We have seen that the result of a division does not change if the same sharing is done many times.
All produce the same result.
Using the same wildholder gives the same result as .
In this case the number of boards to be divided must be a multiple of Three, that is any number in the Three times table and the number of counters is the same multiple of Two.
We now have a method of deciding how many boards to play with, given two different fractions.
For example the hand
gives the same result as
Instead of dividing the three counters over five boards we can use any multiple of Five provide we use the same multiple of Three for the counters.
The same is true for Four over Seven.
To be working with the same number of boards we want a multiple of Five that is also a multiple of Seven. This is easy since seven Fives is a multiple of Five and five Sevens is a multiple of Seven and seven Fives is the same as five Sevens which is Thirty Five. So change both to using thirty five boards.
which gives the same results as and now we can add Thirty Fifths.
Thirty Five is not the only multiple that the Five and Seven times tables have in common, Seventy and One Hundred and Five are others. Thirty Five is the smallest multiple that Five and Seven have in common. When looking for common multiples of two numbers we are most interested in the least, we look for the Least Common Multiple or LCM.
In this case the LCM of Five and Seven is found by multiplying them together. This is not always the case. For example the LCM of Fifteen and Twelve is Sixty not One Hundred and Eighty which is fifteen Twelves.
Why is this? Fifteen and Twelve have a common factor Three. 15 = 3 x 5 and 12 = 3 x 4. Using the common factor only once produces the LCM.
15 x 12 = 3 x 5 x 3 x 4 uses 3 twice
60 = 5 x 3 x 4 = 15 x 4 = 5 x 12 using 3 just once still produces both 15 and 12
We are now nearly ready to give a step by step method to add different fractions. just need to do some naming of parts.