Maxi Product

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GCSE Maths Coursework - Maxi Product

Introduction

In this investigation, I am going investigate the Maxi Product of numbers. I am going to find the Maxi Product for selected numbers and then work out a general rule after individual rules are worked out for each step. I am going to find the Maxi Product for double numbers, I will find two numbers which added together equal the number selected and when multiplied will equal the highest number possible that can be retrieved from two numbers multiplied together. I am also going to find the Maxi Product for triple numbers, I will find three numbers which added together equal the number selected and when multiplied will equal the highest number possible that can be retrieved from three number multiplied together. And finally, I am going to find the Maxi Product for quadruplet numbers, I will find four numbers which added together equal the number selected and when multiplied will equal the highest number possible that can be retrieved from four numbers multiplied together. After working out the individual rules for these three sectors of numbers, I will then work out the general rule for any amount of numbers it can be split into. For example, it can be split up into five numbers and I will be able to find the Maxi Product of any number given by splitting it up into five numbers. I will be using whole numbers, decimal numbers and fractional numbers.

Double Numbers

Examples: 12

(5,7)= 12 --> 5+7 --> 5x7=35

(6,6)= 12 --> 6+6 --> 6x6=36

(4,8)= 12 --> 4+8 --> 4x8=32

I have found that 36 is the highest number so far that can be retrieved from 6 and 6 when the number is 12, in whole numbers. I will now try in decimal numbers if I can get a number higher than 36.

(6.5,5.5)=12 --> 6.5+5.5 --> 6.5x5.5=35.75

(6.7,5.3)=12 --> 6.7+5.3 --> 6.7x5.3=35.51

(6.3,5.7)=12 --> 6.3+5.7 --> 6.3x5.7=35.91

(6.2,5.8)=12 --> 6.2+5.8 --> 6.2x5.8=35.96

(6.1,5.9)=12 --> 6.1+5.9 --> 6.1x5.9=65.99

I still have not yet found a number higher than 36 in decimal numbers. I will try now in fractional numbers if I can get a number higher than 36.

(6 1/3,5 2/3)=12 --> 6 1/3+5 2/3 --> 6 1/3x5 2/3=35.88

(6 2/5,6 3/5)=12 --> 6 2/5+6 3/5 --> 6 2/5x6 3/5=35.84

(6 2/7,5 5/7)=12 --> 6 2/7+5 5/7 --> 6 2/7x5 5/7=35.92 (2dp)

(6 2/9,5 7/9)=12 --> 6 2/9+5 7/9 --> 6 2/9x5 7/9=35.95 (2dp)

I have found that 6 and 6 are the two numbers which added together make 12 and when multiplied together make 36 which is the highest possible answer which is retrieved when two numbers added together equal 12 are multiplied.

3

(1,12)=13 --> 1+12 --> 1x12=12

(2,11)=13 --> 2+11 --> 2x11=22

(3,10)=13 --> 3+10 --> 3x10=30

(4,9)= 13 --> 4+9 --> 4x9 =36

(5,8)= 13 --> 5+8 --> 5x8 =40

(6,7)= 13 --> 6+7 --> 6x7 =42

I have found that 42 is the highest number so far that can be retrieved from 6 and 7 when the number is 13, in whole numbers. I will now try in decimal numbers if I can get a number higher than 42.

(6.1,6.9)=13 --> 6.1+6.9 --> 6.1x6.9=42.09

(6.3,6.7)=13 --> 6.3+6.7 --> 6.3x6.7=42.21

(6.5,6.5)=13 --> 6.5+6.5 --> 6.5x6.5=42.25

(6.6,6.4)=13 --> 6.6+6.4 --> 6.6x6.4=42.24

I have found a number higher than 42 in decimal numbers. I will try now in fractional numbers if I can get a number higher than 42.24.

(6 1/3, 6 2/3)= 13 --> 6 1/3+6 2/3 --> 6 1/3x6/2/3 =42.22 (2dp)

(6 1/15, 6 14/15)=13 --> 6 1/15+6 14/15 --> 6 1/15x6 14/15=42.06 (2dp)

(6 2/13, 6 11/13)=13 --> 6 2/13+6 11/13 --> 6 2/13x6 11/13=42.13 (2dp)

I have found that 6.5 and 6.5 are the two numbers which added together make 13 and when multiplied together make 42.25 which is the highest possible answer which is retrieved when two numbers added together equal 13 are multiplied.

4

(1,13)=14 --> 1+13 --> 1x13=13

(2,12)=14 --> 2+12 --> 2x12=24

(3,11)=14 --> 3+11 --> 3x11=33

(4,10)=14 --> 4+10 --> 4x11=44

(5,9)= 14 --> 5+9 --> 5x9 =45

(6,8)= 14 --> 6+8 --> 6x8 =48

(7,7)= 14 --> 7+7 --> 7x7 =49

I have found that 49 is the highest number so far that can be retrieved from 7 and 7 when the number is 14, in whole numbers. I will now try in decimal numbers if I can get a number higher than 49.

(7.1,6.9)= 14 --> 7.1+6.9 --> 7.1x6.9 =48.99

(7.2,6.8)= 14 --> 7.2+6.8 --> 7.2x6.8 =48.96

(7.01, 6.99)= 14 --> 7.01+6.99 --> 7.01x6.99=48.9999

I still have not yet found a number higher than 49 in decimal numbers. I will try now in fractional numbers if I can get a number higher than 49.

(7 1/10, 6 9/10)= 14 --> 7 1/10+6 9/10 --> 7 1/10x6 9/10 =48.99

(7 4/15, 6 11/15)= 14 --> 7 4/15+6 11/15 --> 7 4/15x6 11/15=48.929 (3dp)

(7 1/15, 6 14/15)= 14 --> 7 1/15+6 14/15 --> 7 1/15x6 14/15=48.996 (3dp)

I have found that 7 and 7 are the two numbers which added together make 14 and when multiplied together make 49 which is the highest possible answer which is retrieved when two numbers added together equal 14 are multiplied.

5

(1,14)= 15 --> 1+14 --> 1x14=14

(2,13)= 15 --> 2+13 --> 2x13=26

(3,12)= 15 --> 3+12 --> 3x12=36

(4,11)= 15 --> 4+11 --> 4x11=44

(5,10)= 15 --> 5+10 --> 5x10=50

(6,9)= 15 --> 6+9 --> 6x9 =54

(7,8)= 15 --> 7+8 --> 7x8 =56

I have found that 56 is the highest number so far that can be retrieved from 7 and 8 when the number is 15, in whole numbers. I will now try in decimal numbers if I can get a number higher than 56.

(7.1,7.9)= 15 --> 7.1+7.9 --> 7.1x7.9=56.09

(7.2,7.8)= 15 --> 7.2+7.8 --> 7.2x7.8=56.16

(7.3,7.7)= 15 --> 7.3+7.7 --> 7.3x7.7=56.21

(7.4,7.6)= 15 --> 7.4+7.6 --> 7.4x7.6=56.24

(7.5,7.5)= 15 --> 7.5+7.5 --> 7.5x7.5= 56.25

I have found a number higher than 56 in decimal numbers. I will try now in fractional numbers if I can get a number higher than 56.25.

(7 2/9, 7 7/9)= 15 --> 7 2/9+7 7/9 --> 7 2/9x7 7/9= 56.173 (3dp)

(7 5/9, 7 4/9)= 15 --> 7 5/9+7 4/9 --> 7 5/9x7 4/9= 56.247 (3dp)

I have found that 7.5 and 7.5 are the two numbers which added together make 15 and when multiplied together make 56.25 which is the highest possible answer which is retrieved when two numbers added together equal 15 are multiplyed.

6

(1,15)= 16 --> 1+15 --> 1x15=15

(2,14)= 16 --> 2+14 --> 2x14=28

(3,13)= 16 --> 3+13 --> 3x13=39

(4,12)= 16 --> 4+12 --> 4x12=48

(5,11)= 16 --> 5+11 --> 5x11=55

(6,10)= 16 --> 6+10 --> 6x11=60

(7,9)= 16 --> 7+9 --> 7x9 =63

(8,8)= 16 --> 8+8 --> 8x8 =64

I have found that 64 is the highest number so far that can be retrieved from 8 and 8 when the number is 16, in whole numbers. I will now try in decimal numbers if I can get a number higher than 64.

(8.1,7.9)= 16 --> 8.1+7.9 --> 8.1x7.9 =63.99

(8.2,7.8)= 16 --> 8.2+7.8 --> 8.2x7.8 =63.93 (2dp)

(8.01,7.99)= 16 --> 8.01+7.99 --> 8.01x7.99=63.999

I still have not yet found a number higher than 64 in decimal numbers. I will try now in fractional numbers if I can get a number higher than 64.

(8 1/10,7 9/10)= 16 --> 8 1/10+7 9/10 --> 8 1/10x7 9/10 =63.99

(8 4/15,7 11/15)= 16 --> 8 4/15+7 11/15 --> 8 4/15x7 11/15=63.93 (2dp)

(8 1/15, 7 14/15)= 16 --> 8 1/15+7 14/15 --> 8 1/15x7 14/15=63.996 (3dp)

I have found that 8 and 8 are the two numbers which added together make 16 and when multiplied together make 64 which is the highest possible answer which is retrieved when two numbers added together equal 16 are multiplied.

Results of Numbers

Number

Two numbers used

Two numbers added

Two numbers multiplied

Maxi Product

2

5 and 7

5+7

5x7

35

2

6 and 6

6+6

6x6

36- Maxi Product

2

4 and 8

4+8

4x8

32

2

6.5 and 5.5

6.5+5.5

6.5x5.5

35.75

2

6.7 and 5.3

6.7+5.3

6.7x5.3

35.51

2

6.3 and 5.7

6.3+5.7

6.3x5.7

35.91

2

6.2 and 5.8

6.2+5.8

6.2x5.8

35.96

2

6.1 and 5.9

6.1+5.9

6.1x5.9

35.99

2

6 1/3 and 5 2/3

6 1/3+5 2/3

6 1/3x5 2/3

35.88

2

6 2/5 and 5 3/5

6 2/5+5 3/5

6 2/5x5 3/5

35.84

2

6 2/7 and 5 5/7

6 2/7+5 5/7

6 2/7x5 5/7

35.92 (2dp)

2

6 2/9 and 5 7/9

6 2/9+5 7/9

6 2/9x5 7/9

35.95 (2dp)

Number

Two numbers used

Two numbers added

Two numbers multiplied

Maxi Product

3

and 12

+12

x12

2

3

2 and 11

2+11

2x11

22

3

3 and 10

3+10

3x10

30

3

4 and 9

4+9

4x9

36

3

5 and 8

5+8

5x8

40

3

6 and 7

6+7

6x7

42

3

6.1 and 6.9

6.1+6.9

6.1x6.9

42.09

3

6.3 and 6.7

6.3+6.7

6.3x6.7

42.21

3

6.5 and 6.5

6.5+6.5

6.5x6.5

42.25- Maxi Product

3

6.6 and 6.4
Join now!


6.6+6.4

6.6x6.4

42.24

3

6 1/3 and 6 2/3

6 1/3+6 2/3

6 1/3x6 2/3

42.24 (2dp)

3

6 1/15 and 6 14/15

6 1/15+6 14/15

6 1/15x6 14/15

42.06 (2dp)

3

6 2/13 and 6 11/13

6 2/13+6 11/13

6 2/13x6 11/13

42.13 (2dp)

Number

Two numbers used

Two numbers added

Two numbers multiplied

Maxi Product

4

and 13

+13

x13

3

4

2 ...

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