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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GCSE Foundation and Intermediate Level

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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...

2

"

x 12 = 12

22 - 12 = 10

1

2

1 x 2 = 22

4

5

"

4 x 15 = 60

70 - 60 = 10

4

5

4 x 5 = 70

88

89

"

88 x 99 = 8712

8722 - 8712 = 10

98

99

98 x 89 = 8722

From these results I can conclude that every 2 x 2 rectangle has a difference of 10. I am now going to try rectangles of size 2 x m (where m is the other side).

2 x 3:

3

"

33 - 11 = 20

1

3

4

4

"

84 - 64 = 20

4

6

7

9

"

53 - 133 = 20

7

9

This shows that all 2 x 3 rectangles have a difference of 20

2 x 4:

4

"

44 - 14 = 30

1

4

5

8

"

20 - 90 = 30

5

8

21

24

"

744 - 714 = 30

31

34

This shows that all 2 x 4 rectangles have a difference of 30

2 x 5:

"

55 - 15 = 40

1

5

6

0

"

60 - 120 = 40
Join now!


6

20

21

25

"

775 - 735 = 40

31

35

This shows that all 2 x 5 rectangles have a difference of 40

After carefully reviewing my above results I predict that for any 2 x m rectangle the formula for finding the difference is:

0 ( m - 1 )

I will now test this prediction by finding the difference of rectangles 2 x 6 and 2 x 7.

2 x 6: This should be 10(6 - 1) which equals 50

6
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