Quiz 1 Math Review

Here is some more review information about proportionalities.

First: direct proportionality.

An example of a direct proportionality is the equation:

This is the easiest case. What it means is that as you change A by multiplying it by any factor, B must change by the same factor. That is, if A gets changed to (3 x A), then B must change to be (3 x B).

For example, if you get paid by the hour, your total pay is directly proportional to the number of hours you work (in the equation above, A is total pay and B is hours worked). If you make $10 an hour, and if you work 20 hours a week, then you make $200. If you work 40 hours a week, you make $400 (I double the number of hours, so the total pay doubled). If I triple the number of hours that you work to 60 hours a week, you make $600, your total pay tripled, too. You can think of this as 20 hours is to $200 as 40 hours is to $400.

An example quiz question might be something like:

"If you triple the number of hours you work per week, how much more total pay will you earn at the end of the week compared to your usual number of hours?"

a) 2 times more

b) 1/3 as much

c) 3 times more

d) half as much.

The answer is c.

Second: inverse proportionality.

An example of inverse proportionality is the equation:

In this case, when you change A by multiplying it by some factor, B changes by 1 over that same factor (A increased, so B must decrease). That is, if A gets changed to (3 x A), then B must change to be (1/3 x B). The opposite is true as well. If B gets changed to (3 x B), then A must change to be (1/3 x A).

For example, if you are driving 120 miles (or any fixed distance), your speed is inversely proportional to the time it takes you to arrive (A is speed, and B is time to arrive). If you drive 30 mph, it will take you four hours to arrive at your destination. If I double the speed to 60 mph, it takes you 2 hours to arrive. That is, if I double your speed, the time it takes you to arrive is half as long. If I double the time it takes you to arrive to 8 hours, then your speed must be multiplied by one half, or 15 miles per hour. To sum this one up: 15 mph is to 8 hours as 30 mph is to 4 hours as 60 mph is to 2 hours.

An example quiz question here might be something like,

"The distance to a star is inversely proportional to its parallax angle. If star A has a parallax angle twice as large as star B, how does star A's distance compare to star B's?"

a) star A is half as far away as star B

b) star A is twice as far away as star B

c) star A is four times farther away than star B

d) star A is one fourth as far away as star B

The answer is choice a.

Third: squares and inverse squares.

Examples of proportionalities involving squares are:

In these cases, the main idea is the same. For the first example (A proportional to B squared), if you increase A, B increases. The inverse proportionality also works the same way (A proportional to 1/B squared) - if you increase A, B must decrease. If you increase B, A must decrease. The difference between these examples and the ones above is that the changes are larger.

For example, in the first case, if I double B to (2 x B), since B is squared, then the right hand side of the equation becomes (2 x B x 2 x B) = (4 x B). So if I double B, then I've quadrupled the right hand side of the equation. So A must quadruple as well. If I triple B to (3 x B), then the right hand side becomes (3 x B x 3 x B) = (9 x B), so A must increase by a factor of 9.

In the second case, if I double B to (2 x B), since B is squared, then the right hand side of the equation becomes 1 / (2 x B x 2 x B) or 1 / (4 x B), so if I double B, then A must be multiplied by 1/4th. An example of the second case is the force of gravity, which is a "one over r squared law" -- that is, if you double the distance between two objects, the force of gravity is 1/4th as strong. Below is this relationship illustrated in a plot of force versus distance (note, when r=1, f=1, and when r=2, f=1/4):