How to logically interpret the division of small number by large number on a number line?

Question

When we divide a number by another number ($x \div y$), we can interpret it in two ways:

1. $x$ is divided in equal groups, where each group consists of $y$
2. $x$ is divided in $y$ equal groups

Suppose we divide $500 \div 5$. We can interpret this in two ways:

1. $500$ is divided in equal groups of $5$
2. $500$ is divided in $5$ equal groups

Now, if we divide $5 \div 500$, how can we interpret it in the same way? I am a beginner at math and I want to understand this in a simple manner on the number line. I would be grateful for your assistance. Thank you

Answer 1

Let's interpret $5\div 500$:

1. $5$ is divided into $500$ equal groups. Here you need to think of the $5$ as something that is not limited to being an integer. Don't think apples, or people. Maybe kilos of gold, or liters of water, or cups of flour, or something. In that case, it is easier to imagine dividing it up into many small groups, and we see that each group is alotted $5\div 500 = 0.01$.

2. $5$ is divided into groups, each group containing $500$. This one is, as you say, a little tricky. But it's still doable. You are unable to finish a single group. How much of a group do you get? You get $5\div 500 = 0.01$ of a group.

Maybe interpretation 2 is a little easier to imagine when the numbers are a bit closer. Like, for instance, $5\div 10$. If you try to make groups of 10, and you only have 5 available, it is difficult to dispute that you will get half a group. And one half in decimal is $0.5$. Thus $5\div 10 = 0.5$.

Note that these two interpretations are complementary. If you say that $10\div 5 = 2$ according to one of the interpretations, then you automatically have $10\div 2 = 5$ according to the other interpretation. This is also true for something like $5\div 500 = 0.01$ versus $5\div 0.01 = 500$.

Answer 2

Humans have a tendency to quantify most of the things,For example,We knew how much Food we need for a man to suffice his hunger,We quantified food,We quantified all sorts of things in human history,When We as humans started to think That How to divide $50$ apples in $50$ people,We naturally got to the conclusion that Each one should Get a whole apple,How about $60$ apples in $12$ people,Each one would then get $5$ apples

But the question then was how to divide An apple into two people so that no one fights for it?Not so easy, We take a knife and divide the whole 1 apple into 2 equals,What is the intuition here,What do I mean?How can We divide an Apple into two equals?in what sense are we talking about equals?

Here equals can mean, in equal volume,Equal geometrical shape,Equal parts such that each part completely suffices each individual's hunger,The intuition here is that what we can see that things are equal might not be equal in different senses,What if apple had a small void on one side and had dense flesh on one side,Would you still call the two geometrically equal parts,Equal?

Hence,We can Either think Numbers and their operations in terms of some abstract ideal objects that are continuous or we can use some more useful definitions

Premise

As humans,intuitively,We made concepts like length only because we had senses which gave us external stimuli,Like When we walked a particular distance,We sensed that we can move back and forth,Giving us a sense of length, Now how do we had intuition that it was quantifiable?Easy,Walk a mile first and walk two miles,You will feel more tired walking two miles,Which means length is indeed quantifiable,Earliest quantities to measure them was distance between shoulders and tip of Finger,Which varied differently for different people,These differences and unfair measurements is what led us to Unified systems of measurement,But that's another topic

We can now abstractly use concept of length to define numbers without associating it with some ideal Quantised quantity like matter, as it really has Fundamental particles like atoms and more,Hence we can Imagine one dimension,Which has measurable length,Called the number line

Take a reference point as origin,Now define any length as 1 unit of that line,imaginarily mark one by one Quantised 1 unit parts,Like 1 units,2 units ,3 units etc..,Here we go,We have defined all Natural numbers,Can you guess, We can mark any number of points and name it a different number,As all lengths are still different,What now, How does this relate to Division, Well now, You can mark exact half distance of 1 unit, mark it as another non-natural number, What is it then? A fraction maybe, Let's name it hither,How do we define a Hither now,Hither is a number defined such that It is the marked point whose Distance on defined number line when traversed two times led us to mark point "$1$"

In fact hither is indeed $\frac{1}{2}$, Which is a mark point on number line whose length is such,When traversed two times led us to mark point "$1$"

EXPLANATION

Now when performing mathematical operation of division of two natural numbers a and b,a÷b is a number such that the distance of its mark point on number-line when traversed "b" times led us to mark point a

Now,I know this gives a general sense that Multiplication is just a repetitive addition,and you can be further confused how will be then define mathematical operation of 4÷6.5, note,we cannot travell a distance 6.5 times, but that is discussion for another time, But you can have an idea that dividing a smaller number by bigger number is indeed meaningful, For your case, 5÷500, is a mark of number on number line such that it's distance when traversed 500 times led us to mark point of 5, Hope this helps you

That's where moved from wholes to parts, The definition of division then was simple, Quantity required to

Answer 3

The English-language utterance “five hundred divided by five” is not only the way we speak the mathematical expression “$500 \div 5$”, it is also the way we implicitly interpret it—often unawares.

Imagine a group of five people encountering a stack of, say, five hundred 1-pound notes. If they decided to share that cash evenly, we would say that that money was divided by the people who found it. Of course this dividing, or sharing, would separate the notes into five groups of one hundred.

Likewise, if a company employing 500 people decided to liquidate its assets to be shared equally by its workforce, and the value of those assets amounted to exactly 5, oh I don’t know, million pounds, then the 500 employees would divide all 5 of the million-pound portions among themselves. This would effectively create 500 portions of value £10000 each.

Answer 4

Interpretation 1:

a. You have $500$ items going into boxes that hold $5$ items each. You need $100$ boxes to accommodate the $500$ items. So, $500 \div 5 = 100$

b. You have $5$ items going into boxes that hold $500$ items each. You need one one-hundredth of a box to accommodate the $5$ items. So $5\div 500 = 0.01$.

Interpretation 2:

a. You have $500$ pounds of flour to divide equally among $5$ people. Each person gets $500 \div 5 = 100$ pounds of flour.

b. You have $5$ pounds of flour to divide equally among $500$ people. Each person gets $5 \div 500 = 0.01$ pounds of flour.

Answer 5

Suppose we divide 500÷5. We can interpret this in two ways:

a) 500 is divided in equal groups of 5 b) 500 is divided in 5 equal groups

Lets do a practical examples of the two statements:
a) 500 apples (split into groups of) 5 apples (is equal to) 100 groups.
b) 500 apples (shared equally between) 5 people (is equal to) 100 apples for each person.

Lets do the same for 5÷500 = 1/100 = 0.01:
a) 5 apples (split into groups of) 500 apples (is equal to) 0.01 groups.
b) 5 apples (shared equally between) 500 people (is equal to) 1/100 of an apple for each person.

When expressed like this you can see that the (b) interpretation is much more natural and intuitive interpretation that works well either way round while the (a) interpretation is unnatural and not intuitive. Eg what does having 0.01 of a group mean?

I think its best to think in terms of the (b) definition and discard the (a) definition to avoid confusion.

There may be some confusion with the term division in the context of integer arithmetic. For example we might say 5 divides 500, 100 times which means how many groups of 5 are in 500 and the answer is 100 groups. In integer arithmetic we can only accept integer results of groups and possibly a remainder. If we ask the question does 500 divide 5, the answer is not 0.01 or 1/100 but is "no" or "false". If the divisor is greater than the numerator then the answer is that 5 is not divisible by 500. If we ask "Does 5 divide 500 ?" in integer arithmetic the answer is yes and if we ask if 5 divides 23 the answer is no, because we cannot divided 23 into an integer number of equal integer groups of 5.