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 similarly the second group 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 4/6 = 10/15 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/b = c/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:
3/5 = x/100 ⇒ 3 × 100 = x × 5 ⇒ x = 3 × 100/5 ⇒x = 60

Continued...

Averages

• The averageof a set of n numbers is the sum of those numbers divided by n. The technical name for these kind of averages is Arithmetic Mean.
Average = Sum of n Numbers/n or simply A=Sum/n

• 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.

Continued...

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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, 5x^3, 1.5xy^2 are all monomials.

• The number that appears in front of a variable in a monomial is called the
coefficient. The coefficient of 5x^3 is 5. If there is no number, the coefficient is either 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. 5x ^3 and −2x^3 are like terms, whereas, 5x^3 and 5x^2 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
3x^2 + 4x + 5x − 2x^2 − 7 is equivalent to the polynomial x^2 + 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 5x^2 + 10x − 7 and 3x^2− 4x + 2 ?
Solution:

(5x^2 + 10x − 7) + (3x^2− 4x + 2)
= 5x^2 + 10x − 7 + (3x^2− 4x + 2
= 8x^2 + 6x − 5

• To subtract two polynomials, reverse the signs of subtrahend, and add two
polynomials as done before.

Example:
Subtract 3x^2− 4x + 2 from 5x^2 + 10x − 7
Solution:

(5x^2 + 10x − 7) − (3x^2− 4x + 2)
= (5x^2 + 10x − 7) + (−3x^2 +4x −2)
= 5x^2 + 10x − 7 − 3x^2 +4x −2
= 2x^2 + 14x +9

• To multiply monomials, first multiply their coefficients, and then multiply
their variables by adding the exponents.

Example:
What is the product of 3x^2.yz and −2x^2.y^2?
Solution:

(3x^2.yz)(−2x^2.y^2)
= (3 × -2)(x^2 × x^2)(y × y^2)(z)
= -6.x^4.y^3.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 and 3.x^2 −6.x.y^2 +2?

Solution:
(3x)(3.x^2 −6.x.y^2 +2)
= (3x × 3.x^2 ) − (3x × 6x.y^2) + (3x × 2)
= 9x^3 − 18x^2.y^2 + 6x

• 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 and 3.x^2 −6.x.y^2?
Solution:

(3x + y)(3.x^2 −6.x.y^2)
= (3x × 3.x^2) + (3x × (-6x.y^2)) + (y × 3.x^2) + (y × (-6x.y^2))
= (9x^2) + ( −18.x^2.y^2) + (3.x^2.y) + ( −6.x.y^3)
= 9x^2 −18.x^2.y^2 3.x^2.y −6.x.y^3


• The three most important binomial products are:

o (x − y)(x + y) = x^2 + x.y − x.y + y^2 = x^2 + y^2
o (x − y)(x − y) = x^2 − x.y − x.y + y^2 = x^2 −2xy + y^2
o (x + y)(x + y) = x^2 + x.y + x.y + y^2 = x^2 −2xy + y^2

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 32.x^2.y + 12.x.y^3.z is divided by 8xy?

Solution:

(by reducing the terms)
32.x^2.y + 12.x.y^3.z/8xy
= 32.x^2.y/8xy + 12.x.y^3.z/8xy
= 4x + 3/2.y^2.z

Continued....

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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 1/2.x +3(x-2)=2(x+1)+1, what is the value of x?

Step 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 .

Step 2:

Get rid of all parentheses by solving them i.e. x+6x-12 = 4x+4+2

Step 3:

Combine like terms on each side. 7x -12 = 4x + 6

Step 4:

By adding and subtracting get all the variables on one side (mostly left).
Subtract 4x from each side to get: 3x - 12 = 6

Step 5:

By adding or subtracting get all plain numbers on the other side.
Add 12 to each side to get: 3x = 18

Step 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.) i.e. 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:
if a=3b-c what is the value of b in terms of a and c?
Solution:
Step 1: There are no fractions and decimals.

Step 2: There are no parentheses.

Step 3: There are no like terms.

Step 4: By adding and subtracting get all the variables on one side i.e. Remember there is only one variable b, which is on one side only.

Step 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

Step 6: Divide both sides by the coefficient of the variable.
Divide both sides by 3 to get: a+c/3 = 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.

• 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.

• Another type of equation is that in which the variable appears in exponent.
These equations are basically solved by inception.

If 2 ^ x+32 =32 ( ^ stands for exponent means raise to power), what is the value of 3^x+2?

Solution:
2^x+3=32 ⇒ 2^x+3=2^5 ⇒ x+3=5 ⇒ x=2
Now as x=2 you can get x=2 ⇒ x+2=4 ⇒ 3^x+2 = 3 ^4 =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 then what is the value of y?

Solution:
Add these two equations: (x+y=10) + (x-y=10) gives us 2x=12 i.e. x=6
Now replacing x with 6 in the first equation : 6+y=10 gives us 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.

Continue...

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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.

• Following English to Algebra dictionary will be helpful in translating word
problems to algebraic expressions.

English words
1. Is, was, will be, had, has, will have, is equal to, is the same as
Mathematical meaning = Equals, =

2. Plus, more than, sum, increased by, added to, exceeds, received, got, older than, farther than, greater than
Mathematical meaning = Addition, +

3. Minus, fewer, less than, difference, decreased by, subtracted from, younger than, gave, lost
Mathematical meaning = Subtraction −

4. Times, of, product, multiplied by
Mathematical meaning = Multiplication ×

5. Divided by, quotient, per, for
Mathematical meaning = Division ÷ or a/b

6. More than, greater than
Mathematical meaning = Inequality >

7. At least
Mathematical meaning = Inequality ≥

8. Fewer than, less than
Mathematical meaning = Inequality <

9. At most
Mathematical meaning = Inequality ≤

10. What, how many, etc.
Mathematical meaning = 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.

• 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 * 60 minutes = 3/2 * 60 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.

Continued...

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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°.
• 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.
a^2 + b^2 = c^2

• 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 all the
sides of that triangle. perimeter = a+b+c

• The area of a triangle is calculated by the formula: area = 1/2.b.h
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.

Continued...
.

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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.

• 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 opposite sides are equal. AB = CD and AD = BC
o Measures of opposite 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 = s.s , where s is the side of the square.

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

Continued...

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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 formed by connecting the end points of two radii, is an isosceles.

• A line segment 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).

• π = C/d ⇒ C = πd ⇒ C = 2πr 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.

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

• If x is the degree measure of an arc, its length can be calculated as x/360 * C where C is the circumference.

• The area of a circle can be calculated as πr^2 .

• The area of a sector formed by the arc and two radii can be calculated as x/360 * A, where A is the area of a circle.

Continued...

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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.

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.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 store owner 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 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.

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.

Continued...

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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 you need to attempt the questions of this format.

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. 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
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

m^2 m^3

• In this example divide both the quantities by m^2. 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.

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-Average after 4 tests
B-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-The time it takes to drive 40 miles at 35 mph
B-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-13y
B-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)

Continued...

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