-2

For intuition, I reference objects. Imagine making a dessert with: $a$ as apples and $c$ as chestnuts.
Question. How and why is

$\dfrac{a}{c} \qquad (3) \quad = \quad\dfrac{\color{red}{1}}{\dfrac{c}{a}} \qquad (4) \qquad ?$

I ask only about intuition; please omit formal arguments and proofs (eg: Intuition is not generated by the explanation that rationalising (4)'s denominator produces (3)).

Community
  • 1
  • 9
    You can gain intuition by thinking about feeding apples to children and seeing what happens when you feed a fraction of a children apples. This is why you don't learn this in high-school. It's a bit scary. – Pedro Apr 23 '16 at 05:04
  • If I have $5$ apples per $10$ children then I have $1$ apple per $2$ children. i.e. $\frac{5}{10}=\frac{1}{\frac{10}{5}}$. – John Douma Apr 23 '16 at 05:09
  • 1
    In the US they measure fuel consumption in miles per gallon. In Europe we talk liters (consumed) per hundred kilometers. It is still fuel consumption whichever way you look at it. The US way is geared towards making it easy to guesstimate the distance within which you need to find a gas station (a concern in sparsely populated states), the European way is more suited for calculations like how much fuel I need for this 300 km trip? (and also a lower number means lower consumption). Both ways get the job done (though translating one to the other is not as straightforward as you might wish). – Jyrki Lahtonen Aug 03 '16 at 10:55
  • I find it amusing that "children" was changed to "chestnuts", and now it is a dessert with these ingredients that is being described. If one doesn't read carefully, it almost seems like a dessert made from apples and children now. Hilarious. – MPW Aug 09 '16 at 13:05

5 Answers5

4

The fraction $\frac{a}c$ is the number of apples per chestnut, and $\frac{c}a$ is the number of chestnuts per apple.

Suppose that you have $x$ chestnuts per apple, where $x$ can be any positive real number. No matter what $x$ is, each one of those $x$ chestnuts must be getting $\frac1x$ of an apple, so there are $\frac1x$ apples per chestnut.

Thus, in the particular case that $x=\frac{c}a$, it must be the case that

$$\frac{a}c=\frac1{\frac{c}a}\;.$$

Brian M. Scott
  • 616,228
  • I only edited your answer to modify 'children' to 'chestnut', because my choice of 'children' was macabre in hindsight as they should not be described as fractions. –  May 15 '16 at 22:40
  • 1
    @LePressentiment: No problem. (Though I don’t find the original version macabre, since I automatically understand ‘half a child per apple’ as ‘two apples per child’!) – Brian M. Scott May 15 '16 at 22:42
  • Thanks. Oh right; I see that it is not macabre then. –  May 15 '16 at 22:44
1

You're making it less intuitive by introducing apples and chestnuts.

Generally, "$\frac 1x$" is that number by which you need to multiply $x$ in order to obtain the product $1$.

So, isn't it clear that $\frac bc$ is the correct candidate for $\frac 1{(\frac cb)}$, since $\frac cb \times \frac bc=1$?

Addendum: There's nothing special about the "$1$". So, even more generally, "$\frac px$" is that number by which you need to multiply $x$ in order to obtain the product $p$.

This is the very definition of division.

MPW
  • 43,638
0

I think the best way to think about this is as a nested fraction. Let's let $k=c/a$.Your number is

$$a/c=1/(c/a)=1/k=1/\text{chestnuts per apple}$$

Intuitively, you can think about this in one of two ways as:

  1. the flip (i.e. Inverse) of the thing on the bottom (making it the number of apples per chestnut) or

  2. you have one apple and you can feed people at a rate of $k$ chestnuts per apple, and so if you have to feed $1$ chestnut, how much of an apple do you need? Reminder that the equation is for feeding people is $c=ka$.

Stella Biderman
  • 31,155
  • 1
    I only edited your answer to modify 'children' to 'chestnut', because my choice of 'children' was macabre in hindsight as they should not be described as fractions. –  May 15 '16 at 22:40
  • Thanks. Is there a typo? You wrote "if you have to feed 1 chestnut"? –  May 09 '18 at 04:42
0

Let $\frac{a}{c}=\frac{1}{x}$

Then, $\frac{a}{c}.x=1$

$x=\frac{1}{\frac{a}{c}}$

NEW SUN
  • 311
  • 1
  • 21
-1

I believe that your ratio in (3) represents how many apples per child. Then $\frac{c}{a}$ is the ratio of how many children eat the same apple. Finally, your (4) I guess it is the ratio when comparing the situation: $1$ apple is eaten by $1$ child, which is $\frac{1}{1}$, with your current ratio of children per apple. Does that make sense? hm..

Ioannis Tsartsaflis
  • 65