First of all sorry if this question sounds too stupid or offends anyone.
One apple divide by two you get half an apple. $\large{\frac{1}{2} = 0.5}$
I couldn't get my head around with dividing fractions...
Half an apple divide by half an apple you get one apple? $\large{\frac{0.5}{0.5} = 1}$
How do I make any sense out of it? Any real life scenarios / applications that I can relate or explain dividing fractions?
On
The English idiom "divide by half" is confusing you; there is nothing that requires division by a fraction here. Let's rephrase in a clearer way:
$$“\text{half of}” \;X \;=\;\frac{1}{2}\cdot X \;=\; \frac{5}{10}\cdot X \;=\;0.5\cdot X$$ Therefore $$“\text{half of}” \;(\text{apple}) \;=\;\frac{1}{2} \cdot \text{apple}\;=\; \frac{5}{10}\cdot \text{apple} \;=\;0.5 \cdot \text{apple}$$ and therefore $$\begin{align*} “\text{half of}” \;(\text{“half of” (apple)}) \;&=\;\frac{1}{2} \cdot \left(\frac{1}{2}\cdot\text{apple}\right)\;=\; \frac{5}{10}\cdot \left(\frac{5}{10}\cdot\text{apple}\right)\;=\;0.5 \cdot \left(0.5\cdot\text{apple}\right)\\\\ &=\;\frac{1}{4} \cdot \text{apple}\;=\; \frac{25}{100}\cdot \text{apple} \;=\;0.25 \cdot \text{apple} \end{align*}$$
On
Well for your first example. If we divide one apple on two person , each one will get half an apple , that's when you get $$\frac{\color{red}{an \space apple}}{two \space persons} = half \space an \space\color{red}{apple} \space each$$
Now to understand $\large{\frac{0.5}{0.5} = 1}$ , consider that there is 100 rooms in a student residence building and a 100 students. Assuming that each student can only get one room , if we divide $50$ rooms (half the number of rooms) on $50$ students (half the number of students) , then each student will still get $1$ room. Does it make sense now ?
It is basically same as saying divide the number of rooms on the students.
That's why it is 1.
On
Stay within the set of natural numbers first. You must be familiar with the operation called addition, and its inverse operation called subtraction. Addition is clearly defined. The subtraction however is trickier. When we say $$5-3=x$$ we ask a question: What do we have to add to $3$ in order to get $5$?
The same can be told if we define the inverse operation of multiplication. What does the following script mean:
$$36 \div 4=x?$$
It means that we are looking for a number which will give $36$ if we multiply it by $4$. The answer is $9$. There is another convention to ask the same question: $$\frac{36}{4}=x.$$ The convention is as follows: In the denominator we have the number with which we will have to multiply $x$ in order to get the number we have in the numerator.
Now, what if we step out from the set of natural numbers? We may have similar equations. For instance we may ask the following one: by what number do we have to multiply $0.5$ in order to get $0.5$? The answer is obviously $1$. However if we follow the convention introduced above this latter question is asked the following way: $$0.5 \div 0.5 = x \text{ or } \frac{0.5}{0.5}=x.$$
The answer is still $1$. If we write $\frac{1}{2}$ for $0.5$ then our humble equation may look a little more frightening:
$$\frac{\frac{1}{2}}{\frac{1}{2}}=x.$$
First we have to understand the meaning of $$\frac{1}{2}=y.$$ In other words, first, we have to answer the following question: By what number do we have to multiply $2$ in order to get $1$. It turns out that this number is $\frac{1}{2}$. So the meaning of $\frac{1}{2}$ splits now. (a) multiplying $2$ by $\frac{1}{2}$ we get $1$ or (b) dividing $2$ by $2$ gives $1$. The best is if you try to keep in mind that division is the inverse operation of multiplication. If you stick to the "take apart" concept of division then it becomes nonsense if the divisor is not a natural number.
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One of the main problems people have in understanding math (and I am one of them) is that those who do understand math sometimes forget how to translate between mathematical language and regular spoken language. I find this in computer techs too; they get so used to working with computers, they forget how to communicate with people. I worked in computers all my life trying to help bridge that gap.
I was failing math until I met a teacher in high school who knew how to speak in both languages... math and people. I have tried all my life to remember that lesson-- and taught many people how to use computers by remembering how to speak plain English.
In the initially cited example the important point is (as someone mentioned prior) we aren't dividing 1/2 apple by 1/2 apple. We are dividing 1/2 apple by 1/2. That is much easier to visualize when we think of it reversed: what is 1/2 of 1/2 an apple? The answer becomes immediately clear: 1/4 of an apple. We can do that in our head. But that is using multiplication: 1/2 apple x 1/2. It's understandable that 1/2 apple / 1/2 is likely going to give a different result... even if it doesn't make sense in every-day application. (And there's a reason for that, which follows.)
Doing math in our head isn't always possible, especially when working with numbers that aren't as simple. So "inversion of fractions" is the written method we use to reach the goal. While we can easily visualize half of a half of an apple, what about 5/10ths of 1/15th of an apple? That's when it becomes tricky. So we solve it by using the "inversion" method: inverting the second fraction and multiplying it by the first. When dividing the key to remember is we aren't dividing one object by another object; we are dividing an object by a number. That's where most people get confused. When we realize we're not dividing apples by apples but rather, apples by a number, the concept falls into place.
That's the difference between being able to DO math and TEACH math (and why good teachers really should be better paid). :D
I think part of the trick here too is the very concept of division of fractions... which some will argue is in reality multiplication of inverted fractions. Semantics, semantics... it just doesn't always fit into our brains looking for a common-sense example. In the case of the apple we have to see beyond the obvious... that it can be either 1/2 of an apple, or 1 whole PIECE of an apple. When we convert it from 1/2 to 1 through simple observation... the reason for the seemingly wonky math becomes easier to understand. 1/2 apple / 1/2 doesn't miraculously become 1 whole apple... it becomes 1 unit of apple. Then we start studying quantum reality-- how can it be 1/2 an apple and 1 whole piece at the same time? Questions upon questions...
You don't divide the onw half apple by one half of an apple. You divide one half of an apple by one half, just like you don't divide one apple by two apples, you divide one apple by two.
That said, the easiest way to understand division by fractions is to re-look at what division really is. When you are dividing $1$ by $2$, you are asking
Meaning you are solving the equation $2\cdot x = 1$ for $x$, and the solution of that equation is $x=\frac12$.
Similarly, calculating $\frac{\frac12}{\frac12}$ is the same as solving the equation
$$\frac12 \cdot x = \frac12$$ and it is clear that $x=1$ solves this equation.
As far as dividing fractions in real life, think about traveling with a car. If you drive $2$ miles per hour, how long will it take to travel one mile? It will take $\frac{1}2$ hours, of course. The time $t$ it takes to travel a distance $s$ is equal to the distance, divided by the speed at which you are traveling:
$$t=\frac sv$$