2. Dividing Fractions
- How many bottles can you divide?
- To begin with, please look at the Figure 1.
Coach Dragon brought one tank containing 12. into isotonic drink, and many bottles containing 3/2(1.5). for today's friendly match. And he said, "Let's fill these bottle with drink."
Figure 1
Novy "I am glad with your kindness, Coach Dragon. But if 12. divided by 3/2, then it can fill only two persons with a drink. Because, twelve divided by 3/2 makes two."
Snake "You are so silly, Novy. It's equal 8 parsons. because it should just 12. divided by 1.5."
Selica "I also think I am sure the answer is 8 parsons. Though, it is easier to calculate with a fraction like --- 12 div 3/2 = 12 by 2/3 --- rather than it calculates it, after it reduces 12 div 1.5 to a common denominator to 120 div 15."
Novy "Yes. Of course I know it. But, I was only mistaken in some calculation methods just for a moment! It's 12 div 3/2 equal 2... What? Well..."
Jason "The dividing fractions needs to make a denominator and a numerator reverse, and needs to make them multiplication. Probably, you have learned from your teacher. And he have said you must make it like that."
Novy "But, why does it make a denominator and a numerator reverse and make them multiplication? Why?"
Jason "You are so noisy! That is such a rule."
Novy "Why? I don't understand. Help me coach Dragon!"
- Why does it make a denominator and a numerator reverse and make them multiplication?
- Why does it make a denominator and a numerator reverse and make them multiplication? In order to understand this thing, you have to know "two things" in advance.
Firstly, A div 1/B is equal to A by B. No matter A and B may be what numbers, they are OK.
Please look at the the Figure 2.
Fig. 2 is the example of -- 6 div 1/3 = 6 by 3 --. But, even if this expression changes a number, it is the same thing. Let's actually write to a scale and try many things. Even it is a number which cannot be divided, you can understand that it is about right, I think.
Figure 2
Secondly, A div C/B is equal to A div 1/B div C. No matter A, B, and C may be what numbers, they are OK.
Please look at the Figure 3.
Both 6 div 2/3 and 6 div 1/3 div 2 are equal to a number which divided 6 div 1/3 by 2, aren't they?
Figure 3
- The reason for making a denominator and a numerator reverse
- It means that preparation was completed now. Do you understand it already? Why does it make a denominator and a numerator reverse and make them multiplication? Please consider that a little by yourself.
What is the reason why is A div C/B equal to A by B/C?
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Well, I explain it.
a. As mentioned the 2nd reason of above, 12 div 3/2 is equal to 12 div (1/2 + 1/2 + 1/2). Therefore it is equal to 12 div (1/2) div 3.
b. As mentioned the 1st reason of above, 12 div (1/2) is equal to 12 by 2. Therefore 12 div (1/2) div 3 is equal to 12 by 2 div 3.
c. 12 by 2 div 3 is same as 12 by (2/3). Therefore, 12 div 3/2 of the 1st expression and 12 by 2/3 is equal.
Do you understand my explanation? That is, what is necessary is just to change dividing fractions into that method "as a result". If you know that it becomes so, you may calculate the expression mechanically. However, if you sometimes look back upon why it becomes so and do not consider, you will forget an essential thing!
Although you read at once, if you did not understand, please read one line at the time carefully once again and understand it.
One-Point Advice (1) The number which made the denominator and the numerator reverse is called "a reciprocal number" of an original number. The reciprocal of 3/2 is 2/3. And the reciprocal of 5 is 1/5. If the reciprocal of A is A', it becomes the relation of A by A' = 1.
One-Point Advice (2) It can be shown algebraically that dividing fractions and multiplication by the reciprocal are equal.
12 div 3/2 = 12 div 3/2 div 1
If the reciprocal of A is A', it becomes the relation of A by A' = 1. Then,
= 12 div 3/2 div 3/2 by 2/3
Since 3/2 div 3/2 is the division of the same values, this result is 1, isn't it? Therefore,
= 12 div 1 by 2/3
= 12 by 2/3
Brilliant! You were able to get the answer for which you wanted to ask.
- Let's solve the Exercise.
- Then, let's solve the Exercise 1.
Exercise 1
[Question]
Solve the following dividing fractions.
(1) 15 div 5/3
(2) 15/1 div 5/3
(3) 15/3 div 5/3
(4) 15/5 div 5/3
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[Answer]
(1) 15 div 5/3 = 15 by 3/5 = 9
(2) 15/1 div 5/3 = 15/1 by 3/5 = 45/5 = 9
(3) 15/3 div 5/3 = 15/3 by 3/5 = 45/15 = 3
(4) 15/5 div 5/3 = 15/5 by 3/5 = 45/25 = 9/5
How was it? Even if the 1st number which is divided is either an integer or a fraction, it is a same calculation method, isn't it?
If you didn't just come to you, please compare the expression and the answer of from (1) to (4) in Exercise 1.
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Novy "I see. So, dividing fractions is able to convert into multiplication. That is, since 12 div 3/2 is equal to 12 by 2/3, 24 div 3 is ... Well, what is that?"
Snake "It's better for Novy to study standard division before dividing fractions!"
Jason "HAHAHA! It does not change but Novy is silly."
Novy "Boohoo. Coach Dragoooooooooon !"
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