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Number grids

Extracts from this document...

Introduction

Saad Zahid

Maths coursework

(Number grids)

This is a number grid which we have been asked to, draw a box round four number, find the product of the top left number and the bottom right number in this box, then again do the same with the top right and bottom left numbers and then calculate the difference between the products.

I will also be adding some variables to this number grid and them solving them with formulas.  The variables are going to be:

  1. If I don’t use a square instead we use a rectangle (M x N)
  2. If I change the length of the grid instead of it being 10 we could change it to another number.
  3. If I change the step length, at the moment it is 1 I could change it to 2.

For all these variables I will be working out formulas and at the end of it I will come up with one general formula.

image00.png

This is the number square we are going to work on at the moment.

12x23=276        

13x22=286

286-276=10

The answer is 10

Now let’s take another four;image01.png

81x92=7452

82x91=7462

7462-7452=10

The answer is 10

There looks as though there is a pattern going on here the answer will always be 10.

Now I am going to choose a general position of my square, the top right will be “M”.

Mx(M+11)=M2+11M

(M+1)x(M+10)= M2+11M+10

M2+11M+10 - M2+11M=10

...read more.

Middle

40

90

I have decided to work out a formula for this:

In this formula “N” will be the box size and “M” is the number of a square in the top right hand corner.

M                M+N-1

M+10N-10                M+10(N-1)+N-1

=M+10(N-1)        =M+11(N-1)

M x M+11(N-1)= M2+11(N-1)

M+10(N-1) x M+N-1 = M2+11(N-1)+10(N-1) 2

M2+11(N-1)+10(N-1) 2 - M2+11(N-1) = 10(N-1) 2

This proves that the following formula is correct and working for squared numbers.  10(N-1) 2

Now I have done this I am going to work out the formula when you use a number grid which is M x N.

This formula is simple to work out as 10(N-1) 2 is for squared numbers you can multiply this out so its 10((N-1) x (N-1)).

In this “N” is the grid size but we can change this so it is length and width instead of always being the same number.

So the formula is 10((L-1)(W-1))

To prove this…….

M                M+L-1

M+10W-10                M+10(W-1)+L-1

=M+10(W-1)        

M x M+10(W-1)+(L-1)= M2+10M(W-1)+M(L-1)

M+10(W-1) x M+L-1 = M2+10M(W-1)+M(L-1)+10((L-1)(W-1))

M2+10M(W-1)+M(L-1)+10((L-1)(W-1)) - M2+10M(W-1)+M(L-1)= 10((L-1)(W-1))

So this has been proven.

Now I have decided to change the size of the rows in the grid.

...read more.

Conclusion

        I now want to find out a general formula which would let me work out the difference.  The formula will work out the difference even if the all the following variables are changed:

  • Step size
  • Changing the length of the grid.
  • Any size of rectangle (M x N)

This formula will be my general formula for working out all the variables with one formula.

As before “M” will be the number in the top right.

“L” is the length

“W” is the width

“T” will be for the step size

“S” will be for the size of one column in the grid.

M                M+(TL-T)

M+S(TW-T)                M+S(TW-T)+(TL-T)

M x M+S(TW-T)+(TL-T) = M2+SM(TW-T)+M(TL-T)

M+(TL-T) x M+S(TW-T) = M2+SM(TW-T)+M(TL-T)+S((TW-T)(TL-T))

M2+SM(TW-T)+M(TL-T)+S(TW-T)(TL-T) - M2+SM(TW-T)+M(TL-T) = S((TW-T)(TL-T))

This is the general rule for working out the differences when finding the product of the top right and bottom left, and take the product of the top left and bottom right from it. The formula is for all the variables which I have been working out through out this piece of work.

S((TW-T)(TL-T))

“M” will be the number in the top right.

“L” this is the length of the box

“W” this is the width of the box

“T” will be for the step size

“S” will be for the size of one column in the grid.

By SAAD ZAHID

...read more.

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