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When I see a m/n equation, I thought it as the value of each part of m if m gets divided n number of times. For, 10/2, 10 to be divided into 2 parts so each part holds value equal to 5. 10/1 implies 10 to be divided into only 1 part so that single part holds all the value that is 10.

But, for 10/0.01, I cannot decipher the logic of dividing into 0.01 parts. Can someone help me to explain these in words along the same lines of dividing the number into those many parts.

wholesome
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  • Related question on another stackexchange: [How to explain the flipping of division by a fraction?](https://matheducators.stackexchange.com/questions/7837/how-to-explain-the-flipping-of-division-by-a-fraction/7868#7868) – Stef Aug 23 '23 at 16:44
  • And this is a different question, but its answers appear to address your interrogation as well: [How do you explain division by zero?](https://matheducators.stackexchange.com/questions/5648/dividing-by-zero/5947) – Stef Aug 23 '23 at 16:49
  • "The population of Luxembourg is 0.01 of the population of France. If I give one cake to France and 10 cakes to Luxembourg, how much cake will a Luxembourger eat relatively to a French?" The answer is 10/0.01 = 1000. The share of cake eaten by a Luxembourger is 1000 bigger than the share of cake eaten by a French. – Stef Aug 23 '23 at 16:56

4 Answers4

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X / Y = Z and X / Z = Y ought to be equivalent. Y and Z play a symmetric role in these equations, because these equations are also equivalent to X = Y × Z and X = Z × Y.

Depending on how you look at it, maybe Y is the number of parts and Z is the size of each part, or maybe Y is the size of each part and Z is the number of parts.

So, instead of reading "10/0.01 is 10 divided into 0.01 parts", you can say: "10/0.01 is the number of parts of size 0.01 that can fit into 10".

And indeed, it is true that if you have an empty bottle of volume 10L, and a small cup of volume 0.01L, then you can pour 10/0.01 = 1000 cups of water into the bottle.

Alternatively, if you insist on 10 being a number of cakes and 0.01 being a number of shares:

The population of Luxembourg is 0.01 of the population of France. If I give one cake to France and 10 cakes to Luxembourg, how much cake will a Luxembourger eat relatively to a French?

The answer is 10/0.01 = 1000. The share of cake eaten by a Luxembourger is 1000 bigger than the share of cake eaten by a French.

Stef
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  • Absolutely awesome. Not going good with the cake example but yeah from the explanation above that. Now I can frame the sentence of M/N as = Number of parts that will be formed if M is divided so that every part has a value as N(for N<=M) and M/N = Value of each part if M is divided into N parts (for N>M). Does that make sense? – wholesome Aug 23 '23 at 17:18
  • @wholesome Well, my point was that "number of shares" and "size of share" are interchangeable. They're both quantities. Imagine X is the area of a rectangle. The rectangle can have base Y and height Z, or base Z and height Y, you can say that it's made of Y columns of size Z, or that it's made of Z rows of size Y, it's all equivalent. – Stef Aug 23 '23 at 17:57
  • @wholesome How about: 1 Luxembourger-population = 0.01 French-population; 1 Luxembourger-cake = 10 French-cakes (Luxembourger-cakes are **big**!!). How much cake does one Luxembourger eat? Answer is 1 Luxembourger-cake / 1 Luxembourger-population = 10 French-cakes / 0.01 French-population = 1000 French-shares. – Stef Aug 23 '23 at 17:59
  • Yeah got it that number of shares and size of share are interchangeable. That's why I put it that way to make it more analytical. – wholesome Aug 24 '23 at 09:46
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Since 0.01 = 1/100, we have

10/0.01 = 10 / (1/100)

Now we know that X/Y = X*(1/Y). Hence X/(Y/Z) = X*(Z/Y)

This gives us 10/(1/100) = 10*(100/1)

Since 100/1 is obviously 100, we have

10/0.01 = 10*100 = 1000
user1934428
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  • I know the math. Just wanna know any explanation of dividing a number into 0.01 parts. – wholesome Aug 23 '23 at 06:26
  • But the mathematical rules **are** the explanation! This is how multiplication and its inverse (division) are defined. What else are you missing? I think your misunderstanding starts by thinking of "a whole dividing into parts". This is how we teach basic arithmetic in primary school, so that the kids can understand it. But as soon as we teach to our children in high school rational and irrational numbers, this simplified picture is not used anymore. – user1934428 Aug 23 '23 at 06:30
  • For instance, we learned in high school the division first by defining inverse relations, and then more formally (in the context of set theory) as an [algebraic ring](https://en.wikipedia.org/wiki/Division_ring), which satisfies certain properties. There was no need to picture a "whole and its parts". – user1934428 Aug 23 '23 at 06:33
  • *"Now we know that X/Y = X*(1/Y). Hence X/(Y/Z)"*... = X*(1/(Y/Z)), but that doesn't seem to help ;-p – Stef Aug 23 '23 at 16:36
  • @user1934428 doesn't specifically answer my question though. – wholesome Aug 23 '23 at 16:41
  • @wholesome: I think the question does not make sense. Dividing a physical thing into parts is one idea. Talking about mathematical division is a different idea, which is related to your "divide-into-parts" only in that far, that if you have a mathematical division M/N, **and** M and N are whole numbers, **and** M>N, the mathematical division happens to solve the "divide-into-parts" problem. For a similar reason, you can describe the square root of a number geometrically by a square of a certain size, but this description doesn't work if anymore if you take the square root of a negative number – user1934428 Aug 24 '23 at 05:50
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A division by a number between 0 and 1 is equivalent to a multiplication by its reciprocal.

Example:

10 / 0.01 = 10 / (1 / 100) = 10 * 100

Petra Minaler
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  • Dividing a number into 0.01 parts can this be logically explained? – wholesome Aug 23 '23 at 06:26
  • I don't know if this helps, but a real world question which uses this kind of calculation could be: If one person in a group of 100 has 10 cookies, how many cookies in total would be required for everybody in the group to have 10 cookies? – Petra Minaler Aug 23 '23 at 07:11
  • Yeah got it. But I want to think it that way into how many parts I can divide. – wholesome Aug 23 '23 at 16:40
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both of them multiply 100 10 / 0.01 = 1000 / 1 = 1000

Jiu_Zou
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  • I know the math, but I am trying to understand is there any logical explanation of dividing a number into 0.01 parts. – wholesome Aug 23 '23 at 06:25