> Start from deep points TOP> Divide by a fraction
2. When we divide a number by a fraction,
why must we turn that fraction upside down and after then multiply them?
THAT IS,
what we are doing here is to get 1, as a standard unit size.
(Caution: even if you can not understand this sentence, do not give up here.
And please continue to read down.)
(About a standard unit size, please read 1. What is 1 as ratio? in advance.)
In fact, divisions have two meanings.
Many people know one of two, I think.
For example, a calculation in a question,
"If we put 13 balls into baskets by 5, how many baskets will we make?"
13 / 5 = 2 R 3 (R means a division reminder.)
The answer is "We have two baskets and three balls remind."
I show this situation in a figure as below.
Another meaning is to get 1 (a standard unit size).
When we calculate 12 / 3 = 4,
we think 12 as 3 times of a standard unit size
like below figure.
I show a standard unit size as 
(in a left circle, "1 (one)" is there, not "I").
Then a standard unit size is found as 4.
This situation means,
"If we divide something into 3,
we can see one of size."
Let's think about a situation of dividing by a fraction.
How can we think about
?
We can understand 12 as 
.
Then we can find by thinking like below figure.
So, as shown here,
we can turn a fraction upside down and after then multiply them
to find this answer.
An idea seed of this problem is
"We make it into a smallest unit, after then we bring together to 1."
< Additional Writings >
When we want to get in a situation that 12 as ,
we have another way to think about getting a standard unit size
for whom have studied ratios, such as A : B.
So, Anyway,
when we want to seek as a standard unit size,
we also divide.
No. 16 in high school questions,
I think about division from another aspect again.
If you want to know, let's go.
(I recommend you to think about it by yourself first.)
2017.09.29 (GMT)
"4 third" in < Additional Writings > was changed to an icon.
