The Maximum Product of Any Given Number When Split into Parts
The objective of this coursework is to find the formula that gives the maximum product of any given number when split into parts.
For the first part of the coursework I am going to investigate the splitting of a number into two parts, then three parts and into four and five parts. For each number of parts, I am going to come up with a formula that gives the maximum product.
Starting number 12
PAIR WHICH TOTAL 12 PRODUCT OF THE PAIR
12, 0 0
11,1 11
10,2 20
9,3 27
8,4 32
7,5 35
6,6 36
5,7 35
4,8 32
3,9 27
2,10 20
1,11 11
0,12 0
TEST FOR THE MAXI PRODUCT
I will assume my maxi product is in the range between 7,5 and 5,7 because this where the arch is at it's highest. So,
6.9 × 5.1=35.19
5.9 × 6.1=35.99
Neither of these are higher than 36. So for my starting number 12, I have used the pairs 6 and 6 and my maxi product is 36.
Starting number 10
PAIR WHICH TOTAL 10 PRODUCT OF THE PAIR
10,0 0
9,1 9
8,2 16
7,3 21
6,4 24
5,5 25
4,6 24
3,7 21
2,8 16
1,9 9
0,10 0
TEST FOR THE MAXI PRODUCT
Again, I am looking in the range between 6,4 and 4,6.
6 × 4=24
4.5 × 5.5=24.75
5 × 5=25
Neither of the products are higher than 25. For my starting number 10, I have used the pair 5,5 and my maxi product is 25.
Starting number 14
PAIR WHICH TOTAL 14 PRODUCT OF THE PAIR
14,0 0
13,1 13
12,2 24
11,3 33
10,4 40
9,5 45
8,6 48
7,7 49
6,8 48
5,9 45
4,10 40
3,11 33
2,12 24
1,13 13
0,14 0
TEST FOR THE MAXI PRODUCT
Between 8,6 and 6,8 as this is where the number is highest.
7 ×7=49
7.9 × 6.1=48.19
6.9 × 7.1=48.99
For my starting number 14, I have used 7,7 and my maxi product is 49.
Starting number 16
PAIR WHICH TOTAL 16 PRODUCT OF THE PAIR
16,0 0
15,1 15
14,2 28
13,3 39
12,4 48
11,5 55
10,6 60
9,7 63
8,8 64
7,9 63
6,10 60
5,11 55
4,12 48
3,13 39
2,14 28
1,15 15
0,16 0
TEST FOR THE MAXI PRODUCT
Between 9,7 and 7,9 as this is where the number seems highest.
7.9 × 8.1=63.19
8 × 8=64
8.9 × 7.1=63.99
For my starting number 16, I have used the pair 8,8 and my maxi product is 64.
Starting number 9
PAIR WHICH TOTAL 9 PRODUCT OF THE PAIR
9,0 0
8,1 8
7,2 14
6,3 18
5,4 20
4,5 20
3,6 18
2,7 14
1,8 8
0,9 0
TEST FOR THE MAXI PRODUCT
The maxi product lies somewhere between 5,4 and 4,5.
5 × 4=20
4.5 × 4.5= 20.25
4.9 × 4.1=20.09
20.25 is the highest, therefore my maxi product is 20.25. The pair I used was 4.5 and 4.5.
Starting number 13
PAIR WHICH TOTAL 13 PRODUCT OF THE PAIR
13,0 0
12,1 12
11,2 22
10,3 30
9,4 36
8,5 40
7,6 42
6,7 42
5,8 40
4,9 36
3,10 30
2,11 22
1,12 12
0,13 0
TEST FOR THE MAXI PRODUCT
The maxi product is somewhere between 7,6 and 6,7.
7 × 6 = 42
6.9 × 6.1 = 42.09
6.5 × 6.5 = 42.25
For my starting number 13, my maxi product is 42.25. The pair I have used is 6.5 and 6.5.
STARTING NUMBERS PAIR WHICH GIVE MAXIPRODUCT VALUE OF MAXI PRODUCT
12 6 х 6 36
10 5 х 5 25
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