0.5 Divided by 1, stages to answer?

Question

It may seem pretty simple, but it just doesn't make any sense to me. Dividing any positive decimal under 1 by 1 but just getting the same decimal back? I would like it if someone explained to me how this works.

Answer 1

Students are told:

1. when you divide by a number greater than 1, the answer is smaller than what you started with, and

2. when divide by a number less than 1, the answer is larger than what you started with.

If you're sharp, you'll notice that these statements omit 1 itself. What happens if you divide by a number exactly 1? It's the same as multiplying by its reciprocal. But the reciprocal of 1 is 1! So dividing by 1 is the same as multiplying by 1. And by now you should know that 1 is called the multiplicative identity, which is how graduate students express the fact that multiplying by 1 does not change the other number: $a \times 1 = a$. Therefore, since multiplying by 1 is the same as dividing by 1, then dividing 0.5 (or any other real number) by 1 will be 0.5.

Does that help?

Answer 2

Think of division as a question: "$a\over b$" is asking, "What do I need to multiply by $b$ to get $a$?"

Now, $1$ has the lovely property that it makes multiplication boring: $1\cdot x=x=x\cdot 1$. So anytime I ask "What do I need to multiply $1$ by to get $x$?", the answer is always . . . $x$! If I multiply $1$ by something $z$ other than $x$, I'll get $z$, not $x$.

So ${0.5\over 1}$ is the number you need to multiply $1$ by to get $0.5$. That's just $0.5$ itself!

Answer 3

Division is the opposite of multiplication, so the two equations $$ c = a \times b $$ and $$ \frac{c}{b} = a $$ say the same thing (as long as $b \ne 0$).

When $b=1$ each equation says $c=a$. There's nothing special happening when $a$ is less than $1$.

Answer 4

Look at the property $${{a\over b}\over {c\over d}}={a\cdot d\over b\cdot c}$$

Then look that $0.5={1\over2}$, so if you replace it, it will be like $${{1\over 2}\over {1\over 1}}={1\cdot1\over 1\cdot2}={1\over 2}$$

Answer 5

Think of it this way. You have $31$ oranges and you need to put them into $6$ bags. How many oranges go into each bag? That is the practical meaning of $31 \div 6$. $x \div y$ means if we divide $x$ into $y$ equal parts, how big is each part?

Okay, what if we want to put $x$ into $1$ bag? Just !$1$! bag. Well, that $1$ bag gets .... everything. $x \div 1 = x$. You have just one bag in which to put everything so... that one bag will have everything.

So you have $1/2$ an orange. You put it into $1$ bag. How many oranges are in that bag? ... Um. $1/2$... right?