Saturday, July 10, 2021

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.

 

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