GMAT data sufficiency strategy - assume,check and adjust

Stumped with data sufficiency...

Data sufficiency is a test of mathematical reasoning.  It tests your ability to evaluate the adequacy of given data in answering a question in the mathematical setting. This involves verifying the sufficiency of data to solve a problem, distinguishing between relevant and irrelevant data, and establishing relationship between variables.

Here’s how the directions for data sufficiency problems appear in the exam

A given question is followed by two statements. You are required to determine whether the statements can be used to answer the question.

Mark (A) if statement I alone is sufficient but statement II alone is not sufficient to answer the question

Mark (B) if statement II alone is sufficient but statement I alone is not sufficient to answer the question

Mark (C) if both statements I and II together are sufficient to answer the question

Mark (D) if each statement alone is sufficient to answer the question

Mark (E) if statement I and II together are not sufficient to answer the question

Let us understand one approach to solve DS questions with lots of variables

Let’s take a question

Is the product abcd = 1?        

Statement 1: ab/cd=1

Statement 2: a,b,c,d are integers

Strategy:

Take a statement. Substitute different sets of numbers and check for consistency. If the results are inconsistent, when different sets of number are substituted, the given statement is insufficient. 

Lets solve this question with this approach

Is the product abcd = 1?        

Statement 1: ab/cd=1

Statement 2: a,b,c,d are integers

Consider statement 1

Substitute numbers which satisfy statement 1

a=2,b=3,c=6 and d=1 satisfy statement 1. But is the product abcd= 1. The answer is “no”

Plug in a different set of numbers to check consistency

a=4,b=3,c=6 and d=2 satisfy statement 1. Is the product abcd= 1. The answer is “no”

But

a=2,b=1/2,c=3 and d=1/3 satisfy statement 1. Is the product abcd= 1. The answer is “Yes”.

As the result is inconsistent, sometimes the answer is “yes” other times it is “no”. The given statement is insufficient.

Similarly analyse statement 2

Plug in numbers which satisfy statement2

a=2,b=3,c=6 and d=1 satisfy statement 1. But is the product abcd= 1. The answer is “no”

a=1,b=1,c=1 and d=1 satisfy statement 1. But is the product abcd= 1. The answer is “yes”

Thus statement 2 is insufficient, since, for certain numbers “yes” is arrived and for others “no” is arrived

Even when both statements are combined, for certain numbers “The product abcd is equal to 1” is arrived and for others “The product abcd is not equal to 1” is arrived

It is necessary to arrive at consistent result before marking an answer.

The answer is E.

Try another question with the same approach

Is (a/b)>(c/d)?

1.a>c

2.b>d

Answer is E.

Understanding GMAT data sufficiency question

The next set of blogs will be focusing on Data sufficiency(DS). Many students feel that they make more errors in DS than problem solving. Let us understand this question type in depth.

Sample question:

What is the value of a?

(1)      3a + 2b = 15

(2)      b = (-3/2) (a – 5)

The Directions:

Each data sufficiency problem consists of a question and two statements, labeled (1) and (2), that give data. You have to decide whether the data given in the statements are sufficient for answering the question.

Use the data given in the questions plus your knowledge of mathematics and everyday fact, you must indicate whether the data given in the statements are sufficient for answering the questions and then indicate one of the following answer choices:

(A) Statement (i) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;

 (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;

(D) EACH statement ALONE is sufficient to answer the question asked;

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

NOTE: In data sufficiency problems that ask for the value of a quantity, the data given in the statements are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Numbers: All numbers used are real numbers.

Figures: A figure accompanying a data sufficiency problem will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight. You may assume that the positions of points, angles, regions, and so forth exist in the order shown and that angle measures are greater than zero degrees. All figures lie in a plane unless otherwise indicated.

These questions require knowledge of the following topics: • Arithmetic • Elementary algebra • Commonly known concepts of geometry

DS problem – Approach flow chart

Next post: Strategies to tackle data sufficiency questions

GMAT math - digits in decimal places

Digits in decimal places:
Note: 1234.567

1 - THOUSANDS
2 - HUNDREDS
3 - TENS
4 - UNITS
. - decimal point
5 - TENTHS
6 - HUNDREDTHS
7 - THOUSANDTHS

10 minute speed test math - 28.10.2015

Crack this challenging math speed test in less than 10 minutes...

"To acquire knowledge, one must study; but to acquire wisdom, one must observe."- Marilyn vos Savant

Answers will be posted in 2 days..

10 min test with G 15.9.2015 with solution

Try these questions in 10 minutes. Email us your answers. Please check answers here
Have fun..

Study these math problems and observe their solutions.

We have presented three math problems here..
These problems can be solved in more than one way.

Study these problems, observe their solutions.

We are sure you can replicate the solution process to other problems.
Version 1: questions 1 to 3



Visit  semanticslearning.com , click on GMAT tips for solutions

How to prepare for GMAT math

Mostly everyone while preparing for the GMAT math, studies official guide and few other books like Manhattan, Kaplan...

Although for some, studying these few GMAT books is enough to crack 99 percentile in the math, others are not able to crack the 600 barrier itself. What might be the reason?

Well, I feel that students who excel in math since school stand a better chance of cracking GMAT math with a minimal preparation.

For the others, who are just starting their math journey, not only they have to read the GMAT OG,they have to read basic math concept books, problem solving books, etc...GMAT OG and other GMAT books, just scratch the surface while preparing for math. First it is required to tune your brain to work with numbers and then numbers and words. The GMAT math books does not cover the entire skills which is required to reach 51 raw score in the GMAT math..

Watch this video. Some of the books mentioned will cover the additional sources which you need to refer to develop
1. math concepts
2. math reasoning
3 GMAT math.

Are you a quant person?

Are you a quant person?
Quantitative thinking ( thinking with numbers) is integral to corporate business careers. Hence MBA entrance tests contain a generous dose of quantitative problems. One’s performance in such problem solving is a manifestation of his overall problem solving ability.
Business Schools perceive quantitative scores as indicative of higher order thinking and decision making skills. They believe that quant thinkers can handle diverse business challenges. They can analyse, diagram, hypothesise, set goals, try permutations and combinations, perceive probabilistic outcomes and synthesis a possible outcome.

Quantitative personality is not necessarily a hardcore math person
For a quantitative thinker, math knowledge is one of the many tools in his quest for excellence in problem solving. It is also possible that one is a good quantitative person but not a math person.
By and large, a quant person is someone who can look at independent ideas and facts, look at a situation and be able to come up with a response irrespective the accuracy of the approach and thereby the solution.  It also means looking at a situation and draw up on one’s own repertoire of tactics for a possible way forward…. a possible answer... In short, a quant person  might have a great memory but is rather someone who reasons very well.

A quant person uses thinking skills approach to problems
So when a quant person looks at a math problem with varied factors, and probably requiring more than one mathematical concept, he  doesn’t get confused; he will pull the question apart and can see where one step leads into the other and can merge and manipulate the combinations to get the final answer. He goes beyond the given data, creates a problem field, assumes himself to be part of the problem, takes various experiences and knowledge points to extrapolate a position and direction. In other words, a quant person is empowered to handle problem situations well; one who says no ‘can’t’, until he has exhausted all possible knowledge, theories, and experiences before asking for help.

A quant person ‘transfers learning’
For a quant person, the idea of doing a lot of problems stems from the need to see the various possibilities of solving problems rather than an expectation of chancing upon an exam like problem. For effective ‘transfer of learning’ making observations while attempting a problem is the key.

The quant person in a nut shell should be inquisitive, innovative, fearless, flexible and an inherent risk taker. “the Science of Thinking” methodology attempts to inculcate quantitative reasoning in addition to quantitative aptitude in test aspirants. Visit www.semanticslearning.com for more details.
Read http://www.semanticslearning.com/beta/gmat-science-of-thinking.asp of thinking for more details

How to study for the GMAT math?

Video covers 1.How do I start? How much time? Should I self study? 2.Sources required to build concepts & higher order reasoning 3.How to build mathematical reasoning 4.How many tests... Visit www.gmatsuperia for more info Email urmentor@semanticslearning.com for responses & queries

GMAT permutation combination learning video

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5 most crucial points while solving. Permutation combinations

ARRANGMENT
N terms can be arranged in N! factorial ways, if each position can be occupied by one term.
N terms can be arranged in N^M where. Each position can be occupied by 1 term or 2 terms or …… N terms. M stands for the number of positions to be filled.

COMBINATION

M terms can be selected from P terms in PCm ways.
In certain situations it is required to first choose the terms and then arrange the terms. i.e. PERMUTATION.

Permutation = combination x arrangement.

4. When N objects are distributed among P positions such that each position can get any number of objects (zero, one, two ……N) then the number of ways of arranging the items is N+P-1Cp-1

5. When N objects are distributed among P positions such that each position can get atleast one objet (one, two ……N) then the number of ways of arranging the items is N-1Cp+1

5 crucial points while solving a probability based problem.
1. Calculate the numerator {Nos. of foverable terms} and the denominator {Total number of terms} separately using the concepts of arrangement, permutation and combination.
2. TAKE IT PERSONAL : Always imagine you are arranging / selecting the items. The action of taking the object and placing it in the relevant position is the key.
3. When two or more items are picked it is easier to compute the probability of picking one element at a time than computing the probability of picking many items at a time.
4. When A and B are selected relate the respective probabilities with multiplicataion. When either A or B is selected relate the respective probabilities with addition.
5. When the multiple outcomes are possible the probability of atleast one of them happening is computed by calculating the reverse probability = 1 – probability of event not happening.




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How to compute squares faster. Speed math for GMAT

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How do we conquer the fear of GMAT math?

Let start with a show of hands

How many of us go blank when we see a math sum?
I can see that most of you are raising your hands, the others must be one of lucky 2%.

Now the big question.

Why some of us go blank and others seem to have the knack of solving math sums?

Well you can blame it on your mathematics teacher @ school or on your genes. But nevertheless while preparing for GMAT or in fact while doing an MBA, you will encounter lot of math.
I can hear lots of groans. :-)

Few years back I read this book ‘How to solve it’ by George Polya. I modified my teaching style from just teaching question answers, question answers, question answers, question answers……to question logic answers, question logic answers, question logic answers…..

I found that I could tutor a person to achieve 45+(raw score in GMAT) within few weeks as instead of few months. Wow!! The best part of it I could see that students are able to solve math problems independently without me intervening.

In the book, Polya gives a detailed step by step process on how to approach math problems in general

I will modify the process and present it to you in context with GMAT math
Keep these steps in mind when you approach a math problem in the future.

Step 1: Understanding the problem
Answer the following questions first

• Do you understand all the words used in stating the problem?
• What are you asked to find or show?
• Can you think of a picture or diagram that might help you understand the problem?
• Is there enough information to find the solution?
• What information, if any, is missing?

The answer to these questions will channelize your thinking towards the answer.

Step 2: Devise a plan
Answer these questions now

• What will be the best approach to address the problem?

Approaches can only be devised. If a tutor explains a sum to you, then you will be able to understand only that problem. But when you encounter a new problem, you will go blank again.
Ideally when you encounter a new problem, you will have to use the existing ideas plus any new ideas you can conjure up. These process are mostly done mentally and involve little computation/calculation.

To get an idea, do any/all of the following.

• Make a systematic list/table
• Write an equation
• Consider special cases
• Use direct reasoning- for example If A>B and B>C then A>C.
• Use indirect reasoning.-Think of an earlier sum where you encountered a similar problem
• Look for a pattern
• Draw a picture
• Solve a simpler problem- break the problem into small parts and solve each part.
• Use a model- Make a general assumption and solve by guessing.
• Work backwards. –work with answer options

Now that you have got an idea. Put pen on paper and solve to get an answer
Stage 3: Carry out the plan

• Solve the problem with great care and patience
• Discard the plan if it does not work and devise a new plan
• Record what you have done to avoid repetitive work – For future use.

While attempting Data sufficiency questions, it is imperative you check your results. So
Stage 4: Looking back or checking

• Have you addressed the problem?
• Is your answer reasonable?
• Can the method applied to other similar problems?
• Is It consistent.

Now go ahead and repeat this thought process on different math problem and the next time when you see a math problem you will not go blank.



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Hypothesis testins as a tool for GMAT math problem solving

Certain problems require you to formulate a hypothesis and verify. The relationships between relevant variables which are yet unknown but promise to offer solution in full or in part forms the basis of this method. Such formulations are tested for validity and accepted or rejected. More than one hypothesis can be formulated in a problem context. These hypothesis have to be examined and reformulated.

Errors which occur during hypothesis testing
(1) Overlooking certain data
(2) Overemphasizing data which give positive conclusions while failing to give sufficient importance on data which falsifies information.

The following sum is a tough mathematical problem where skill of hypothesizing information and testing it comes forth.

Fifteen years back Mrs. John had only three daughters Rachael, Annie, Mary and their combined age was half of hers. During the next 5 years, Thomas was born. At that time Mrs. John’s age equaled the total of all her children’s ages. After some years George was born and then Rachael was as old as Mary and Thomas together. And now, the combined age of all the children is double Mrs. John’s age, which is only equal to that of Rachael and Annie together. Rachael’s age is also equal to the combined age of the two sons’.

What is Mrs. John’s age?

1. 39        2. 34          3. 29         4.24

This sum takes a long time if you solve by writing equations.

It can be solved faster by hypothesizing a data and testing the hypothesis wrt to other conditions.

First let us interpret the problem carefully and diagram it.

PROBLEM ANALYSIS

Information which is direct
• 15 years back Mrs. John had only three children Rachael, Mary and Annie. Mrs. John’s age was double the sum of the ages of her children.
• Sometime between 15 and ten years back, Thomas was born. At that time Mrs. John’s age was equal to the sum of the 3 children
• Between 10 years back and present time, George was born. At that time Rachael was as old as Mary and Thomas together.
• At present the combined age of all the children is double Mrs. John’s age. Mrs. John’s age is equal to the sum of Rachael and Annie. Rachael’s age is equal to the sum of George and Thomas

Implicit information
• All the ages are whole positive number, there are no fractions.
• Thomas’s age must be less than 15 and near 15. As 15 years back Mrs. John had only 3 children
• The last statement states that Rachael’s age + Annie’s age = Mary’s age + Thomas’s age + George’s age.
Rachael’s age = George’s age + Thomas’s age and Mrs. John = Rachael’s age + Annie’s age

PROBLEM CONVERSION

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-      George

T+G+A    -   Present  - T+G    -A      -A      -T         -G

Where T is Thomas age, A is Annies age and G is Georges age
We can conclude that Annie and Mary were twins


We will assume data from the questions
Let us hypothesize that Thomas’s age is 12. (9 is far away from 15).
{If we don’t get the answer using T =12 we can conclude that T = 9. Other options are wrong.}

Lets verify our hypothesis.

Rachael must be the eldest daughter . Let us assume that Rachael age must be 21 other options are close to 15 and as she is the eldest we will assume the biggest number
So George’s age must be 9 ( question 4 seems to be satisfied)

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

                    15 yrs back-6     -

                     12 yrs back-9    -                         -0(Thomas born)

                      10 yrs back-11 -                        - 2

                      9 yrs back - 12 -                        -3     -0(Georges born)

                      5 yrs back  - 16-                        7       - 4

T+G+A              Present  -21 -     A-   A-       12      -   9

Using the info: Between 10 years back and present time, George was born. At that time Rachael was as old as Mary and Thomas together.
It can be concluded that Mary was 9 as 12 = Mary’s age + 3
Hence Annie’s age was also 39

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

                    9 yrs back- 12     -  9      -9     -  3        - 0(Georges born)

Filling our table
All the children’s age 9 yrs back can be calculated.
Their present ages can also be calculated including Mrs. John age

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

39            -  Present   - 21        -18    - 18   -   12      -  9

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

24          -  15 yrs back -  6     -      3  -   3

               -12 yrs back  -  9      -    6   -   6 -  0(Thomas born)

               - 10 yrs back - 11                        -  2

                - 9 yrs back  - 12     -   9    -   9  -  3       -0(Georges born)

               -  5 yrs back   -16                         - 7        -  4

      39       -Present     - 21       -  18    -  18  -12     -   9

Now the table can be completed and the all the answers can be calculated

What is Mrs. John’s age? - 39
What is the age of the eldest daughter? - 21
What is the age of the eldest son, Thomas ? -12
What is the age of the youngest child? - 9



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Pattern recognition as a skill to solve GMAT math problems

Many math problems are based on patterns. These problems may involve a set of numbers or a set of alphabets or maybe even a set of figures.

The skill is in

• Identifying patterns
• Taking a sample and deriving meaning full relationships between the various elements in the pattern
• Expostulating the pattern to encompass the entire series
• Use this new knowledge to arrive at an answer

Lets take a problem

The sum of the even numbers between 1 and n is 79*80, where n is an odd number, then what is the value of n?

This sum involves a set of even numbers from 1 to n.(n is an odd number)
Lets derive the pattern
First let n =5
Then the even numbers involved are 2,4
Hence, Sum = 2+4 = 6
6 can be written as 2 *3( Same pattern as 79*80)

Now let n =7
The even numbers are 2,4,6
Sum = 2+4+6 = 12 i.e 3*4

So you get a pattern 2*3, 3*4…………………….79*80, when n = 5,7……n
Do you observe that 2+3 =5 and 4+3 =7, 4+5 =9

This leads to the answer.

In a nutshell: when you encounter problems which ask you to compute the value for n terms
Take a small sample and analyze.(Relate the analysis to the answer)
Take another sample and analyze
Write the result together and derive a relationship among the numbers
This leads to the answer.




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Math Problem solved using ScoT

I got a query from a in this problem. I solved using Science of Thinking(ScoT) approach.Observe the problem solving process.

The sum of the even numbers between 1 and n is 79*80, where n is an odd number, then n ?

These type of sums can be solved using my thinking skills – “pattern recognition” and “hypothesis testing”


Take sum of even numbers when n =5( N has to be an odd number)
Sum = 2+4 = 6 i.e 2 *3( Same pattern as 79*80 i.e n*(n-1))
Now take sum of even numbers when n = 7
Sum = 2+4+6 = 12 i.e 3*4

So you get a pattern 2*3, 3*4…………………….79*80
When n = 5,7……n
Do you observe that 2+3 =5 and 4+3 =7.

So our hypothesis is that n should be sum of the product of the numbers(in the form n*(n-1) which yields the sum of the even numbers.
Now lets check our hypothesis
When n =9
Sum = 2+4+6+8 = 20 = 4*5
4+5 is equal to n
Hence n can be concluded as 79+80=159

For more details visit http://www.semanticslearning.com/gmat-l3-method.asp
My math Ebook has all the thinking skills tested in GMAT.You can access the demo at
http://www.semanticslearning.com/gmat-home.asp Title GMAT higher order problem solving.

Cheers





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Analyzing GMAT math problems using the Science of Thinking(ScoT) approach

The first step in the problem solving process is problem analysis. Problem analysis comprises

• Problem Definition
• Solution length
• Problem length
• Constraints and conditions

Let me explain the process of classification of problem based on their definition now.

Problems can be classified as a poorly defined or a well defined problem.

A well defined problem everything relevant and required is clearly specified, without any ambiguity or uncertainty, such that a solution, even if it involves complex calculations can be arrived at with accuracy. You can predict the path to take or steps required to solve the problem.

A poorly defined problem much of the data & relationships are hidden or not clear.

Lets take a poorly defined problem.

A says to B: I will be three times as old as you were when I was five years older than you are. I am 5/4th as old as you will be and then you will realize that you will be double the age you were. If the sum of the future ages of A and B is 50, what are their present ages?

The data present in the above question is cryptic. The interpretation of this problem lies in your ability to attach meaning to the verb tense.

To analyze the above problem you have to represent the problem diagrammatically to understand the relationship between the variables.

Try creating a table with the past ages, present ages and future ages as the columns. Given below is a simplified version of the table.

More ScoT approaches follow this link..
http://www.semanticslearning.com/gmat-l3-method.asp



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5 crucial points to be kept in mind while solving a probability based problem in GMAT

1.    Calculate the numerator {Number of favourable terms} and the denominator {Total number of terms}     separately using the concepts of arrangement, permutation and combination.

2.    Be part of the problem : Imagine you are arranging / selecting the items. The action of taking     the object and placing it in the relevant position is the key.

If you have to arrange 10 rings in 4 fingers, you have to imagine yourself picking a ring and placing it on a finger instead of computing the number of rings each finger has.

3.    When two or more items are picked it is easier to compute the probability of picking one     element at a time than computing the probability of picking many items at a time.

4.    When A and B are selected relate the respective probabilities with multiplication. When either     A or B is selected relate the respective probabilities with addition.

5.    When there are multiple outcomes possible the probability of at-least one of them happening is     computed by calculating the reverse probability
 = 1 – probability of event not happening.


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5 crucial points to keep in mind while solving a permutation combination problem

1.ARRANGMENT N terms can be arranged in N! ways, if each position can be occupied by one term. N terms can be arranged in NM ways if each position can be occupied by 1 term or 2 terms or …… N terms.  M stands for the number of positions to be filled.

2.COMBINATION M terms can be selected from P terms in [ (P)combination(m) ] ways.

3.In certain situations it is required to first choose the terms and then arrange the terms. i.e.     PERMUTATION.     Permutation = combination x arrangement.

4.When N objects are distributed among P positions such that each position can get any number     of objects (zero, one, two ……N) then the number of ways of arranging the items is [ (N+P-1) combination (P-1) ]

5.When N objects are distributed among P positions such that each position can get atleast one     objet (one, two ……N) then the number of ways of arranging the items is [ (N-1) combination (P+1) ]

5 most overlooked points while solving GMAT triangle based problems (under Geometry)

1. Similar triangles are triangles whose sides are proportional. If ABC and PQR are two similar triangles then AB/PQ = BC/QR =AC/PR. The angles opposite to the sides(which are in a ratio) are also proportional  
   
2.   Area of a triangle(sides are a,b,c) is based on the hero’s formula 

           

          s= (a+b+c)/2 , R = circum radius and r= inradius

  3.   The sum of two sides of a triangle is greater than or equal to the third side and the difference of any    

      two sides is lesser than or equal to the third side. This property is used extensively in GMAT higher

           difficult problems

    4.    For the equilateral and the isosceles triangle the altitude bisects the base and hence the triangle into
           two equal parts

    5.    The largest triangle (with the maximum area) which can be inscribed in a circle is the equilateral
           triangle.