Investigate the Maxi Product of numbers
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: 12coag agr seagagw orag agk inag foag ag.
(5,7)= 12 à 5+7 à 5x7=35
(6,6)= 12 à 6+6 à 6x6=36
(4,8)= 12 à 4+8 à 4x8=32coge ger segegew orge gek inge foge ge.
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 à 6.5+5.5 à 6.5x5.5=35.75
(6.7,5.3)=12 à 6.7+5.3 à 6.7x5.3=35.51 Marx obfuscated Sixx's functionalism theory.
(6.3,5.7)=12 à 6.3+5.7 à 6.3x5.7=35.91
(6.2,5.8)=12 à 6.2+5.8 à 6.2x5.8=35.96
(6.1,5.9)=12 à 6.1+5.9 à 6.1x5.9=65.99coae aer seaeaew orae aek inae foae ae.
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 à 6 1/3+5 2/3 à 6 1/3x5 2/3=35.88 Visit coursework gb in gb fo gb for gb more project gb Do gb not gb redistribute
(6 2/5,6 3/5)=12 à 6 2/5+6 3/5 à 6 2/5x6 3/5=35.84
(6 2/7,5 5/7)=12 à 6 2/7+5 5/7 à 6 2/7x5 5/7=35.92 (2dp)cobb bbr sebbbbw orbb bbk inbb fobb bb.
(6 2/9,5 7/9)=12 à 6 2/9+5 7/9 à 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 à 1+12 à 1x12=12
(2,11)=13 à 2+11 à 2x11=22
(3,10)=13 à 3+10 à 3x10=30
(4,9)= 13 à 4+9 à 4x9 =36
(5,8)= 13 à 5+8 à 5x8 =40
(6,7)= 13 à 6+7 à 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 à 6.1+6.9 à 6.1x6.9=42.09
(6.3,6.7)=13 à 6.3+6.7 à 6.3x6.7=42.21
(6.5,6.5)=13 à 6.5+6.5 à 6.5x6.5=42.25
(6.6,6.4)=13 à 6.6+6.4 à 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 à 6 1/3+6 2/3 à 6 1/3x6/2/3 =42.22 (2dp)
(6 1/15, 6 14/15)=13 à 6 1/15+6 14/15 à 6 1/15x6 14/15=42.06 (2dp)
(6 2/13, 6 11/13)=13 à 6 2/13+6 11/13 à 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 à 1+13 à 1x13=13
(2,12)=14 à 2+12 à 2x12=24
(3,11)=14 à 3+11 à 3x11=33
(4,10)=14 à 4+10 à 4x11=44
(5,9)= 14 à 5+9 à 5x9 =45
(6,8)= 14 à 6+8 à 6x8 =48
(7,7)= 14 à 7+7 à 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 à 7.1+6.9 à 7.1x6.9 =48.99
(7.2,6.8)= 14 à 7.2+6.8 à 7.2x6.8 =48.96
(7.01, 6.99)= 14 à 7.01+6.99 à 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 à 7 1/10+6 9/10 à 7 1/10x6 9/10 =48.99
(7 4/15, 6 11/15)= 14 à 7 4/15+6 11/15 à 7 4/15x6 11/15=48.929 (3dp)
(7 1/15, 6 14/15)= 14 à 7 1/15+6 14/15 à 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 à 1+14 à 1x14=14
(2,13)= 15 à 2+13 à 2x13=26
(3,12)= 15 à 3+12 à 3x12=36
(4,11)= 15 à 4+11 à 4x11=44
(5,10)= 15 à 5+10 à 5x10=50
(6,9)= 15 à 6+9 à 6x9 =54
(7,8)= 15 à 7+8 à 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 à 7.1+7.9 à 7.1x7.9=56.09
(7.2,7.8)= 15 à 7.2+7.8 à 7.2x7.8=56.16
(7.3,7.7)= 15 à 7.3+7.7 à 7.3x7.7=56.21
(7.4,7.6)= 15 à 7.4+7.6 à 7.4x7.6=56.24
(7.5,7.5)= 15 à 7.5+7.5 à 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 à 7 2/9+7 7/9 à 7 2/9x7 7/9= 56.173 (3dp)
(7 5/9, 7 4/9)= 15 à 7 5/9+7 4/9 à 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 à 1+15 à 1x15=15
(2,14)= 16 à 2+14 à 2x14=28
(3,13)= 16 à 3+13 à 3x13=39
(4,12)= 16 à 4+12 à 4x12=48
(5,11)= 16 à 5+11 à 5x11=55
(6,10)= 16 à 6+10 à 6x11=60
(7,9)= 16 à 7+9 à 7x9 =63
(8,8)= 16 à 8+8 à 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 à 8.1+7.9 à 8.1x7.9 =63.99
(8.2,7.8)= 16 à 8.2+7.8 à 8.2x7.8 =63.93 (2dp)
(8.01,7.99)= 16 à 8.01+7.99 à 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 à 8 1/10+7 9/10 à 8 1/10x7 9/10 =63.99
(8 4/15,7 11/15)= 16 à 8 4/15+7 11/15 à 8 4/15x7 11/15=63.93 (2dp)
(8 1/15, 7 14/15)= 16 à 8 1/15+7 14/15 à 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
...
This is a preview of the whole essay
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
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 and 12
2+12
2x12
24
4
3 and 11
3+11
3x11
33
4
4 and 10
4+10
4x10
40
4
5 and 9
5+9
5x9
45
4
6 and 8
6+8
6x8
48
4
7 and 7
7+7
7x7
49- Maxi Product
4
7.1 and 6.9
7.1+6.9
7.1x6.9
48.99
4
7.2 and 6.8
7.2+6.8
7.2x6.8
48.96
4
7.01 and 6.99
7.01+6.99
7.01x6.99
48.9999
4
7 1/10 and 6 9/10
7 1/10+6 9/10
7 1/10x6 9/10
7 1/10x6 9/10
4
7 4/15 and 6 11/15
7 4/15+6 11/15
7 4/15x6 11/15
48.929 (3dp)
4
7 1/15 and 6 14/15
7 1/15+6 14/15
7 4/15x6 11/15
48.996 (3dp)
Number
Two numbers used
Two numbers added
Two numbers multiplied
Maxi Product
5
and 14
+14
x14
4
5
2 and 13
2+13
2x13
26
5
3 and 12
3+12
3x12
36
5
4 and 11
4+11
4x11
44
5
5 and 10
5+10
5x10
50
5
6 and 9
6+9
6x9
54
5
7 and 8
7+8
7x8
56
5
7.1 and 7.9
7.1+7.9
7.1x7.9
56.09
5
7.2 and 7.8
7.2+7.8
7.2x7.8
56.16
5
7.3 and 7.7
7.3+7.7
7.3x7.7
56.21
5
7.4 and 7.6
7.4+7.6
7.4x7.6
56.24
5
7.5 and 7.5
7.5+7.5
7.5x7.5
56.25- Maxi Product
5
7 2/9 and 7 7/9
7 2/9+7 7/9
7 2/9x7 7/9
56.179 (3dp)
5
7 5/9 and 7 4/9
7 5/9+7 4/9
7 5/9x7 4/9
56.247 (3dp)
Number
Two numbers used
Two numbers added
Two numbers multiplied
Maxi Product
6
and 15
+15
x15
5
6
2 and 14
2+14
2x14
28
6
3 and 13
3+13
3x13
39
6
4 and 12
4+12
4x12
48
6
5 and 11
5+11
5x11
55
6
6 and 10
6+10
6x10
60
6
7 and 9
7+9
7x9
63
6
8 and 8
8+8
8x8
64- Maxi Product
6
8.1 and 7.9
8.1+7.9
8.1x7.9
63.99
6
8.2 and 7.8
8.2+7.8
8.2x7.8
63.93 (2dp)
6
8.01 and 7.99
8.01+7.99
8.01x7.99
63.999
6
8 1/10 and 7 9/10
8 1/10+7 9/10
8 1/10x7 9/10
63.99
6
8 4/15 and 7 11/15
8 4/15+7 11/15
8 4/15x7 11/15
63.93 (2dp)
6
8 1/15 and 7 14/15
8 1/15+7 14/15
8 1/15x7 14/15
63.996 (3dp)
Table of results for double numbers:
Number
Two numbers used
Two numbers added
Two numbers multiplied
Maxi Product of number
2
6 and 6
6+6
6x6
36
3
6.5 and 6.5
6.5+6.5
6.5x6.5
42.25
4
7 and 7
7+7
7x7
49
5
7.5 and 7.5
7.5+7.5
7.5x7.5
56.25
6
8 and 8
8+8
8x8
64
What I notice:
I notice that the Maxi product is retrieved when the two halves of the selected number are multiplied together. I also notice that the two halves are always the same.
For example, 12; twelve will have a Maxi Product of 36 as the half of twelve is six and six multiplied by itself will give you 36 which is the Maxi Product of 12.
Testing my Theory
The maximum product that can be retrieved by the number 8 is 16. You get this by halving 8, which is 4 and then multiplying by itself, which gives you, 16. I will test this now.
(1,7)= 8 à 1+7 à 1x7=7
(2,6)= 8 à 2+6 à 2x6=12
(3,5)= 8 à 3+5 à 3x5=15
(4,4)= 8 à 4+4 à 4x4=16
I have found that 16 is the highest number so far that can be retrieved from 4 and 4 when the number is 8, in whole numbers. I will now try in decimal numbers if I can get a number higher than 16.
(4.7,3.3)= 8 à 4.7+3.3 à 4.7x3.3=15.51
(4.1,3.9)= 8 à 4.1+3.9 à 4.1x3.9=15.99
I still have not yet found a number higher than 16 in decimal numbers. I will try now in fractional numbers if I can get a number higher than 16.
(4 1/20, 3 19/20) =8 à 4 1/20+3 19/20 à 4 1/20x3 19/20 =15.9975
(4 1/50, 3 49/50) =8 à 4 1/50+3 49/50 à 4 1/20x3 49/50 =15.996
(4 1/100, 3 99/100)=8 à 4 1/100+3 99/100 à 4 1/100x3 99/100= 15.9999
I have found that 16 is the Maxi Product of 8 as the two numbers, 4 and 4, when multiplied together make 16. So, my theory was right.
Rule in Words:
Maxi Product equals the number selected divided by two and then the answer squared.
Rule in Algebra:
M=(N/2)²
Key:
M= Maxi Product
N= Number that has been selected
Proving my rule:
Find the Maxi Product of a) 44 b) 55 and c) 66.
A) M=(N/2)² B) M=(N/2)² C) M=(N/2)²
M=(44/2)² M=(55/2)² M=(66/2)²
M=484 M=756.25 M=1089
Triple Numbers
Examples:12
(1,9,2)= 12 à 1+9+2 à 1x9x2=18
(2,8,2)= 12 à 2+8+2 à 2x8x2=32
(3,7,2)= 12 à 3+7+2 à 3x7x2=42
(3,6,3)= 12 à 3+6+3 à 3x6x3=54
(4,5,3)= 12 à 4+5+3 à 4x5x3=60
(4,4,4)= 12 à 4+4+4 à 4x4x4=64
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 64 when three numbers are multiplied together.
(4,4.1,3.9)= 12 à 4+4.1+3.9 à4x4.1x3.9 =63.96
(4,4.2,3.8)= 12 à 4+4.2+3.8 à4x4.2x3.8 =63.84
(4.2,3.9,3.9)= 12 à 4.2+3.9+3.9 à4.2x3.9x3.9=63.882
I am now going to use fractional numbers as I yet have not found a number that is the product of three numbers that leaves a result that is more than 64. I will see in fractional number if I can retrieve a number higher than 64 by multiplying three numbers together.
(4 1/10,4 4/10,3 5/10)= 12 à 4 1/10+4 4/10+3 5/10 à 4 1/10x4 4/10x3 5/10 = 63.14
(4 1/15,4 11/15,3 3/15)= 12 à 4 1/15+4 11/15+3 3/15 à 4 1/15x4 11/15x3 3/15=61.596
(3dp)
(4 1/3,4 1/3,3 1/3)= 12 à 4 1/3+4 1/3+3 1/3 à 4 1/3x4 1/3x3 1/3 =62.59
(2dp)
I have found that 4, 4, and 4 are the three numbers which added together make 12 and when multiplied together make 64 which is the highest possible answer which can be retrieved when three numbers added together equal 12 are multiplied.
3
(1,11,1)= 13 à 1+11+1 à 1x11x1=11
(2,10,1)= 13 à 2+10+1 à 2x10x1=20
(2,9,2)= 13 à 2+9+2 à 2x9x2 =36
(3,8,2)= 13 à 3+8+2 à 3x8x2 =48
(3,7,3)= 13 à 3+7+3 à 3x7x3 =63
(4,6,3)= 13 à 4+6+3 à 4x6x3 =72
(4,5,4)= 13 à 4+5+4 à 4x5x4 =80
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 80 when three numbers are multiplied together.
(4.1,4.8,4.1)= 13 à 4.1+4.8+4.1 à4.1x4.8x4.1=80.688
(4.2,4.6,4.2)= 13 à 4.2+4.6+4.2 à4.2x4.6x4.2=81.144
(4.3,4.4,4.3)= 13 à 4.3+4.4+4.3 à4.3x4.4x4.3=81.356
I will now move onto fractional numbers as there can be no other decimal number that can give a result higher than 81.356 using three numbers. I will see in fractional numbers if I can get a number higher than 81.356 from three fractional numbers.
(4 1/2 ,4 1/4,4 1/4)= 13 à 4 1/2+4 1/4+4 1/2 à 4 1/2x4/14x4 1/4 =81.28125
(4 5/16,4 5/16, 4 6/16)= 13 à 4 5/16+4 5/16+4 6/16 à 4 5/16x4 5/16x 4 6/16=81.364
(3dp)
(4 1/3,4 1/3,4 1/3)= 13 à 4 1/3+4 1/3+4 1/3 à 4 1/3x4 1/3x4 1/3 =81.370
(3dp)
I have found that 4 1/3, 4 1/3, and 4 1/3 are the three numbers which added together make 13 and when multiplied together make 81.370 (3dp) which is the highest possible answer which can be retrieved when three numbers added together equal 13 are multiplied.
4
(1,12,1)= 14 à 1+12+1 à 1x12x1=12
(2,11,1)= 14 à 2+11+1 à 2x11x1=22
(2,10,2)= 14 à 2+10+2 à 2x10x2=40
(3,9,2)= 14 à 3+9+2 à 3x9x2 =54
(3,8,3)= 14 à 3+8+3 à 3x8x3 =72
(4,7,3)= 14 à 4+7+3 à 4x7x3 =84
(4,6,4)= 14 à 4+6+4 à 4x6x4 =96
(5,5,4)= 14 à 5+5+4 à 5x5x4 =100
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 100 when three numbers are multiplied together.
(4.7,4.9,4.4)= 14 à 4.7+4.9+4.4 à4.7x4.9x4.4=101.332
(4.7,4.7,4.6)= 14 à 4.7+4.7+4.6 à4.7x4.7x4.6=101.614
I will now move on to fractional numbers as there can be no other decimal number that can give a result higher than 101.614 using three numbers. I will see in fractional numbers if I can get a number higher than 101.332 from three fractional numbers.
(5 1/3,4 1/3,4 1/3)= 14 à 5 1/3+4 1/3+4 1/3 à 5 1/3x4 1/3x4 1/3= 100.148 (3dp)
(4 2/3,4 2/3,4 2/3)= 14 à 4 2/3+4 2/3+4 2/3 à 4 2/3x4 2/3x4 2/3= 101.6296296
I have found that 4 2/3, 4 2/3, and 4 2/3 are the three numbers which added together make 14 and when multiplied together make 101.6296296 which is the highest possible answer which can be retrieved when three numbers added together equal 14 are multiplied.
5
(1,13,1)= 15 à 1+13+1 à 1x13x1=13
(2,12,1)= 15 à 2+12+1 à 2x12x1=24
(2,11,2)= 15 à 2+11+2 à 2x11x2=44
(3,10,2)= 15 à 3+10+2 à 3x10x2=60
(3,9,3)= 15 à 3+9+3 à 3x9x3 =81
(4,8,3)= 15 à 4+8+3 à 4x8x3 =96
(4,7,4)= 15 à 4+7+4 à 4x7x4 =112
(5,6,4)= 15 à 5+6+4 à 5x6x4 =120
(5,5,5)= 15 à 5+5+5 à 5x5x5 =125
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 125 when three numbers are multiplied together.
(5.1,4.8,5.1)= 15 à 5.1+4.8+5.1 à5.1x4.8x5.1=124.848
(5.3,4.4,5.3)= 15 à 5.3+4.4+5.3 à5.3x4.4x5.3=123.596
I will now move on to fractional numbers as there is no decimal number that can give a result higher than 125 using three numbers. I will see in fractional numbers if I can get a number higher than 125 from three fractional numbers.
(5 1/80,5 70/80,4 9/80)= 15 à 5 1/80+5 70/80+4 9/80 à 5 1/80x5 70/90x4 9/80=121.11
(2dp)
(5 5/90,5 80/90,4 5/90)= 15 à 5 5/90+5 80/90+4 5/90 à 5 5/90x5 80/90x4 5/90=120.74
(2dp)
I have found that 5, 5, and 5 are the three numbers which added together make 15 and when multiplied together make 125 which is the highest possible answer which can be retrieved when three numbers added together equal 15 are multiplied.
6
(1,14,1)= 16 à 1+14+1 à 1x14x1=14
(2,13,1)= 16 à 2+13+1 à 2x13x1=26
(2,12,2)= 16 à 2+12+2 à 2x12x2=48
(3,11,2)= 16 à 3+11+2 à 3x11x2=66
(3,10,3)= 16 à 3+10+3 à 3x10x3=90
(4,9,3)= 16 à 4+9+3 à 4x9x3 =108
(4,8,4)= 16 à 4+8+4 à 4x8x4 =128
(5,7,4)= 16 à 5+7+4 à 5x7x4 =140
(5,6,5)= 16 à 5+6+5 à 5+6+5 =150
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 150 when three numbers are multiplied together.
(5.01,5.90,5.09)= 16 à5.01+5.90+5.09 à 5.01x5.90x5.09=150.45531
(5.25,5.30,5.45)= 16 à5.25+5.30+5.45 à 5.25x5.30x5.45=151.64625
I will now move on to fractional numbers as there can be no other decimal number that can give a result higher than 151.64625 using three numbers. I will see in fractional numbers if I can get a number higher than 151.64625 from three fractional numbers.
(5 1/20,5 7/20,5 12/20)= 16 à 5 1/20+5 7/20+5 12/20 à 5 1/20x5 7/20x5 12/20=
51.298
(5 1/50,5 20/50,5 29/50)= 16 à 5 1/50+5 20/50+5 29/50 à 5 1/50x5 20/50x5 29/50=
51.26264
(5 1/3,5 1/3,5 1/3)= 16 à 5 1/3+5 1/3+5 1/3 à 5 1/3x5 1/3x5 1/3=151.704
(3dp)
I have found that 5 1/3, 5 1/3, and 5 1/3 are the three numbers which added together make 16 and when multiplied together make 151.704 (3dp) which is the highest possible answer which can be retrieved when three numbers added together equal 16 are multiplied.
Result of Numbers:
Number
Three numbers used
Three numbers added
Three numbers multiplied
Maxi Product
2
,9 and 2
+9+2
x9x2
8
2
2,8 and 2
2+8+2
2x8x2
32
2
3,7and 2
3+7+2
3x7x2
42
2
3,6 and 3
3+6+3
3x6x3
54
2
4,5 and 3
4+5+3
4x5x3
60
2
4,4, and 4
4+4+4
4x4x4
64- Maxi Product
2
4,4.1 and 3.9
4+4.1+3.9
4x4.1x3.9
63.96
2
4,4.2 and 3.8
4+4.2+3.8
4x4.2x3.8
63.84
2
4.2,3.9 and 3.9
4.2+3.9+3.9
4.2x3.9x3.9
63.882
2
4 1/10,4 4/10 and 3 5/10
4 1/10+4 4/10+ 3 5/10
4 1/10x4 4/10x 3 5/10
63.14
2
4 1/15,4 11/15 and 3 3/15
4 1/15+4 11/15 +3 3/15
4 1/15x4 11/15 x3 3/15
61.596 (3dp)
2
4 1/3,4 1/3 and 3 1/3
4 1/3+4 1/3+ 3 1/3
4 1/3x4 1/3x 3 1/3
62.59 (2dp)
Number
Three numbers used
Three numbers added
Three numbers multiplied
Maxi Product
3
,11 and 1
+11+1
x11x1
1
3
2,10 and 1
2+10+1
2x10x1
20
3
2,9 and 2
2+9+2
2x9x2
36
3
3,8 and 2
3+8+2
3x8x2
48
3
3,7 and 3
3+7+3
3x7x3
63
3
4,6 and 3
4+6+3
4x6x3
72
3
4,5 and 4
4+5+4
4x5x4
80
3
4.1,4.8 and 4.1
4.1+4.8+4.1
4.1x4.8x4.1
80.688
3
4.2,4.6 and 4.2
4.2+4.6+4.2
4.2x4.6x4.2
81.144
3
4.3,4.4 and 4.3
4.3+4.4+4.3
4.3x4.4x4.3
81.356
3
4 1/2,4 1/4 and 4 1/4
4 1/2+4 1/4+ 4 1/4
4 1/2x4 1/4x 4 1/4
81.281 (3dp)
3
4 5/16, 4 5/16 and 4 6/16
4 5/16+4 5/16+ 4 6/16
4 5/16x4 5/16x 4 6/16
81.365 (3dp)
3
4 1/3, 4 1/3 and 4 1.3
4 1/3+4 1/3+ 4 1.3
4 1/3x4 1/3x 4 1.3
81.370 (3dp)- Maxi Product
Number
Three numbers used
Three numbers added
Three numbers multiplied
Maxi Product
4
,12 and 1
+12+1
x12x1
2
4
2,11 and 1
2+11+1
2x11x1
22
4
2,10 and 2
2+10+2
2x10x2
40
4
3,9 and 2
3+9+2
3x9x2
54
4
3,8 and 3
3+8+3
3x8x3
72
4
4,7 and 3
4+7+3
4x7x3
84
4
4,6 and 4
4+6+4
4x6x4
96
4
5,5 and 4
5+5+4
5x5x4
00
4
4.7,4.9 and 4.4
4.7+4.9+4.4
4.7x4.9x4.4
01.332
4
4.7,4.7 and 4.6
4.7+4.7+4.6
4.7x4.7x4.6
01.614
4
5 1/3,4 1/3 and 4 1/3
5 1/3+4 1/3+ 4 1/3
5 1/3x4 1/3x 4 1/3
00.1481481
4
4 2/3,4 2/3 and 4 2/3
4 2/3+4 2/3+ 4 2/3
4 2/3x4 2/3x 4 2/3
01.630 (3dp)- Maxi Product
Number
Three numbers used
Three numbers added
Three numbers multiplied
Maxi Product
5
,13 and 1
+13+1
x13x1
3
5
2,12 and 1
2+12+1
2x12x1
24
5
2,11 and 2
2+11+2
2x11x2
44
5
3,10 and 2
3+10+2
3x10x2
60
5
3,9 and 3
3+9+3
3x9x3
81
5
4,8 and 3
4+8+3
4x8x3
96
5
4,7 and 4
4+7+4
4x7x4
12
5
5,6 and 4
5+6+4
5x6x4
20
5
5,5 and 5
5+5+5
5x5x5
25- Maxi Product
5
5.1,4.8 and 5.1
5.1+4.8+5.1
5.1x4.8x5.1
24.848
5
5.3,4.4 and 5.3
5.3+4.4+5.3
5.3x4.4x5.3
23.596
5
5 1/80,5 70/80 and 4 9/80
5 1/80+5 70/80 +4 9/80
5 1/80x5 70/80 x4 9/80
21.1066992
5
5 5/90,5 80/90 and 4 5/90
5 5/90+5 80/90 +4 5/90
5 5/90x5 80/90 x4 5/90
20.7403978
Number
Three numbers used
Three numbers added
Three numbers multiplied
Maxi Product
6
,14 and 1
+14+1
x14x1
4
6
2,13, and 1
2+13+1
2x13x1
26
6
2,12 and 1
2+12+1
2x12x1
48
6
3,11 and 2
3+11+2
3x11x2
66
6
3,10 and 3
3+10+3
3x10x3
90
6
4.9 and 3
4+9+3
4x9x3
08
6
4,8 and 4
4+8+4
4x8x4
28
6
5,7 and 4
5+7+4
5x7x4
40
6
5,6 and 5
5+6+5
5x6x5
50
6
5.01,5.90 and 5.09
5.01+5.90+ 5.09
5.01x5.90x 5.09
56.45531
6
5.25,5.30 and 5.45
5.25+5.30+ 5.45
5.25x5.30x 5.45
51.64625
6
5 1/20,5 7/20 and 5 12/20
5 1/20+5 7/20+ 5 12/20
5 1/20x5 7/20x 5 12/20
5.298
6
5 1/50,5 20/50 and 5 29/50
5 1/50+5 20/50 +5 29/50
5 1/50x5 20/50 x5 29/50
51.26264
6
5 1/3,5 1/3 and 5 1/3
5 1/3+5 1/3+5 1/3
5 1/3x5 1/3x5 1/3
51.704(3dp)- Maxi Product
Table of results for triple numbers:
Number
Three numbers used
Three numbers added
Three numbers multiplied
Maxi Product
2
4,4 and 4
4+4+4
4x4x4
64
3
4 1/3,4 1/3 and 4 1/3
4 1/3+4 1/3+ 4 1/3
4 1/3x4 1/3x 4 1/3
81.370 (3dp)
4
4 2/3,4 2/3 and 4 2/3
4 2/3+4 2/3+ 4 2/3
4 2/3x4 2/3x 4 2/3
01.300 (3dp)
5
5,5 and 5
5+5+5
5x5x5
25
6
5 1/3,5 1/3 and 5 1/3
5 1/3+5 1/3+ 5 1/3
5 1/3x5 1/3x 5 1/3
51.704 (3dp)
What I Notice:
I notice that the Maxi Product is retrieved when the number is split up into three parts and then multiplied together.
For example, 12, twelve will have a Maxi Product of 64 as a third of 12 is 4, and 4 multiplied by itself three times gives you 64. I will now test my theory.
Testing my theory:
The maximum product that can be retrieved by 8 is 18.963 (3dp). You get this by finding one third of 8 and then multiplying it by itself three times, which is 18.963 (3dp). I will prove this now.
(1,6,1)= 8 à 1+6+1 à 1x6x1=6
(2,5,1)= 8 à 2+5+1 à 2x5x1=10
(2,4,2)= 8 à 2+4+2 à 2x4x2=16
(3,3,2)= 8 à 3+3+2 à 3x3x2=18
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 18 when three numbers are multiplied together.
(3.2,2.6,2.2)= 8 à 3.2+2.6+2.2 à 3.2x2.6x2.2=18.304
(3.6,3.6,0.8)= 8 à 3.6+3.6+0.8 à 3.6x3.6x0.8=10.368
(3.1,2.8,2.1)= 8 à 3.1+2.8+2.1 à 3.1x2.8x2.1=18.228
(2.6,2.6,2.8)= 8 à 2.6+2.6+2.8 à 2.6x2.6x2.8=18.928
I will now move onto fractional numbers as there can be no other decimal number that can give a result higher than 18.928 using three numbers. I will see in fractional numbers if I can get a number higher than 18.928 from three fractional numbers.
(2 1/3,2 1/3,3 1/3)= 8 à 2 1/3+2 1/3+3 1/3 à 2 1/3x2 1/3x3 1/3=18.148 (3dp)
(2 2/3,2 2/3,2 2/3)= 8 à 2 2/3+2 2/3+2 2/3 à 2 2/3x2 2/3x2 2/3=18.963 (3dp)
I have found that 2 2/3,2 2/3 and 2 2/3 are the three numbers which added together make 8 and when multiplied together make 18.963 (3dp) which is the highest possible answer which can be retrieved when three numbers added together equal 8 are multiplied.
Rule in words:
Maxi Product equals the number divided by three and then the answer cubed.
Rule in Algebra:
M=(N/3)³
Key:
M=Maxi product
N=Number that is highlighted
Proving my rule:
Find the Maxi Product of a)44 b)55 and c)66
a) M=(N/3)³ b) M=(N/3)³ c) M=(N/3)³
M=(44/3)³ M=(55/3)³ M=(55/3)³
M=3154.963 (3dp) M=6162.037 M=10648
Quadruplet Numbers:
2
(1,1,1,9)= 12 à 1+1+1+9 à 1x1x1x9=9
(2,1,1,8)= 12 à 2+1+1+8 à 2x1x1x8=16
(2,2,1,7)= 12 à 2+2+1+7 à 2x2x1x7=28
(2,2,2,6)= 12 à 2+2+2+6 à 2x2x2x6=48
(3,2,2,5)= 12 à 3+2+2+5 à 3x2x2x5=60
(3,3,2,4)= 12 à 3+3+2+4 à 3x3x2x4=72
(3,3,3,3)= 12 à 3+3+3+3 à 3x3x3x3=81
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 81 when three numbers are multiplied together.
(3.1,3.1,3.1,2.7)= 12 à3.1+3.1+3.1+2.7 à 3.1x3.1x3.1x2.7=80.4357
(3.2,3.2,3.2,2.4)= 12 à3.2+3.2+3.2+2.4 à 3.2x3.2x3.2x2.4=78.6432
I will now move onto fractional numbers as there no decimal number that can give a result higher than 81 using three numbers. I will see in fractional numbers if I can get a number higher than 81 from three fractional numbers.
(3 1/4,3 1/4,3 1/4,2 1/4)= 12 à 3 1/4+3 1/4+3 1/4+2 1/4 à 3 1/4x3 1/4x3 1/4x2 1/4=
77.23828125
(3 2/5,3 1/5,3 1/5,2 1/5)= 12 à 3 2/5+3 1/5+3 1/5+2 1/5 à 3 2/5x3 1/5x3 1/5x2 1/5=
76.5952
I have found that 3,3 and 3 are the three numbers which added together make 12 and when multiplied together make 81 which is the highest possible answer which can be retrieved when three numbers added together equal 12 are multiplied.
3
(1,1,1,10)= 13 à 1+1+1+10 à 1x1x1x10=10
(2,1,1,9)= 13 à 2+1+1+9 à 2x1x1x9 =18
(2,2,1,8)= 13 à 2+2+1+8 à 2x2x1x8 =32
(2,2,2,7)= 13 à 2+2+2+7 à 2x2x2x7 =56
(3,2,2,6)= 13 à 3+2+2+6 à 3x2x2x6 =72
(3,3,2,5)= 13 à 3+3+2+5 à 3x3x2x5 =90
(3,3,3,4)= 13 à 3+3+3+4 à 3x3x3x4 =108
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 108 when three numbers are multiplied together.
(3.1,3.1,3.1,3.7)= 13 à3.1+3.1+3.1+3.7 à 3.1x3.1x3.1x3.7 =110.2267
(3.2,3.2,3.2,3.4)= 13 à3.2+3.2+3.2+3.4 à 3.2x3.2x3.2x3.4 =111.4112
(3.3,3.2,3.2,3.3)= 13 à3.3+3.2+3.2+3.3 à 3.3x3.2x3.2x3.3 =111.5136
(3.25,3.25,3.25,3.25)=13 à3.25+3.25+3.25+3.25 à 3.25x3.25x3.25x3.25=111.5664063
I will now move onto fractional numbers as there can be no other decimal number that can give a result higher than 111.5664063 using three numbers. I will see in fractional numbers if I can get a number higher than 111.5664063 from three fractional numbers.
(3 1/15,3 5/15,3 4/15,3 5/15)= 13 à 3 1/15+3 5/15+3 4/15+3 5/15 à 3 1/15x3 5/15x
3 4/15,3 5/15=111.308642
(3 1/4,3 1/4,3 1/4,3 1/4)= 13 à 3 1/4+3 1/4+3 1/4+3 1/4 à 3 1/4x3 1/4x3 1/4x
3 1/4=111.5664063
I have found that 3 1/4,3 1/4,3 1/4 and 3 1/4 are the three numbers which added together make 13 and when multiplied together make 111.5664063 which is the highest possible answer which can be retrieved when three numbers added together equal 13 are multiplied.
4
(1,1,1,11)= 14 à 1+1+1+11 à 1x1x1x11=11
(2,1,1,10)= 14 à 2+1+1+10 à 2x1x1x10=20
(2,2,1,9)= 14 à 2+2+1+9 à 2x2x1x9 =36
(2,2,2,8)= 14 à 2+2+2+8 à 2x2x2x8 =64
(3,2,2,7)= 14 à 3+2+2+7 à 3x2x2x7 =84
(3,3,2,6)= 14 à 3+3+2+6 à 3x3x2x6 =108
(3,3,3,5)= 14 à 3+3+3+5 à 3x3x3x5 =135
(3,3,4,4)= 14 à 3+3+4+4 à 3x3x4x4 =144
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 144 when three numbers are multiplied together.
(3.1,3.1,3.9,3.9)= 14 à3.1+3.1+3.9+3.9 à 3.1x3.1x3.9x3.9=146.1682
(3.3,3.3,3.7,3.7)= 14 à3.3+3.3+3.7+3.7 à 3.3x3.3x3.7x3.7=149.0841
(3.5,3.5,3.5,3.5)= 14 à3.5+3.5+3.5+3.5 à 3.5x3.5x3.5x3.5=150.0625
I will now move onto fractional numbers as there can be no other decimal number that can give a result higher than 150.0625 using three numbers. I will see in fractional numbers if I can get a number higher than 150.0625 from three fractional numbers.
(3 4/10,3 4/10,3 6/10,3 6/10)= 14 à 3 4/10+3 4/10+3 6/10+3 6/10 à 3 4/10x3 4/10x
3 6/10x3 6/10=149.8176
(3 1/2,3 1/2,3 1/2,3 1/2)= 14 à 3 1/2+3 1/2+3 1/2+3 1/2 à 3 1/2x3 1/2x3 1/2x3 1/2
=150.0625
I have found that 3.5,3.5,3.5 and 3.5 or 3 1/2,3 1/2,3 1/2 and 3 1/2 are the three numbers which added together make 14 and when multiplied together make 150.0625 which is the highest possible answer which can be retrieved when three numbers added together equal 14 are multiplied.
5
(1,1,1,12)= 15 à 1+1+1+12 à 1x1x1x12=12
(2,1,1,11)= 15 à 2+1+1+11 à 2x1x1x11=22
(2,2,1,10)= 15 à 2+2+1+10 à 2x2x1x10=40
(2,2,2,9)= 15 à 2+2+2+9 à 2x2x2x9 =72
(3,2,2,8)= 15 à 3+2+2+8 à 3x2x2x8 =96
(3,3,2,7)= 15 à 3+3+2+7 à 3x3x2x7 =126
(3,3,3,6)= 15 à 3+3+3+6 à 3x3x3x6 =162
(4,3,3,5)= 15 à 4+3+3+5 à 4x3x3x5 =180
(4,4,3,4)= 15 à 4+4+3+4 à 4x4x3x4 =192
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 192 when three numbers are multiplied together.
(3.5,3.5,3.5,4.5)= 15 à3.5+3.5+3.5+4.5 à 3.5x3.5x3.5x4.5=192.9375
(3.75,3.75,3.75,3.75)= 15 à3.75+3.75+3.75+3.75 à 3.75x3.75x3.75x3.75
=197.7539062
I will now move onto fractional numbers as there can be no other decimal number that can give a result higher than 197.7539062 using three numbers. I will see in fractional numbers if I can get a number higher than 197.7539062 from three fractional numbers.
(3 1/4,3 1/4,3 1/4,5 1/4)= 15 à 3 1/4+3 1/4+3 1/4+5 1/4 à 3 1/4x3 1/4x3 1/4x5 1/4=
80.2226563
(3 3/4,3 3/4,3 3/4,3 3/4)= 15 à 3 3/4+3 3/4+3 3/4+3 3/4 à 3 3/4x3 3/4x3 3/4x3 3/4=
97.7539062
I have found that 3.75,3.75,3.75 and 3.75 or 3 3/4,3 3/4,3 3/4 and 3 3/4 are the three numbers which added together make 15 and when multiplied together make 197.7539062 which is the highest possible answer which can be retrieved when three numbers added together equal 15 are multiplied.
6
(1,1,1,13)= 16 à 1+1+1+13 à 1x1x1x13=13
(2,1,1,12)= 16 à 2+1+1+12 à 2x1x1x12=24
(2,2,1,11)= 16 à 2+2+1+11 à 2x2x1x11=44
(2,2,2,10)= 16 à 2+2+2+10 à 2x2x2x10=80
(3,2,2,9)= 16 à 3+2+2+9 à 3x2x2x9 =108
(3,3,2,8)= 16 à 3+3+2+8 à 3x3x2x8=144
(3,3,3,7)= 16 à 3+3+3+7 à 3x3x3x7=189
(4,3,3,6)= 16 à 4+3+3+6 à 4x3x3x6=216
(4,4,3,5)= 16 à 4+4+3+5 à 4x4x3x5=240
(4,4,4,4)= 16 à 4+4+4+4 à 4x4x4x4=256
I am now going to use decimal numbers as I have found the highest possible result in whole numbers. I will see in decimal numbers if I can retieve a higher result than 256 when three numbers are multiplied together.
(4.1,4.1,3.9,3.9)= 16 à4.1+4.1+3.9+3.9 à 4.1x4.1x3.9x3.9=255.6801
(4.2,4.2,3.8,3.8)= 16 à4.2+4.2+3.8+3.8 à 4.2x4.2x3.8x3.8=254.7216
I will now move onto fractional numbers as there no decimal number that can give a result higher than 256 using three numbers. I will see in fractional numbers if I can get a number higher than 256 from three fractional numbers.
(4 1/5,4 1/5,4 2/5,3 1/5)= 16 à 4 1/5+4 1/5+4 2/5+3 1/5 à 4 1/5x4 1/5x4 2/5x3 1/5=
248.3712
(4 2/9,4 5/9,4 1/9,3 1/9)= 16 à 4 2/9+4 5/9+4 1/9+3 1/9 à 4 2/9x4 5/9x4 1/9x3 1/9=
246.0124981
I have found that 4,4,4 and 4 are the four numbers which added together make 16 and when multiplied together make 256 which is the highest possible answer which can be retrieved when three numbers added together equal 16 are multiplied.
Result of Numbers:
Number
Four numbers used
Four numbers added
Four numbers multiplied
Maxi Product
2
,1,1 and 9
+1+1+9
x1x1x9
9
2
2,1,1 and 8
2+1+1+8
2x1x1x8
6
2
2,2,1 and 7
2+2+1+7
2x2x1x7
28
2
2,2,2 and 6
2+2+2+6
2x2x2x6
48
2
3,2,2 and 5
3+2+2+5
3x2x2x5
60
2
3,3,2 and 4
3+3+2+4
3x3x2x4
72
2
3,3,3 and 3
3+3+3+3
3x3x3x3
81- Maxi Product
2
3.1,3.1,3.1 and 2.7
3.1+3.1+3.1+2.7
3.1x3.1x3.1x2.7
80.4357
2
3.2,3.2,3.2 and 2.4
3.2+3.2+3.2+2.4
3.2x3.2x3.2x2.4
78.6432
2
3 1/4,3 1/4, 3 1/4 and 2 1/4
3 1/4+3 1/4+ 3 1/4+2 1/4
3 1/4x3 1/4x 3 1/4x2 1/4
77.23828125
2
3 2/5,3 1/5, 3 1/5 and 2 1/5
3 2/5+3 1/5+ 3 1/5+2 1/5
3 2/5x3 1/5x 3 1/5x2 1/5
76.5952
Number
Four numbers used
Four numbers added
Four numbers multiplied
Maxi Product
3
,1,1 and 10
+1+1+10
x1x1x10
0
3
2,1,1 and 9
2+1+1+9
2x1x1x9
8
3
2,2,1 and 8
2+2+1+8
2x2x1x8
32
3
2,2,2 and 7
2+2+2+7
2x2x2x7
56
3
3,2,2 and 6
3+2+2+6
3x2x2x6
72
3
3,3,2 and 5
3+3+2+5
3x3x2x5
90
3
3,3,3 and 4
3+3+3+4
3x3x3x4
08
3
3.1,3.1,3.1 and 3.7
3.1+3.1+3.1+3.7
3.1x3.1x3.1x3.7
10.2267
3
3.2,3.2,3.2 and 3.4
3.2+3.2+3.2+3.4
3.2x3.2x3.2x3.4
11.4112
3
3.3,3.2,3.2 and 3.3
3.3+3.2+3.2+3.3
3.3x3.2x3.2x3.3
11.5136
3
3.25,3.25,3.25 and 3.25
3.25+3.25+3.25+
3.25
3.25x3.25x3.25x
3.25
11.5664063- Maxi product 1
3
3 1/15,3 5.15,3 4/15 and 3 5/15
3 1/15+3 5.15+
3 4/15+3 5/15
3 1/15x3 5.15x
3 4/15x3 5/15
11.308642
Number
Four numbers used
Four numbers added
Four numbers multiplied
Maxi Product
4
,1,1 and 11
+1+1+11
x1x1x11
1
4
2,1,1and 10
2+1+1+10
2x1x1x10
20
4
2,2,1 and 9
2+2+1+9
2x2x1x9
36
4
2,2,2 and 8
2+2+2+8
2x2x2x8
64
4
3,2,2 and 7
3+2+2+7
3x2x2x7
84
4
3,3,2 and 6
3+3+2+6
3x3x2x6
08
4
3,3,3 and 5
3+3+3+5
3x3x3x5
35
4
3,3,4 and 4
3+3+4+4
3x3x4x4
44
4
3.1,3.1,3.9 and 3.9
3.1+3.1+3.9+3.9
3.1x3.1x3.9x3.9
46.1681
4
3.3,3.3,3.7 and 3.7
3.3+3.3+3.7+3.7
3.3x3.3x3.7x3.7
49.0841
4
3.5,3.5,3.5 and 3.5
3.5+3.5+3.5+3.5
3.5x3.5x3.5x3.5
50.0625- Maxi product
4
3 4/10,3 4/10,
3 6/10 and
3 6/10
3 4/10+3 4/10+
3 6/10+3 6/10
3 4/10x3 4/10x
3 6/10x3 6/10
49.8176
Number
Four numbers used
Four numbers added
Four numbers multiplied
Maxi Product
5
,1,1 and 12
+1+1+12
x1x1x12
2
5
2,1,1 and 11
2+1+1+11
2x1x1x11
22
5
2,2,1 and 10
2+2+1+10
2x2x1x10
40
5
2,2,2 and 9
2+2+2+9
2x2x2x9
72
5
3,2,2 and 8
3+2+2+8
3x2x2x8
96
5
3,3,2 and 7
3+3+2+7
3x3x2x7
26
5
3,3,3 and 6
3+3+3+6
3x3x3x6
62
5
4,3,3 and 5
4+3+3+5
4x3x3x5
80
5
4,4,3 and 4
4+4+3+4
4x4x3x4
92
5
3.5,3.5,3.5 and 4.5
3.5+3.5+3.5=4.5
3.5x3.5x3.5x4.5
92.9375
5
3.75,3.75,3.75 and 3.75
3.75+3.75+3.75+3.75
3.75x3.75x3.75x3.75
97.7539062- Maxi Product
5
3 1/4,3 1/4,3 1/4 and 5 1/4
3 1/4+3 1/4+3 1/4+ 5 1/4
3 1/4x3 1/4x3 1/4x 5 1/4
80.2226563
Number
Four numbers used
Four numbers added
Four numbers multiplied
Maxi Product
6
,1,1 and 13
+1+1+13
x1x1x13
3
6
2,1,1 and 12
2+1+1+12
2x1x1x12
24
6
2,2,1 and 11
2+2+1+11
2x2x1x11
44
6
2,2,2 and 10
2+2+2+10
2x2x2x10
80
6
3,2,2 and 9
3+2+2+9
3x2x2x9
08
6
3,3,2 and 8
3+3+2+8
3x3x2x8
44
6
3,3,3 and 7
3+3+3+7
3x3x3x7
89
6
4,3,3 and 6
4+3+3+6
4x3x3x6
216
6
4,4,3 and 5
4+4+3+5
4x4x3x5
240
6
4,4,4 and 4
4+4+4+4
4x4x4x4
256- Maxi Product
6
4.1,4.1,3.9 and 3.9
4.1+4.1+3.9+3.9
4.1x4.1x3.9x3.9
255.6801
6
4.2,4.2,3.8 and 3.8
4.2+4.2+3.8+3.8
4.2x4.2x3.8x3.8
254.7216
6
4 1/5,4 1/5,
4 2/5 and 3 1/5
4 1/5+4 1/5+ 4 2/5+3 1/5
4 1/5x4 1/5x 4 2/5x3 1/5
248.3712
6
4 2/9,4 5/9,
4 1/9 and 3 1/9
4 2/9+4 5/9+
4 1/9+3 1/9
4 2/9x4 5/9x
4 1/9x3 1/9
246.0124981
Table of results for quadruplet numbers:
Number
Four numbers used
Four numbers added
Four numbers multiplied
Maxi Product
2
3,3,3 and 3
3+3+3+3
3x3x3x3
81
3
3.25,3.25,3.25 and 3.25
3.25+3.25+3.25+3.25
3.25x3.25x3.25x3.25
11.5664063
4
3.5,3.5,3.5 and 3.5
3.5+3,5+3.5+3.5
3.5x3.5x3.5x3.5
50.0325
5
3.75,3.75,3.75 and 3.75
3.75+3.75+3.75+3.75
3.75x3.75x3.75x3.75
97.7539062
6
4,4,4 and 4
4+4+4+4
4x4x4x4
256
What I Notice:
I notice that the Maxi Product is retrieved when the four parts of the number are multiplied together.
For example, 12, twelve will have a Maxi Product of 81 as a fourth of 12 is 3 and 3 multiplied by itself four times gives you 81. I will now prove this.
Testing my theory:
The maximum product that can be retrieved by 8 is 16. You get this by finding a fourth of 8 and then multiplying itself four times, which is 16. I will prove this now.
(1,1,1,5)= 8 à 1+1+1+5 à 1x1x1x5=5
(2,1,1,4)= 8 à 2+1+1+4 à 2x1x1x4=8
(2,2,1,3)= 8 à 2+2+1+3 à 2x2x1x3=12
(2,2,2,2)= 8 à 2+2+2+2 à 2x2x2x2=16
I will now move on to decimals as I have found the highest result in whole numbers. I will see in decimal numbers if I can retriev a result higher than 16 when four decimal numbers are multiplied.
(2.1,2.1,2.1,1.7)= 8 à 2.1+2.1+2.1+1.7 à 2.1x2.1x2.1x1.7=15.7437
(2.2,2.2,2.2,1.4)= 8 à 2.2+2.2+2.2+1.4 à 2.2x2.2x2.2x1.4=14.9072
I will now move on to use fractions as I have yet not found a number higher than 16. I will see in fractional numbers if I can retriieve a higher number than 16 when four numbers are multiplied together.
(2 5/10,2 2/10,2 1/10,1 2/10)= 8 à 2 5/10+2 2/10+2 1/10+1 2/10 à 2 5/10x2
2/10x2 1/10x1 2/10=13.86
(2 40/100,2 30/100,2 22/100,1 8/100)= 8 à 2 40/100+2 30/100+2 22/100+1 8/100 à
2 40/100+2 30/100+2 22/100=13.234752
I have found that 2,2,2 and 2 are the four numbers, which added together make 8 and when multiplied together make 16 which is the highest possible answer which can be retrieved.
Rule in words:
Maxi Product equals a fourth of the selected number multiplied by itself four times.
Rule in Algebra:
M=(N/4)
Key:
M= Maxi Product
N=Number Selected
Proving my rule:
Find the Maxi Product of a)44 b)55 and c)66.
a) M=(N/4) b) M=(N/4) c) M=(N/4)
M=(44/4)
I notice that what ever the number is divided by, it is always then powered by the same number.
For example, if the number were to be split into 6 parts (sextuplets), then the equation for that would be: M=(N/6)
Key:
M=Maxi Product
N=Number selected
X=Parts it has been split up in e.g. Doubles, triples, etc.
Proving My Rule:
Find the Maxi Product of a) 44 when it's split up into 7 parts; b) 55 when it's split up by 8 parts and c) 66 when it's split up into 9 parts.
a) M=(N/X)
M=(44/7)
M=387688.0863
b) M=(N/X)
M=(55/8)
M=4990931.629
c) M=(N/X)
M=(66/9)
M=61335630.63
Evaluation:
If I was to do this again, I would organise my time better as I rushed to type it all up one day before it was due in, including this evaluation. I think what took the most time was re-writing all of my results in a table again. Other than that, I wrote up everything in rough and then typed it up neatly on the computer. On the whole, I think I put in a good effort and I have produced the work to the best of my ability.
And that was my investigation on Maxi Products.