GMAT math thinking skills 10

 GMAT tests your logical skills as well as your knowledge of math concepts.  To score high, you need to remember various formulas, theorems. Also you need to master critical problem-solving skills.

Today I am going to  take you through one problem -solving skill –

Counting skills

Take this problem .

This sum requires higher order thinking

There are two ways to solve this question

 

Method 1:Using pattern recognition

 Lets say,  you didn’t know the concept of counting and permutation and combination

First focus on the 8x8 grid...

Take the smallest unit of chessboard. It’s a 2x2 grid

How many rectangles can you count?

All the squares are rectangles too.

First count squares  there are 5

How many rectangles can you count? There are 4.

In total there are 9 . 

If you observe........ 9  = 1+8

these numbers follow the cube series

 

Now take a  3x3 grid

How many rectangles can you count?

First count squares  there are 14

How many rectangles can you count? There are 22

In total there are 36 . 

If you observe ..........36  = 1+8+27

these numbers follow the cube series

 

Always remember 

the number of rectangles in a grid follow the cubic series 1³ 2³ …3³…

So in a chess board.. the number of rectangles is the  sum of the cubes from 1³ to 8³

 Hence 1³ +2³ + 3³+4³ +5³ +6³ +7³ +8³ =1296

  

 

 

Method 2:Using principles of counting

The chess board has 8 rows and 8 columns. a 8x8 grid.

All squares are rectangles. So you need to count the squares also.

 

Okay.. First look at the gird..

Can you observe the number of horizontal lines? There are 9 lines

Similarly

Can you observe the number of vertical lines? There are 9 lines

 

If you observe… to draw a rectangle. You need to select two horizontal lines and two vertical lines.

The point of intersection of these lines form a rectangle.

So how do we choose two lines out of 9. 

Use combination.

To select 2 horizontal lines = ⁹C₂

To select 2 vertical lines = ⁹C₂

Now as per the rules of counting…you need to multiply

⁹C₂ x ⁹C₂ = ((9x8)/ (1x2) ) x ((9x8)/ (1x2) ) = 1296

There are 1296 rectangles in a chessboard.

 

To know more about Math problem solving skills.. Feel free to contact me

 My contact link is here:

LinkedIn profile : https://www.linkedin.com/in/georgeanand/

Facebook learning group: https://www.facebook.com/groups/semanticsGMAT/

Lets try another sum

Solution is here

http://letsgmat.blogspot.com/2020/11/gmat-math-thinking-skills-7.html

 

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

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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Science of high performance in the GMAT -1 : Is GMAT Official guide sufficient?

Official guide- published by the GMAC has a list of 800+ questions (11th edition). There are around 230 problem solving (math) questions.. However I feel that this list represents the easier problems in the GMAT.

To break into the 720+ it is imperative that you have to solve tougher higher difficulty problems. (This is true for the verbal section also)

What is a higher order problem?
A higher order problem is problem which has
• A situation which can otherwise be solved by identifying the concept/formula and applying the same directly
• A complexity present in the problem which acts as a stumbling block, thereby preventing you from getting an answer directly.

When approaching a higher problem first

1. Identify the concept involved
2. Identify the complexity in the problem
3. Remove the complexity and if possible arrive at a result
4. Modify the complexity in such a manner the complexity gets integrated into the problem and this results in  a   newer problem
5. Solve the new problem by directly applying the concept.

Let me highlight a higher order problem. Watch how I analyzed the problem and how I solved.

A car moving at 45 kmph and is chasing a two wheeler that is moving at 30 kmph. The distance between the car and the two wheeler at 10:00 am is 48 kms. The car stops at 11:30 am for 15 mins to fill fuel and moves at 45 kmph. When will the car meet the two-wheeler?

1.12.42 pm      2.1:42 pm           3.1:57 pm          4. 1:47 pm  5. 1:30pm

Try this problem independently first, then read further.
Let me take you through the problem in the science of thinking* approach toward higher order problem solving.

The concept – Time, speed and distance, Relative velocity involving two bodies moving towards each other.
The complexity – The stoppage time of the car. At 11:30 am the car stops for 15 mins.

Eliminate the complexity first
If the car didn’t stop at 11:30 then the time taken by both the bodies to meet is determined using the relationship
Time taken to meet = Initial distance between the bodies/ relative velocity
= 48/(45 -30) { Relative velocity when two bodies move in the same direction = difference of their speeds, hence 45-30 =15)
= 48/15= 3.2hrs

Modification of the complexity
As you would have observed if the complexity is eliminated the problem can be solved directly. As per the problem. The car travels till 11:30 and then stops for 15mins. So you might calculate the distance travelled by each body from 10:00 till 11:30 and then calculate the distance travelled by the two wheeler for that extra 15mins and then proceed. This complicates the problem.
Instead you can restructure the problem in such a way that the complexity gets integrated into the problem and doesn’t get noticed.

Here you can shift the 15min time interval from 11:30 to 10:00 such that the car starts only at 10:15 instead of 10:00. Hence the initial distance increases from 48 to 48 + (distance travelled by two wheeler for 15mins) = 48 + 7.5 =55.5kms

Hence now there is no stoppage time at 11:30.
Time taken to meet = Initial distance between the bodies/ relative velocity
= 55.5/(45 -30) = 55.5/15= 3.7hrs = 3hrs 42minutes
Meeting time =10:15 + 3 : 42 = 13: 57
You would get higher order problems only if the adaptive algorithm decides that you deserve questions of this difficulty.
So for those of you who aim to crack the 720+ barrier. Practice on higher order problems.


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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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5 crucial points to keep in mind while attempting a Data sufficiency problem

of the equation don’t match. Hence x only has to be 0.

4. Do not make any assumptions of the figure drawn. If a four sided figure is drawn with straight line, do not assume it’s a square or if a point is marked in the middle of a circular region, don’t assume it’s the centre of the circle.

5. Although Data sufficiency tests your decision making skills (choosing which statement is sufficient) it is advisable to spend some time arriving at an answer and checking whether the answer derived is always true or always false.



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5 crucial points to keep in mind while attempting a Data sufficiency problem

of the equation don’t match. Hence x only has to be 0.

4. Do not make any assumptions of the figure drawn. If a four sided figure is drawn with straight line, do not assume it’s a square or if a point is marked in the middle of a circular region, don’t assume it’s the centre of the circle.

5. Although Data sufficiency tests your decision making skills (choosing which statement is sufficient) it is advisable to spend some time arriving at an answer and checking whether the answer derived is always true or always false.



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5 crucial points to keep in mind while attempting a Data sufficiency problem

of the equation don’t match. Hence x only has to be 0.

4. Do not make any assumptions of the figure drawn. If a four sided figure is drawn with straight line, do not assume it’s a square or if a point is marked in the middle of a circular region, don’t assume it’s the centre of the circle.

5. Although Data sufficiency tests your decision making skills (choosing which statement is sufficient) it is advisable to spend some time arriving at an answer and checking whether the answer derived is always true or always false.



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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) ]