How to recognize basic proportionality problems
In the simplest form, proportional situations are direct relationships, where if one value doubles, the other value also doubles. In other instances there are variations, like doubling one value causes an inversely proportional response, of 1/2 of the other variable. Other variations exist, like having the factor squared (double one variable is related to 2^2, or 4 times the other variable).
These relationships ONLY work this way because all other factors involved are held constant. You can often recognize them because they will tell about two conditions are discussed for one variable and the question asks for the effect on the other variable.
Simple Example: You work twice as many hours this week as you did last week, how much more money will you earn this week? This problem can only work if the person working has a set hourly rate. That can be expressed by the equation of $=Rt (where $ is the money earned, R is a constant pay rate, and t is the number of hours worked). Notice the two conditions - last week you worked one value of hours, and this week you are working twice as many.
Simple proportional relationships also have a common graphical representation.
Key factors of a proportional graph, like the one above, are that they go through the origin and are linear. Your standard form of this equation, y=mx+b would have a b=0 value, so y=mx. Since we have $ on Y axis, and t on x, replace them in the equation y=mx, and you get $=mt where the slope of the line, m, is equal to the pay rate, R. Understanding this can really help you use slope and other features of a graph in more meaningful ways.
Non-Examples: In reality, there are many jobs that don't work that way; waitresses work with a small hourly rate and then tips, which would have the form of $=Rt+T (where T would be tips earned); or a job with overtime, where $=Rt+R't' (where t would be the hours up to 40 hours and R' would be the overtime rate of pay and t' would be the hours above 40). These are not proportional, because there is not the simple relationship between two values where if one triples, the other triples as well.
Summary of how to recognize basic proportionality problems: So, you can recognize a simple proportionality problem because it will be discussing a simple relationship (m=y/x, or y=mx, or x=y/m) and you see the problem asks about the result of changing one of the variables. Type A may say a value is modified by a factor (halved, doubled, etc) or Type B may tell you two values, a before and after, for the same variable and ask about an effect on one of the related variables.
How to solve a proportional problem
I use a variety of methods to solve proportionality problems because not every student thinks about them the same way.
Your Math Teacher's Way: dividing equations
I asked your math teacher how to solve these and they recommended this method:
- identify a relationship that links your 2 variables with everything else being constant. (sometimes you need to use substitution between two equations to achieve this step)
- Write 2 equations, one for the first set of conditions and one for the new conditions
- dividing the two sides of the equations.
For the example of doubling the hours worked discussed above, I will use subscripts 1 for last week's values and 2 for this week.
Pros to this method: Should be familiar from math class.
Cons to this method: This way is slower than some other ways. If you are not strong in math, you might have a hard time dealing with equations with fractions within fractions (can happen with this method).
Re-arranging the basic equation for constants on one side
For this method, you will 1) re-arrange the equation to group constants on one side of the equation with variables on the other side. 2) set the variables for the first condition (week 1) equal to the variables for the second condition (week 2) and solve.
In this example, we will use PV=nRT from chemistry.
An enclosed balloon has a pressure of P and Volume of V before it is placed in a vacuum chamber and allowed to expand until the volume is twice the original amount. This is done without changing temperature. (Remember that n is the number of moles of the gas (constant in an enclosed container) and that R is the ideal gas constant.)
Pros to this method: Can be a little faster, because you may not need to solve all the way to the end to catch how the variables relate. Might avoid fractions in fractions from math teacher's method.
Cons: Still not the fastest way.
Proportionality symbol method
For this method you will
- Identify a relationship that links your 2 variables with everything else being constant. (sometimes you need to use substitution between two equations to achieve this step)
- Solve the expression for the desired variable (answer),
- Remove any constants and the equal sign and replace with proportionality symbol.
- Identify direct, inverse, quadratic, etc relationship and apply factors identically to both sides.
Let's try this with the simple scenario. A car travels a certain distance in specific amount of time at a constant speed. How long will it take the car to travel the same distance if it travels at twice the speed?
Pros to this method: This way lends itself to giving a name to the type of relationship (direct, inverse, inverse square, etc) better than the others.
Cons: Requires different thinking to interpret the correct answer.