Rational expressions

Rational expressions and rational equations can be useful tools for representing real life situations and for finding answers to real problems. In particular, they are quite good for describing distance-speed-time questions, and modeling multi-person work problems.

Solving Work Problems

Work problems often ask us to calculate how long it will take different people working at different speeds to finish a task.  The algebraic models of such situations often involve rational equations derived from the work formula, W = rt.  The amount of work done (W) is the product of the rate of work (r) and the time spent working (t). The work formula has 3 versions:

Some work problems have multiple machines or people working on a project together for the same amount of time but at different rates. In that case, we can add their individual work rates together to get a total work rate. Let’s look at an example:

How can equations and inequalities be used in real life situations?

One example is the Squeeze theorem where we can squeeze a complicated function between two simpler functions & use results on the simpler functions to prove bounds on the more complicated function. This applies to all sorts of maths including infinite sums, even finding the area of a circle.

See The Beautifully Simple Method Archimedes Used To Find The First Digits Of Pi

I usually try to think of inequalities physically / geometrically / spartially as either squeezing things together, or as constraints / boundaries.

It’s surprising what we can do when we combine constraints

The game of Sudoku is really just a set of constraints. Each number must occur in each row, column and square exactly once, and each cell must contain one number. We can actually set up a system of inequalities (multiple inequalities that must be met at the same time) to represent sudoku, and feed this to an inequality solver. See Linear programmingand Integer programming & https://pypi.python.org/pypi/PuLP

(so this is very useful because, we can create the inequalities we know about, at let the computer algorithms created by smart people figure out solutions to the inequalities - i.e. some of these problems are more complex that sudoku and coding solutions for them is hard, but if we can feed the inequalities into a program created by many PHD students, (magic black box) - then it can give us an answer.

In two dimensions, inequalities look like lines which (if straight lines), define convex polygons. (x>0, x<1, y>0, y<1 would be a 1x1 square for example). (so if we’re thinking about inequalities in 2D, any point within the polygon would satisfy the inequality. Example could be, what are the pokemon that are strong and common. One axis could be strength of the pokemon, the other axis could be how many pokemon of that type exist, and the polygon would be the region that is above a certain strength and certain population count threshold)

In higher dimensions we have convex polytopes defined by hyperplanes but we think about them in the same way as we do with polygons. (these polygons or polytopes are convex because the inequality lines or hyperplanes are straight (There are probably ways of dealing with concave regions but it would probably be a lot more complicated))

If you think of dimensions as the axis of variables on a graph, we’re not restricted to x and y. Variables could be anything we could measure. And restrictions can be any form of heuristic. E.g. I’m baking a chocolate cake using milk, butter and chocolate, but I want a restriction on the amount of fat. The ingredients could be seen as variables (e.g. x, y, x), and the fat restriction could be seen as total_fat = a*x + b*y + c*x <= max_fat_restriction. (in this example, y could represent the measures of butter, and ‘b’ could represent the amount of fat in each measure of butter)

If we were making burgers, the dimensions could be the following ingredients:
chicken, beef, lettuce, tomato, salt, Pepper, Chilli, bread, potato..

We can find our burgers in polytopes. Each hyperplane (face) of the polytope restricts a certain heuristic of ingredients.. though that might be a bit of a mouthful..

What are the heuristics?
Price, Spiciness, Weight, Volume, Carbs (can't have too many carbs!), Meatiness, Variety.

How do inequalities on heuristics create inequalities on ingredients?

- The heuristics can be represented by servings of ingredients. Thus restrictions on heuristics will create restrictions on ingredients!

For example, the serving of each ingredient has a price.
We say Price = Chicken Servings * Price Per serve of chicken + Beef Servings * Price per service of Beef...
So if we want our burger at under 10$, we are creating a hyperplane that will restrict us to only the affordable burgers.

Spiciness = Spiciness of Pepper * Pepper Servings + Spiciness of Chilli * Chilli servings.

Again we can create inequalities on spiciness depending on if we want mild, medium or hot!

We might not want beef and chicken in the same burger, we can say servings of beef + servings of chicken < 2. (note, what if we wanted the possibility of 3 servings of chicken, then we need to create a new variable and use integer programming - which is beyond the scope of this answer).

There are software packages where we can easily list our variables (ingredients) and our equations (hyperplane restrictions). The software package will then figure out the region and allow you to ask questions about heuristics in the region such as "given the constraints, what is the spiciest burger" or "what is the most profitable burger" etc.

Think about real world problems such as supply chain management - we want to get supplies from suppliers around the world to various factories at certain times adhering to certain demands, volatilities, buffers etc. Supplies (ingredients) have costs, transport costs etc. We could create inequalities to match the demand, requirements etc and then minimise costs across supplies and shipping etc. It would be much to difficult to do by hand.

The hard part is figuring out how to formulate a real world problem with inequalities. What are the dimensions or variables? what are the inequalities on the combinations of these variables?

Reference:

https://en.wikipedia.org/wiki/Squeeze_theorem

http://www.businessinsider.com.au/archimedes-pi-estimation-2014-3?r=US&IR=T

https://en.wikipedia.org/wiki/Linear_programming

https://pypi.python.org/pypi/PuLP