NTS pass the test Tips and guide for better grades

National Testing Service (NTS) is an organization in Pakistan that administers the academic performance assessment tests.
NTS offers two main types of tests, the national aptitude test (NAT) and the Postgraduate Assessment Test (GAT). NAT is for students seeking admission to universities and colleges. GAT is aimed at graduates seeking admission to higher education. NTS tests are also used to determine the qualifications of students seeking higher education abroad.
Today Pakistan NTS perform the test for government and Sami-government jobs with these admissions tests. It is mandatory for students to clear these tests performed by NTS. NTS test is also used to determine Pakistani students who wish to pursue advanced level studies abroad.
When considering NTS students try hundreds of multiple choice questions in 2 hours and all questions in the form of multiple choice questions. In these 100 questions, there are 40% of the questions in the English section, 30% from analysis and 30% in the mathematics section. For other topics, there may be a slight variation.
Test of NTS English section usually includes antonyms, synonyms, fill in the blanks and reading the comprehension questions. In mathematics, which is essentially high school level, the questions relate to basic geometry, assemblies, basic arithmetic, and basic algebra. The analysis section consists mainly of logic puzzles and logical reasoning questions.
If you apply to the NTS test and then want to do it here is the right place for preparing all the admissions and recruiting NTS test. We offer professional advice and help students succeed at all stages of life. If you are interested in passing the NTS test at the maximum number, follow our tips and guidance for the NTS approved test.

NTS pass the test Tips and guide for better grades

NTS Pass the Test Tips & Guide:

Tip # 1:
You must prepare the test pattern for the morning on the day of the exam. While some students read up to the last minute in our knowledge is not the best way to approach the test. First, make the program for the subjects of books and days you have to sit down and start studying accordingly. Do not prepare one subject per day, in the morning reading on one subject, afternoons on another and nights on a third.
In general, the GAT contains 40% to 30% of quantitative analysis subjects and 30% English, so to balance and manage their study of division marks. You can see the details of these split marks in the latest pattern.
Tip # 2:
To prepare for any NTS test you need to gather knowledge. Your first attempt to improve your general knowledge in English and Urdu. In our first priority and short time preparation, you should read "Who is who" and "what is what" books.
The questions are usually related to the following areas:

• Urdu General
• Islamiat (Islamic Knowledge)
• Quantitative reasoning
• Pakistan Affairs
• Everyday Science
• Your specific field or related materials
• English grammar
• Communication and verbal skills
• Post related questions
• English Essay
• General knowledge
• IQ Test Questions
• Mathematics
• Computer and IT
• Current Affairs
• Analytical thinking

Gather evidence on previous records, sample sheets, sample documents and be prepared before one week of your exam. Without these helping tools little chance of getting good results in an exam.
NTS test agencies provide the paper pattern, size, previous documents, content, weight-age and model documents for the preparation of the test, so you first need to get help from these resources. These resources are 100% free, so try to take advantage.
These books can be used as a tool rather than completely trusting them. As GAT has no fixed curriculum, in the end, a book only serves at much as a model for the actual test.
Tip # 3:
It is important to maintain a good attitude and remember that you will do everything possible for this test. If you are in a panic, take some deep breaths. For better preparation, do not talk to other students or anyone right before stress can be contagious.
Tip # 4:
Try to sleep early before the day of the test. Get to the test center early enough and locate your room, seat. Stay relaxed. On the day of the test, we are mention again do not read anything. GAT has no curriculum so you do not have to complete the test sample last night. Time management in the GAT test is really important.
Tip 5:
On the day of the test, you can test your own methods to solve the test by reading all the questions carefully and avoiding careless mistakes. First easy part to build some confidence. Then remove the obvious bad answers. Do not be scared if you have no idea.
To consume time, start solving the test with the math section, and analytics, and finally move on to the English section. Do not try to spend too much time on a particular topic.
Divide your time limit on section wise in your NTS exam. If you choose the first paper in the English section and give a period of only 20-30 minutes and the next analytical section, which includes a table or draws a chart, first read each question carefully. We hope this section will spend your almost paper time but not more than 50 minutes. In the last part of the math section and solve any questions that are simple and can be done in 25 minutes. With this strategy, last two to three double check your answer sheet to make sure that no question is not checked.
Tip # 6:
Dividing the time is very important. In this way, enough time is obtained for each question. Read the questions and think of a response to yourself before looking at the given options results in lesser chances of the mistake. Skipping doubtful questions will prevent you from getting negative marks.
If you do not know the answer to any question then try to guess the right answer to your common sense. If this is not possible, then do not let the test to disturb you mentally. You should avoid the pitfalls of cheap ways to bluff and try smartly.
Tip # 7:
Questions related to the position, job, and ministry are also asked in the NTS written test. So try to gather as much knowledge about this department as you have applied for.
Tip # 8:
Usually, one hundred multiple choice questions are asked and each question carries one mark. Candidates must obtain a minimum of fifty percent of NTS scores to qualify. But the final selection is not measured in the grades and assists. The final selection should include making marks in the interview so that your preparation is not just for testing also for the final interview.
Tip # 9:
NTS written test papers distributed in 5 totally different versions colors that every question paper different queries like its color from each other.
Normally question papers of following colors area unit distributed.

• blue
• green
• pink
• White
• Yellow

Therefore, evident to all that the trap is almost impossible in these writings NTS tests due to strict monitoring and different versions of the questionnaires.
Tip # 10:
In the last tip, if you want to seriously start preparing the NTS test, then you should also appear in CSS, PMS and especially in all PCS exams because once your general knowledge and English will be in better you will be able to attempt these exams too easily. Try to read newspapers in English and follow English news channels for better preparation.
We share the best and greatest tips for preparing NTS and GAT exams and get maximum numbers in a short period of time. If you need related information, keep in touch. Must share your views on how to pass the NTS, write your precious lines in the comments section below.

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Preparation NTS Test: Data Interpretation Questions

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.

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

How To Attempt

• 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
  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 (15/25) x 100 = 60%
• 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.
• 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 all eliminate such unreasonable choices, and then choose from the remaining ones.

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Test NTS Preparation: Quantitative Comparison Questions

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.

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.

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:
  • The very best numbers to use are –1, 0 and 1.
  • Often fractions between 0 and 1 are useful (e.g. 1/2, 3/4 etc.).
  • Occasionally, “large” numbers such as 10 or 100 can be used
  • If there is more than one variable, it is permissible to replace each with the same number.
  • 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
  m², m³
• 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.

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
(√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: {(√20)/2}² = 20/4 = 5 and (5/√5)² = 25/5 = 5
So both columns are equal and the answer is choice (C).

Example 4:

A	B
13/y	15/y

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).

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NTS Test Preparation: Discrete Quantitative Questions

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.

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.

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.
  • The answer must be positive but some of the choices are negative so eliminate all the negative ones.
  • The answer must be even but some of the choices are odd so eliminate all the odd choices.
  • 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.
  • The answer must be a whole number, but some of the choices are fractions so eliminate all fractions.
  • 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.

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NTS Pattern And Practice: Algebra

Algebra

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 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. 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:
  (5x² + 10x - 7) + (3x² - 4x + 2)
  = 5x² + 10x - 7 + 3x² - 4x + 2
  = 8x² + 6x - 5
• 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
  Solution:
  (5x² + 10x - 7) - (3x² - 4x + 2)
  = (5x² + 10x - 7) + (-3x² + 4x - 2)
  = 5x² + 10x - 7 - 3x² + 4x - 2
  = 2x² + 14x - 9
• To multiply monomials, first multiply their coefficients, and then multiply their variables by adding the exponents.
  Example:
  What is the product of 3x²yz from -2x²y² ?
  Solution:
  (3x²yz)(-2x²y²)
  = (3 x -2)(x² x x²)(y x y²)(z)
  = -6x⁴y³z
• 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:
  (3x)(3x² - 6xy² + 2)
  = (3x x 3x²) - (3x x 6xy²) + (3x x 2)
  = 9x³ - 18x²y² + 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 from 3x² - 6xy²?
  Solution:
  (3x + y)(3x² - 6xy²)
  = (3x x 3x²) + (3x x (-6xy²)) + (y x 3x²) + (y x (-6xy²))
  = (9x²) + (-18x²y²) + (3x²y) + (-6xy³)
  = 9x² - 18x²y² + 3x²y - 6xy³
• The three most important binomial products are:
  • (x − y)(x + y) = x² + xy − xy + y² = x² + y²
  • (x + y)(x + y) = x² + xy + xy + y² = x² + 2xy + y²
  • (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 32x²y + 12xy³z is divided by 8xy?
  Solution:
  (32x²y + 12xy³z) / (8xy)
  = (32x²y) / 8xy + (12xy³z) / 8xy
  = 4x + (3/2)y²z (by reducing the terms)

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.
  if (1/2)x + 3(x - 2) = 2(x + 1) + 1, 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:
  if 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
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.
  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 2^x+3 = 32 , what is the value of 3^x+2 ?
  Solution:
  2^x+3 = 32 ⇒ 2^x+3 = 25 ⇒ x + 3 = 5 ⇒ x = 2.
  Now as x = 2 , you can get x = 2 ⇒ x + 2 = 4 ⇒ 3^x+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:
  x + y = 10
  x - y = 10
  Add two equations: 2x = 12 ⇒ x = 6
  Now replacing x with 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.

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	X
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:
• The sum of 5 and some number is 13. 5 + x = 13
• Javed was two years younger than Saleem. J = S − 2
• Bilal has at most Rs.10,000. B ≤ 10000
• 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:
  
  • Distance = Speed × Time
  • Speed = Distance / Time
  • 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:
time1 = 1/40 hours = 1/40 x 60 minutes = 3/2 x 60 seconds = 90 seconds
The time to drive at 60 miles per hour can be calculated as
time2 = 1/60 hours = 1/60 x 60 minutes = 1 x 60 seconds = 60 seconds
difference = time1 - time2 = 90 - 60 = 30 seconds.

• Next Geometry
• Previous Arithmetic
• START TEST

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NTS Pattern And Practice: Quantitative Ability

Quantitative Ability

The Quantitative section measures your basic mathematical skills, understanding of elementary mathematical concepts, and the ability to reason quantitatively and solve problems in a quantitative setting. There is a balance of questions requiring basic knowledge of arithmetic, algebra, geometry, and data analysis. These are essential content areas usually studied at the high school level.

The questions in the quantitative section can also be from

• Discrete Quantitative Question
• Quantitative Comparison Question
• Data Interpretation Question etc.

The distribution in this guide is only to facilitate the candidates. This distribution is not a part of test template, so, a test may contain all the questions of one format or may have a random number of questions of different formats.

This chapter is divided into 4 major sections. The first discusses the syllabus/contents in each section of the test respectively and the remaining three sections address the question format, guide lines to attempt the questions in each format and some example questions.

General Mathematics Review

Arithmetic

The following are some key points, which are phrased here to refresh your knowledge of basic arithmetic principles.

Basic arithmetic

• For any number a, exactly one of the following is true:
• a is negative
• a is zero
• a is positive
• The only number that is equal to its opposite is 0 (e.g. a = −a only if a = 0 ).
• If 0 is multiplied to any other number, it will make it zero ( a×0 = 0 ).
• 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.

+	÷ or ×	+	=	+
+	÷ or ×	-	=	-
-	÷ or ×	+	=	-
-	÷ or ×	-	=	+

• 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 1/a.
• The product of a number and its reciprocal is always one a x 1/a = 1.
• Dividing by a number is the same as multiplying by its reciprocal a ÷ b = a x 1/b.
• 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.
• a is a divisor of b
• a is a factor of b
• b is a divisible by a
• 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.
• 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 x 9 = 54
  3 x 8 = 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:
• b^m x bⁿ = b^{m + n}
• b^m ÷ bⁿ = b^{m - n}
• (b^m)ⁿ = b^{m x n}
• b^m x c^m = (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 x² = 9: x = 3 and x = −3. The positive one, 3, is called the (principal) square root of 9 and is denoted by √9 . Clearly, each perfect square has a square root:
  √0 = 0 , √9 = 3, √36 = 6 , √169 = 13 , √225 = 25 etc.
• 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:
• √ab = √a × √b and √a/b = √a/√b
• √a+b != √a + √b
• as 5 = √25 = √9+16 != √9 + √16 = 3 + 4 = 7
• Although it is always true that (√a)² = a , √a² = a is true only if a is
  positive as √(-5)² = √25 = 5 != -5
• For any number a, √aⁿ = a^n/2
• For any number a, b, and c:
  
• a(b + c) = ab + ac, a(b − c) = ab − ac and if a != 0
• (b + c) / a = b / a + c / a, (b - c) / a = b / a - c / a

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 = b means 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 ≥ 5 means that x can be 5 or any number greater than 5.The statement 2 < x < 5 is 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.
  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 positive number, then a × c < b × c and a/c < b/c.
• 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 a/c > b/c
• If sides of an inequality are both positive and both negative, taking the reciprocal reverses the inequality.
• If 0 < x <1 and a is positive, then xa < a.
• If 0 < x < 1 and m and n are integers with m > n , then x^m < xⁿ < x.
• If 0 < x <1, then √x > x.
• If 0 < x <1, then 1/x > x and 1/x > 1.

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 : 0ⁿ = 0.
• For every number a (including 0) : a ÷ 0 and a / 0 are undefined symbols.
• For every number a (other than 0) : 0 ÷ a = 0 /a = 0.
• 0 is the only number that is equal to i ts opposite : 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 a/1 = a.
• For any number n : 1ⁿ = 1.
• 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.
• 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.
  3/10 = 0.3, 3/4 = 0.75, 8/8 = 1, 48/16 = 3, 100/8 = 12.5.
• 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.
  2/3 = 0.66666..., 3/11 = 0.272727..., 5/12 = 0.416666..., 17/15 = 1.133333...

• To compare two decimals, follow these rules:
• 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).
• 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:
• 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. 2/5 = 0.4 and 1/4 = 0.25. Now, as 0.4 > 0.25, 2/5 > 1/4.
• Cross multiply the fractions. For example to compare 2/5 and 1/4, cross multiply: 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 3/5 > 2/5.
• If the fractions have the same numerator, the fraction with the smaller denominator is greater. For example 3/5 > 3/10.
• Two fractions are called equivalent fractions if both of them have same decimal value.
• For example, 1/2 = 5/10 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-multiply. 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, 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 3/5 x 4/7 = 3 x 4 / 5 x 7 = 12/35.
• To multiply a number to a fraction, write that number as a fraction whose denominator is 1. For example 3/5 x 7 = 3/5 x 7/1 = 3 x 7 / 5 x 1 = 21/5.
• 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: 2/5 x 200 = 2/5 x 200/1 = 400/5 = 80.
• The reciprocal of a fraction a/b is another fraction b/a since a/b x b/a = 1.
• To divide one fraction by the other fraction, multiply the reciprocal of divisor with the dividend. For example, 22/7 ÷ 11/7 = 22/7 x 7/11 = 2/1 = 2.
• To add or subtract the fractions with same denominator, add or subtract numerators and keep the denominator. For example 4/9 + 1/9 = 5/9 and 4/9 - 1/9 = 3/9.

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% = 20/100 = 0.20.
• 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: 1/2 = 5/10 = 0.50 = 50%, 1/5 = 2/10 = 0.20 = 20%
  1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%
• 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)/(original amount) x 100%. 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 20/80 x 100% = 1/4 x 100% = 25%.
• 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%, multiply it by 1 + k%, and to decrease a number by k%, multiply 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 result of increasing another number by k%, to find the original number divide i t by 1 + k%, and if a number is the result of decreasing another number by k%, to find the original number, divide it 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.) = Current Salary / (1 + percent increase) = 18,000/(1+20%) = 18,000/1.20 = 15,000.

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/(b+c) 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 x 100 = x multiply 5 ⇒ x = (3 x 100) / 5 ⇒ x = 60.

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.

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