# To investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a one hundred square

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Introduction

GCSE Foundation and Intermediate Level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 O P P O S I T E 50 51 C O R N E R S 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Simon Langley 9RL I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a one hundred square. For example: 54 55 56 54 and 66 64 and 56 64 65 66 54 x 66 = 3564 " 3584 - 3564 = 20 64 x 56 = 3584 The difference of this rectangle is 20. Firstly I am going to look at rectangles of size 2 x 2 as these are the smallest possible rectangles with four corners... 1 2 " 1 x 12 = 12 22 - 12 = 10 11 12 11 x 2 = 22 4 5 " 4 x 15 = 60 70 - 60 = 10 14 15 14 x 5 = 70 ...read more.

Middle

is: 20 ( m - 1 ) I will now test this formula in the same way as I tested the previous formula. 3 x 6: This should be 20(6 - 1) which equals 100 1 6 " 126 - 26 = 100 21 26 15 20 " 700 - 600 = 100 35 40 33 38 " 2014 - 1914 = 100 53 58 3 x 9: This should be 20(9 - 1) which equals 160 1 9 " 189 - 29 = 160 21 29 12 20 " 640 - 480 = 160 32 40 51 59 " 4189 - 4029 = 160 71 79 This, again, shows that my prediction was correct for a rectangle of size 3 x m. As I now have two formulas that work, for 2 x m and 3 x m, I can predict that the formula for a rectangle of size 4 x m will be: 30 ( m - 1 ) I will now test this formula in the same way as the previous two formulas. 4 x 2: This should be 30(2 - 1) which equals 30 1 2 " 62 - 32 = 30 31 32 5 6 " 210 - 180 = 30 35 36 47 48 " 3696 - 3666 = 30 77 78 4 x 7: This should be 30(7 - 1) ...read more.

Conclusion

Using this I can find a new formula for the difference between the corners... (a+x)(a+y)-(a)(a+x+y) > a�+ax+ay+xy - a�+ay+ax > a�+ax+ay+xy - a�+ay+ax * x y As I have shown above that x and y can be substituted, x = b - a and y = c - a, I can rewrite this formula as: ( b - a ) ( c - a ) Although I have proven this formula to be correct I will try some random rectangles in the 100 square to check my above working... Rectangle Difference using first method (b-a) (c-a) Difference using formula Check 34 35 44 45 10 (35-34)(44-34) 10 52 57 72 77 100 (57-52)(72-52) 100 01 04 81 84 240 (4-1)(81-1) 240 03 10 93 100 630 (10-3)(93-3) 630 21 22 71 72 50 (22-21)(71-21) 50 01 10 91 100 810 (10-1)(91-1) 810 This confirms that my working was correct which means that I have proved my formula to be correct for any rectangle within a 100 square. Conclusion From carefully studying patterns of different rectangles and from then using algebra I can say that the formula for finding the difference of the products of the opposite corners of any rectangle within a one hundred square is: ( b - a ) ( c - a ) Marks given: 6-6-6 GCSE grade: B ...read more.

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