# Maxi Product

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Introduction

Maxi Product

Introduction

I am doing a coursework called maxi product. We have to find the maxi product of some numbers, we do this by doing the partition,then times both numbers this will be the product, from this i will get a maximum value. This will my maxi product, from this i will have to gather my results and plot it onto a table. The reason for this is to make my results clear and also it would be easy to find the rule easily and also help me to spot the pattern. I will also use all these same rules to do triplets.

Doubles

Number | Product |

12,0 12x0=0 | |

11,1 11x1=1 | |

10,2 10x2=20 | |

9.5,2.5 9.5x2.5=23.75 | |

12 | 9,3 9x3=27 |

8.5,3.5 8.5x3.5=29.75 | |

8,4 8x4=32 | |

7.5,4,5 7.5x4.5=33.75 | |

7,5 7x5=35 | |

6.5,5.5 6.5x5.5=35.75 | |

6,6 6x6=36 |

I have stopped my table because the answers of the product have started to repeat itself.

My Maxi Product is (6, 6) = 36. Both numbers of the product are even. I got this by dividing the main number by 2 and then multiplying it by itself again to get the maxi product.

Number | Product |

8,0 8x0=0 | |

7,1 7x1=9 | |

6,2 6x2=12 | |

8 | 5.5,2.5 5.5x2.5=13.75 |

5,3 5x3=15 | |

4.5,3.5 4.5x3.5=15.75 | |

4,4 4x4=16 | |

3.5,4.5 3.5x4.5=15.75 | |

Middle

Now i have stopped my table because the answers of the product keep coming the same now.

My Maxi Product is (6.5, 6.5) which equals to 42.25.

My Table

Below i have put my data onto a table so it easier for me to see and easier to find the rule.

Numbers | Partition | Maxi-product |

12 | (6,6) | 36 |

8 | (4,4) | 16 |

16 | (8,8) | 64 |

13 | (6.5,6.5) | 42.25 |

My Rule:

To find the maxi product you have to take the original number and divide it by 2 that will give you a partition. When you get the partitions you have to multiply both numbers together from this you will get the maxi-product. For instance if i have 16, then i will divide it by 2 then square the number I get by dividing 16 and this is how I will get my answer.

Below is another example i am going to show you.

Number | Product |

10,0 10x0=0 | |

9,1 9x1=9 | |

8.5,1.5 8.5x1.5=12.75 | |

8,2 8x2=16 | |

10 | 7.5,2.5 7.5x2.5=18.75 |

7,3 7x3=21 | |

6.5,3.5 6.5x3.5=22.75 | |

6,4 6x4=24 | |

5.5,4.5 5.5x4.5=24.75 | |

5,5 5x5=25 |

Now i have stopped my table because the answers of the product keep coming the same now.

My Maxi Product is (5, 5) which equals to 64.

Rule in Words:

Maxi Product equals the number selected divided by two and then the answer squared.

Rule in Algebra:Key: M= Maxi Product N= Number

M=

Predict my rule and proving it:

To show that my rule was correct I will do it on a new number. By doing this I can check if my formula is correct. I will also draw a table to find my maxi product in an easier way.

Firstly i will do this with my algebric rule and i predict that the maximum product is 7. Then after i will use the number 7 into my formula m=

M=(7/2) M=(3.5) M=3.5x3.5 M=12.25

M=(3.5)

i have done this with my algebric rule and I can now predict that the maximum product for the number 7 is 12.25 with the partition (3.5,3.5).

Below is a table proving my rule

Partition | Product |

0,7 | 0 |

0.5,6.5 | 3.25 |

1,6 | 6 |

1.5,5.5 | 8.25 |

2,5 | 10 |

2.5,4.5 | 11.25 |

3,4 | 12 |

3.1,3.9 | 12.09 |

3.2,3.8 | 12.16 |

3.3,3.7 | 12.12 |

3.4,3.6 | 12.24 |

3.5,3.5 | 12.25 |

3.49,3.51 | 12.2499 |

Conclusion

Rule in Algebra:Key:M=Maxi product N=Number

M=(N/3)³

RULE FOR MY RECORDINGS:

Number | Partitioning Pair | Maxi- Product | Rule | |

Doubles | 8 | 4+4=12 | 4x4=16 | (n/2) |

triples | 21 | 7+7+7=21 | 7x7x7=343 | (n/3) |

Quadruplet | 15 | 3.75+3.75+3.75+3.75 | 3.75x3.75x3.75x3.75=197.7539 | (n/4) |

Predicting the rule for quadruple.

I predict that in quadruples you have to divide the main number by 4. This will give you a partition. Then multiply it to the power of 4. This will give you the maxi product.

Rule in Algebra:Key:M=Maxi product N=Number

M=

Rule for 5 partition

I predict that in 5 partitions you have to divide the main number by 5. This will give you a partition. Then multiply it to the power of 5. This will give you the maxi product.

Rule in Algebra:Key:M=Maxi product N=Number

M=

Rule for 6 partition

I predict that in 6 partitions you have to divide the main number by 6. This will give you a partition. Then multiply it to the power of 6. This will give you the maxi product.

Rule in Algebra:Key:M=Maxi product N=Number

M=

Overall rule and my conclusion.

I conclude this investigation/coursework.The number you divide by is always then powered by the same number.So the over all rule for this investigation is (x/n)

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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