As a math tutor, you’ll likely introduce this concept.  It’s used even more in physics and chemistry.

So often in my university science courses I’d read “the mass is directly proportional to the volume” or “the distance is directly proportional to the time”, etc.  Science people love direct proportionality because predicting the result of a given input is so easy.

Direct proportionality means that if you double the input, the output will also double.  If y is directly proportional to x, it follows that

y=kx

where k is called the constant of proportionality. Once you know k, you can find the result of any input.

Example:  The distance travelled by a long haul train is directly proportional to the time traveled.  The train travels 600 km in 9 hours.
a)  Find the equation to model this situation.
b)  How far will the train travel in 12 hours?

Solution:  y is always the “output”, while x is the “input”.  Some people like to use different letters in order to reflect the actual wording of the question. In that case:

d=kt

where, of course, d stands for distance, t for time.
To find k, we use the idea that the train travels 600 km in 9 hours:

600=k(9)

Dividing both sides by 9, we get

600/9=66.7=k

We know now that the equation to model the train’s travel is

d=66.7t

To predict the train’s distance over 12 hours, we simply put 12 in for t:

d=66.7(12)=800

We conclude that over 12 hours, the train will travel 800 km.

More will be said about direct proportionality in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.