The division of a fraction - Whole or Part?

Question

My example is 1/2 divide by 8.

The correct answer given : 1/16

However, in my mind this answer is 1 divide by 16, which to me is not what the question is asking.

When I construct the argument, it seems it should equal 1/8 as the question is asking "What is the division of 1/2 into 8 pieces?"

Not

"What is the total division of the whole when 1/2 is divided into 8 pieces?"

Why give the answer to the whole, when it is the fraction that is in question?

If I have a box, and I cut it into half, and then I ask "What is the division of this box into 8?" Why would I give a division of the whole box?

Edit

Thank you for the many answers, I believe it boils down to absolute quantity vs relative measurement, and the mass of the object.

So, 1/3 of half a tea spoon is an absurd but correct measurement.

but, say you have a house divided into 2 apartments, both having 8 rooms.

You would say : "An apartment has 8 rooms" not "The house has 16 rooms"

Edit

Absolute measurement, as a method of reaching an answer that is limited in complexity, is preferable to an infinite number of sets of fractions.

1/18 = Absolute Measurement

1/8 = Sets of fractions into infinity

UPDATE EDIT Just an update for visualisation.

Imagine one whole cube, then cut that cube 8 times, now divide that cube into half and you have 16.

I believe this is what the sum is describing and the problem with my visualisation.

This has really helped me appreciate set theory, thank you for your time.

Answer 1

If I have a box, and I cut it into half, and then I ask "What is the division of this box into 8?" Why would I give a division of the whole box?

If I'm understanding your question correctly, you cut a box in half, giving you two boxes, each with half the size of the original. You then take one of those smaller boxes, and cut it into eight equal pieces. The question is "What is the size of a piece?" The answer is $\frac{1}{16}$ (by which people mean, of course, one sixteeth the size of the original box).

And what you are asking is, "Why don't we say that the answer is $\frac{1}{8}$ (by which you mean, of course, one eighth the size of the half box).

You are saying, "Well we already cut the original box in half. Why are we still comparing things to the original box? Why not to the half box?"

Is this what you are saying?

Answer: It's just different ways of saying the same thing. You are correct that the size is one eighth of the half box, and it is also correct that the size is one sixteenth of the original box. If you accept this, the question becomes why everyone except you wants to say that the answer is $\frac{1}{16}$ (and why you should too).

I would say it's because half boxes are stupid. You already have a perfectly good box, which you originally started with, and to which you can compare anything else that you encounter. You are right that the thing you are looking at is one eighth of a half box, but we already have a perfectly good standard of measurement: the box. Not the half box, but the box.

Imagine insisting that a $300$ lbs person acktuallee weighs $600$ half pounds.

Answer 2

Suppose you have a square which is $1$ unit by $1$ unit, for an area of $1$ square unit. Now you take half of the square, which is $1/2$ square units.

Next you are told to divide this half square into $8$ pieces. Each will have $1/16$ square unit of area. So $1/2$ divided by $8$ is $1/16$. Sure they all have $1/8$ the size of the half square, but that is a relative measure. $1/2$ divided by $8$ is an absolute quantity.

Also, if you claimed $1/2$ divided by $8$ should be $1/8$, then $1/3$ divided by $8$ should also be $1/8$. But then you would have $\frac{1/2}{8}=\frac{1/3}{8}$, so $\frac{1}{2}=\frac{1}{3}$.

Answer 3

Interpret the question as follows: we want to find $x$ such that "$8$ lots of $x$ equals a half", which we can write as: $$8x=\frac{1}{2}$$

This gives us $$x=\frac{1}{2}\cdot \frac{1}{8}=\frac{1}{16}$$

Answer 4

Why give the answer to the whole, when it is the fraction that is in question?

The final answer to the division question is "What is the value each part gets". In your example, each part of the 8 parts gets $0.0625$ (as Algebraically explained by other comments and answers in this post), which may be written as $1/16$.

Answer 5

When using real numbers, we typically call the unit $1$. It convenient to compare things to $1$.

Half of $1$ for example is $\frac12(1)=\frac12$.

Now if we consider a quantity, $a$, where $a\ne1$, then half of $a$ is $\frac12a$.

Clearly $\frac12$ differs from $\frac12a$, both in value and in symbolic representation. As it should! We have been working in a single system of units.

You seem to be arguing that $\frac12a$ could or should be referred to as $\frac12$. But doing so redefines the value of your unit to $a$. That is, you're setting $a\to1$ in your new system! This is a transformation of the space you're working in.