, so assume that there are

20 students in that college. Then the number of students choosing medical

sciences is 12 (3/4 of 20). Of the remaining 8 students, 2 (1/4 of 8) choose

computer sciences. The remaining 6 choose engineering sciences. As 6 is

30% of 20, the answer is E.

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Example

If a school cafeteria needs C cans of soup each week for each student and

there are S students, for how many weeks will X cans of soup last?

(A) CX/S (B) XS/C (C) S/CX (D) X/CS (E) CSX

Solution:

Replace C, S and X with three easy to use numbers. Let C=2, S=5 and X=20.

Now each student will need 2 cans per week and there are 5 students, so 10

cans are needed per week and 20 cans will last for 2 weeks. Now put these

values in choices to find the correct one.

The choices A, B, C, D and E become 8, 50, 1/8, 2 and 200 respectively. So

the choice D represents the correct answer.

2.3 Quantitative Comparison Questions

Some of the questions in the Quantitative section of the test may be

quantitative comparison questions. The Following text will explain you the

format and techniques u need to attempt the questions of this format.

2.3.1 Question format

Such questions consist of two quantities, one in column A and the other in

column B. You have to compare the two quantities. The information

concerning one or both quantities is presented before them. Only the following

four choices will be given:

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

And as it is clear from the choices, only one will be correct at one time. Your job is

to choose one of them after careful comparison. The following text explains some

simple tactics to attempt such questions.

2.3.2 How to attempt

Whenever you encounter a quantitative comparison question, the following

guidelines will help you to find the correct answer quickly.

• If the question involves some variables, replace them with appropriate

numbers. Here are some guidelines in choosing an appropriate number:

o The very best numbers to use are –1, 0 and 1.

o Often fractions between 0 and 1 are useful (e.g. 1/2, 3/4 etc.).

o Occasionally, “large” numbers such as 10 or 100 can be used.

o If there is more than one variable, it is permissible to replace each

with the same number.

o Do not impose any un-specified conditions on numbers. Choose

them randomly.

• Eliminate the choices and choose from the remaining ones. For example If

you found the quantities ever equal, the correct choice could never be A or

B, so, eliminate A and B.

• A quantitative comparison question can be treated as an equation or

inequality. Either:

Column A < Column B, or

Column A = Column B, or

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Column A > Column B

So, you can perform similar operation on both columns to simplify the

problem just as in equations (or inequalities).

Example:

m > 0 and m ≠ 1

• In this example divide both the quantities by m2. This will change

column A to 1 and column B to m. Now the comparison is very simple,

as we know that m is greater than 0 and cannot be 1. So the

relationship is not determinable using the current information. m can

be both greater than 1 or between 0 and less than 1.

2.3.3 Example questions with Answers and

Explanations

Example 1:

A student earned a 75 on each of her first three

math tests and an 80 on her fourth and fifth

tests.

A B

Average after 4 tests Average after 5 tests

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

Remember you want to know which average is higher, not what the averages

are. After 4 tests, the average is clearly less than 80, so an 80 on the fifth

test had to raise the average. So the answer is choice (B).

Example 2:

A B

The time it takes to

drive 40 miles at 35

mph

The time it takes to

drive 35 miles at 40

mph

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

Once again there is no need for calculation, as the speed in column B is higher

than that in column A. It is obvious that it will take less time to travel shorter

distance at a greater speed. So the value in column A is larger. The answer is

option (A).

Example 3:

A B

m2 m3

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20

2

5

5

A. The quantity in column A is greater

B. The quantity in column B is greater

C. The two quantities in both columns are equal

D. The relationship cannot be determined from the information given

Square each column:

2

20 20 5

2 4

= =

and

2 5 255

5 5

= =

. So both columns are

equal and the answer is choice (C).

Example 4:

A B

13y 15y

To solve this question, subtract 13y from both columns to get 13y −13y = 0 for

column A and 15y −13y = 2y for column B. As there are no restrictions, 2y can

be greater than, less than or equal to 0. So the correct choice is (D).

2.4 Data Interpretation Questions

These questions are based on the information that is presented in the form of

a graph, chart or table. Most of the data is presented graphically. The most

common types of graphs are line graphs, bar graphs and circle graphs. The

objective of such questions is to test your ability to understand and analyze

statistical data.

2.4.1 Question Format

Data interpretation questions always appear in sets, you are presented with

some data in any format (chart, graph or table), and you will then be asked

with some questions about that data.

The following example explains the format of such questions.

Example:

Question 1:

What is the average sale, in million Rs., for the period 1994-2000?

(A) 5.5 (B) 6.0 (C) 7.0 (D) 8.0

(E) 8.5

Question 2:

For which year, the percentage increase in sales from the previous year is the greatest.

(A) 1995 (B) 1996 (C) 1999 (D) 2000

(E) 2001

2.4.2 How to attempt

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• Do not try to answer such questions immediately, first of all read the

presented data carefully. You must be very clear about the data and its

meanings even before reading the first question.

• Do not confuse numbers with percents. This confusion is most likely to

occur when data is presented in pie graphs. For example in the

following graph

0

2

4

6

8

10

12

1994 1995 1996 1997 1998 1999 2000 2001

Years

Sales in million Rs.

Now it would be a great mistake here to think that sales of “TVs & VCRs” is

15% more than the sales of Computers in 2001 by XYZ Corporation. To

know this you have to calculate it as

15 100 60%

25

× =

.

• Try to avoid un-necessary calculations. Most of the questions could

easily be solved by observation and estimation. Use estimation to

eliminate the choices, if you are not able to find the correct answer

without calculation. For example to solve “Question 1” presented in the

example at the start of this section, if you are not sure of the correct

answer, you can then try to cut down the number of possible choices by

observation. You are being asked to tell the percentage increase.

Where as, in year 2000, the sale is decreasing instead of increasing, so,

you can immediately eliminate choice (D) in that question.

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• Your answers must be based upon the information presented in the

given charts and graphs. If your knowledge contradicts any of the data

presented, ignore what you know and stick to the presented data. The

presented data should be the only base for your calculations and

estimations.

• Always use the proper units, there may be some questions that ask you

to compare different data items possibly from different data sets. Be

careful about the units used to represent the data.

• Because graphs and charts present data in a form that enables you to

readily see the relationships among values and to make quick

comparisons, you should always try to visualize you answer in the same

format as the original data was presented.

• Be sure that your answer is reasonable. For example, the profit could

never increase the actual sales, or the expenses could never be negative

etc. While answering the question, first of al l eliminate such unreasonable

choices, and then choose from the remaining ones.

2.5 Practice exercise

1 What is the average of positive integers from 1 to 100 inclusive?

(A) 49

(B) 49.5

(C) 50

(D) 50.5

(E) 51

2 If x + y = 6 , y + z = 7 , and x + z = 9 , what is the average of x , y and z ?

(A)

11

3 (B)

11

2 (C)

22

3

(D) 11 (E) 22

3 In the diagram below, lines l and m are not parallel.

If A represents the average measure of all the eight angles, what is the value

of A?

(A) A = 45

(B) 45 < A < 90

(C) A = 90

(D) 90 < A <180

(E) A =180

4 Aslam has 4 times as many books as Salman and 5 times as many as Javed.

If Javed has more than 40 books, what is the least number of books that

Aslam could have?

(A) 200 (B) 205 (C) 210 (D) 220 (E) 24

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5 Aslam is now 3 times as old as Javed, but 5 years ago, he was 5 times as

Javed was. How old is Aslam now?

(A) 10 (B) 12 (C) 24 (D) 30 (E) 36

6 If x%of y is 10, what is y?

(A)

x

10

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