Making the fractions in a proportion

"How do you know how to make the fractions in a proportion?
When making them, how do you know where each number goes making the fraction, like which ones go on top of the fraction?"

Well, actually you can choose which quantity will go on top; the proportion WILL work either way!

But, sometimes people are used to always putting certain quantity on top and certain on the bottom. For example, if the question is about speed and the unit is "miles per hour", that tells you that miles go on top, and hours on bottom, because "per" means division (the fraction line).

However, you could still solve the proportion by putting hours in the numerator of the fractions and miles in the denominator, and the calculation will turn out alright.

Or, if the question is about "dollars per pound", then dollars go to the numerator and pounds in the denominator.

Let's look at this problem for example:

A car drives on constant speed. It can go 80 miles in 90 minutes. How long will it take for it to travel 100 miles?

You can make both fractions to be

miles
-----------
minutes

OR

minutes
------------
miles

Let's try the first way:

80 mi           100 mi
--------  =   ---------
90 min            x

To solve, cross multiply and you get 80x = 100 * 90, and then x = 900/8 = 112.5 minutes.

The other way it will be

90 min         x
--------  =  --------
80 mi         100 mi

To solve, cross multiply and you get 80x = 90 * 100

You see, the final equation ends up being the same, no matter which
quantities were on top of the fractions.

HOWEVER, one way is wrong: that is if you but "miles" on top in one
fraction, and "minutes" on top in the other... then you'll get it wrong:

90 min        100 mi
--------  =  -------
80 mi         x min

=> 90x = 100* 80 (WRONG)

Geometry problem: a tiling

UPDATED!

Today I have a geometry problem for you.

Just yesterday I showed you a proportion problem and its solution.

(I want to insert a note here, especially for parents reading this.

Don't feel that you always need to use the problems I present here with your students. Instead, as a homeschooling parent, consider them part of your "math teacher training" that you never got. Reading through problems and their solutions will help you become a better math teacher. Consider yourself an apprentice observing a master completing a task, in this case mathematics problem solving.

It just takes you a minute or two to read this, think it through, try it - and understand (I hope). Your students' or child's book will have similar problems later on when it's time for geometry studies, but if you read these that I present, you will have learned more and can better tackle those problems at that time.)

OK, here we go. Look at the tiling below. Here you can see the tiling in real life, on a floor in Germany.



The original problem asks you about the area of the small square and of the parallelogram, given that the area of the large square is 1 square meter and that the ANGLE between the large and the small square is 45°.

But, I simply ask you (or your student) to DRAW this picture. Just take out blank paper, a ruler, and a protractor, and DRAW it. Can you draw it exactly how it is in my picture?

Solution:

This tiling will work (it will tile the plane), no matter what the size of the "little square". So, it is NOT possible for you to draw it exactly like I did, unless you knew something more about the side lengths.

Here's an example where use the same information, but made the small square smaller than in the previous picture:



To solve the original problem of finding the area of the little square and of the parallelogram, one would need some more information, such as a side length or ratio of side lengths.

SO, all in all, it's a problem with incomplete information to solve it...

However, someone suggested in the comments a solution to the original problem where we assume that the parallelogram is cut into two right triangles by its diagonal. With that additional information, the area calculation would be solvable.

(BTW, For my first picture, I used golden ratio as the ratio of the side of the large square to the side of the small square.)

A certain proportion

I need help explaining how to solve:

 12      36
---- =  ----
 3w      63

Could you solve this if instead of 3w it was x?

Well you can do just that:

 12      36
---- =  ----
 x       63

Now cross multiply and go about it as usual. The answer is x = 21.

But, x is actually 3w. So 3w = 21. So w = 7.

This changing of variables is a very often used "trick" in mathematics.


Alternate solution.

Upon examining the proportion, we notice both the numbers in the first fraction are divisible by 3, and both numbers in the second fraction are divisible by 9.

 12      36
---- =  ----
 3w      63

So, we can write it as follows and simplify both fractions (the right and left sides):

3 * 4        9 * 4
------  =   -------
 3w          9 * 7


 4     4
--- = ---
 w     7

You can't find a simpler proportion than that, obviously w must equal 7.