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. \u00a0Science people love direct proportionality because predicting the result of a given input is so easy.<\/p>\n
Direct proportionality means that if you double the input, the output will also double. \u00a0If y is directly proportional to x, it follows that<\/p>\n
y=kx<\/p>\n
where\u00a0k\u00a0is called the constant of proportionality. Once you know k, you can find the result of any input.<\/p>\n
Example: \u00a0The distance travelled by a long haul train is directly proportional to the time traveled. \u00a0The train travels 600 km in 9 hours.<\/strong> Solution: \u00a0y is always the “output”, while x is the “input”. \u00a0Some people like to use different letters in order to reflect the actual wording of the question. In that case:<\/p>\n d=kt<\/p>\n where, of course, d stands for distance, t for time. 600=k(9)<\/p>\n Dividing both sides by 9, we get<\/p>\n 600\/9=66.7=k<\/p>\n We know now that the equation to model the train’s travel is<\/p>\n d=66.7t<\/p>\n To predict the train’s distance over 12 hours, we simply put 12 in for t:<\/p>\n d=66.7(12)=800<\/p>\n We conclude that over 12 hours, the train will travel 800 km.<\/p>\n More will be said about direct proportionality in future posts:)<\/p>\n
\na) \u00a0Find the equation to model this situation.<\/strong>
\n b) \u00a0How far will the train travel in 12 hours?<\/strong><\/p>\n
\nTo find\u00a0k, we use the idea that the train travels 600 km in 9 hours:<\/p>\n