## 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? $$\dfrac{??}{7}=3$$ 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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