Question

Can someone explain why we flip fractions when we divide fractions? I know how to and that multiplication is the opposite of division, but I just can't seem to make sense of it no matter where I look.

Answer 1

\(\large \cfrac{\frac{a}{b}}{\frac{c}{{\color{blue}{ d}}}}\implies \cfrac{a}{b}\cdot \cfrac{{\color{blue}{ d}}}{c} ?\)

so...what are you asking again?

Answer 2

what is 5/5? and what is 5*1/5?

Answer 3

so if you have to calculate a/b, you can do this\[\frac{ a }{ b }=\frac{ a*1 }{ b }=a*\frac{ 1 }{b }\]

Answer 4

@jdoe0001
In words, can you describe what you just posted?

@caozeyuan
Same for you.
:)

Answer 5

if a divides b, we can say a times 1 then divides b, since a times 1 is a

Answer 6

and it is the same as a times (1 divides b), which is a time the inverse of b

Answer 7

one of the skills you need to acquire is to think aabstractly, i.e. do not rely on words but on equations, since equation is less confusing

Answer 8

I don't understand the last step.
@caozeyuan

Also, how does this explain why we flip one fraction?

Answer 9

For example, dividing by 2 is the same as multiplying by 1/2.
Since when you flip 2 (written as 2/1) you get 1/2, to divide by 2, you flip 2 into 1/2 and you multiply by1/2.

Answer 10

Start with a number a, and you want to divide by b, .\(b \ne 0\):

\(a \div b\)

A fraction means a division, so \(a \div b\) is the same as \(\dfrac{a}{b} \)
Let's rewrite our division as a fraction:

\(a \div b = \dfrac{a}{b} \)

From the multiplication of fractions, we know that

\(\dfrac{a}{b} = \dfrac{a}{1} \times \dfrac{1}{b} \)

Our original division, \(a \div b\) can also be written as \(\dfrac{a}{1} \div \dfrac{b}{1} \)

Now we can conclude that \(\dfrac{a}{1} \div \dfrac{b}{1} = \dfrac{a}{1} \times \dfrac{1}{b} \)

Answer 11

@mathstudent55
Your route makes sense to me the most.

I'm confused on a few things, but I'm starting to understand now.

For example, dividing by 2 is the same as multiplying by 1/2.
Since when you flip 2 (written as 2/1) you get 1/2, to divide by 2, you flip 2 into 1/2 and you multiply by1/2.


From the multiplication of fractions, we know that

These two parts confused me.

Answer 12

What grade level are you?

Answer 13

I'm in 10th. I know how to divide fractions, but I want to understand why we divide fractions the way we do. Our teacher never taught us.

We were simply taught when you divide fractions you flip the second fraction and multiply, but he didn't explain why we did this.

Answer 14

So, out of curiosity, I'm trying to understand it.

Answer 15

Ok, give me a minute :-)

Answer 16

Oh thanks! Haha, I was worried someone wouldn't continue.

Answer 17

I think it can be explained simply as @caozeuyan did way above.

\(a \div b = \dfrac{a}{b} = \dfrac{a \times 1}{b} = a \times \dfrac{1}{b} \)

Answer 18

We "flip" fractions because of the definition of division:
$$\Huge a \div b = \dfrac{a}{b} = a \cdot \dfrac{1}{b}$$
This means we must multiply by the reciprocal of b.
$$\dfrac{1}{\dfrac{a}{b}} = \dfrac{b}{a}$$
Proof:
a/b • b/a = a • 1/b • b • 1/a by the definition of division

= a • 1/a • b • 1/b by the comm and assoc prop of mult

= 1 • 1 by the property of reciprocals

= 1 by the identity property of mult

But a/b • 1/ (a/b) = 1 by the property of reciprocals

Thus a/b • 1/ (a/b) = a/b • b/a substitution principle from above

1/ (a/b) = b/a by the division property of equality

Answer 19

@skullpatrol

This means we must multiply by the reciprocal of b.

I'm stuck on this part.

Answer 20

You must multiply by the the reciprocal of the denominator, right?

Answer 21

I'm extremely confused. http://prntscr.com/ahvjqs
I'm not sure where you're getting this form from haha.

I see where the 1/a comes in but I don't know how that =b/a and I don't know what the other b there is for.

Answer 22

The a and b in the large part at the top is just the definition, that we all memorize. Ok?

Answer 23

Alright, so what about this part?
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Answer 24

I am considering the large b in the definition as a/b.

Answer 25

I should have used different letters, sorry

Answer 26

http://prntscr.com/ahvo3z

This one?

LOL why don't we just start over?

Answer 27

Your question was "why do we flip the fraction" when we divide, right?

Answer 28

Yep : )

Answer 29

And you know that to divide by a number you multiply by its reciprocal, right?

Answer 30

Yes

Answer 31

So let's say we need to divide 1 by (a/b).

Answer 32

For example 1/(2/3)

Answer 33

Why can we do that?

1/2/3

Answer 34

Do what?

Answer 35

What does 1 / ( 2/3 ) = ?

Answer 36

$$\Huge \dfrac{1}{\dfrac{2}{3}}$$

Answer 37

Put the numbers in your calculator and find the answer please.

Answer 38

Oh, we can't solve by hand?
@skullpatrol

Answer 39

Sure.

Answer 40

What do you get?

Answer 41

1/1 divided by 2/3

Answer 42

Yes

Answer 43

Sure.

Answer 44

Put the numbers in your calculator and find the answer please.

Answer 45

0.16

Answer 46

You should get 1.5 = 3/2

Answer 47

are you trying to do it or to make sense out of it?

Answer 48

I'm trying to make sense out of it, and I'm extremely confused.

Answer 49

That's weird, I got 0.16 again.

Answer 50

Divide 2/3

Answer 51

2 divided by 3 = ?

Answer 52

Got 0.6

Answer 53

0.666...

Answer 54

Ah, I didn't know I was supposed to reveal the whole thing.

Answer 55

Now take 1 and divide it by 0.666...

Answer 56

can you think of an example of dividing by a fraction?
here is a simple one
how many quarters are in two dollars?

Answer 57

4 quarters make one dollar so I assume 8 would make 2

Answer 58

yes
what did you do? you multiplied 2 by 4

Answer 59

Got 0.666

Yep, that's what I did.

Answer 60

so as you see \[2\div \frac{1}{4}=2\times 4=8\]

Answer 61

want another example?