Fractions are Hard!

2.3 Changing the amount of pie per girl

Of course girls get pie too! Consider the sharing problem:


Share \(a\) pies equally among \(b\) girls.


In a sharing problem, how might you double the amount of pie each girl receives?


Answer: Double the number of pies!


Let’s translate this our mathematical notation:


\(\dfrac{a}{b}\) is the amount of pie each girl receives in the sharing problem “share \(a\) pies among \(b\) girls.”

\(2 \times \dfrac{a}{b}\) is double this amount of pie.


But we just concluded that we can double the amount of pie per girl by sharing \(2a\) pies among the \(b\) girls instead. That is, we concluded that \(\dfrac{2a}{b}\)  gives the same answer \(2 \times \dfrac{a}{b}\). We have


\(2 \times \dfrac{a}{b}=\dfrac{2a}{b}\).


In the same way, to triple the amount of pie each girl receives in a sharing problem, just triple the amount of pie:


\(3 \times \dfrac{a}{b}=\dfrac{3a}{b}\).


And so on. (For example, \(4 \times \dfrac{7}{3}=\dfrac{28}{3}\). )


BASIC MULTIPLICATION BELIEF:     \(x \times \dfrac{a}{b} = \dfrac{xa}{b}\) . 

To change the amount of pie each girl receives by a factor \(x\) , change the number of pies by that factor  .


This belief is natural and right for positive whole numbers. It feels like it should be true for all types of numbers.


Comment: We like to believe multiplication is commutative. I guess then we should also choose to believe that \(\frac{a}{b} \times x = x \times \frac{a}{b}\), which equals \(\frac{xa}{b}\).



We can practice the idea of this belief with pictures too.


For example, in looking at two copies of  \(\dfrac{3}{5}\) we really do see \(\dfrac{6}{5}\).


We see three copies of \(2\dfrac{1}{2}\)  as this.


This is literally \(6 + \dfrac{3}{2}\)  (six whole pies and three half pies), and this matches our mathematics so far:


\(\left(2 + \dfrac{1}{2}\right) \times 3 = 6 + \dfrac{1}{2} \times 3 = 6 + \dfrac{3}{2}\).


We can also see that two half pies combine to make a whole pie, and so \(6\dfrac{3}{2}\) is equivalent to \(7\dfrac{1}{2}\), if you like.


Check: Can you think your way through \(5 \times 2\dfrac{1}{3}\)?

EXTRA: What sharing problem does \(2\dfrac{1}{3}\) represent? How many pies? How many girls?

How many pies do you need if you want to ensure that each girl now gets five times the amount of pie?



Thinking Question:

Suppose I told you in a sharing problem with seven boys, each boy received three pies. How many pies, in total, were there to begin with?


Logic tells us that there 21 were  pies. Is it a coincidence that \(3 \times 7 = 21\)?

In general, if \(\dfrac{a}{b}= n\) , must it be the case that \(a = n \times b\)?


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