Division and Factors

A multiplication puzzle like this

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Maths 6x = 12

Can be thought of as a puzzle where you are given a number 12, a factor of Twelve, 6, and you have to find the paired factor x.

For simple division it is enough to know your times tables to see that x =2.

However the puzzle below is not so straight forward as we are unlikely to know our Eighteen times table.

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Maths 18x = 54

A knowledge of factors and multiples can help.
First let us examine the simpler 6x = 12 by sharing apples again.

Twelve apples shared between six boxes.



This problem can be shared between three people each carrying four apples and two boxes.




Each person has four apples to share among two boxes.
This is possible because the number of apples 12, and the number of boxes 6 have the factor 3 in common.

The result is two apples in each box .

Let us return to the problem 18x = 54 by sharing out apples.

Here we have fifty four apples to distribute over eighteen boxes.



We might not know the Eighteen times table but could know that Fifty Four is in the Six times table and the Nine times table and also that Eighteen is in the Six times table and the Nines times table.

Six and Nine are common factors of Fifty Four and Eighteen.

Since Nine is larger than Six we work with Nine as the common factor.

9 x 6 =54 and 9 x 2 = 18

Get nine people to each carry six apples and two boxes.
   
   
   

Each person shares six apples over the two boxes giving three apples in a box

So Fifty Four divided by Eighteen is Three, 54 ÷ 18 = 3.

The Game of Maths

We start with a game with eighteen boards and play fifty four counters shared over the eighteen boards.



This produces the same result as



Then instead of one player playing Fifty Four over eighteen boards allow nine players to split the game between them so now each person is playing Six over two boards.

which gives the result

An important point to note is that the hand can be obtained from by removing the Nine holders for the counter cards and the board cards. This is called cancelling the multiplication.

We record cancelling

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Maths