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Proportional Reasoning

Most of the quantities we will be dealing with in physics are ratios - like the velocity (what distance is covered in a certain amount of time) or the acceleration (how much the speed changes in a certain amount of time) or the electric field (how big the force is for each charge). In all of these examples two quantities are involved (distance and time or force and charge). These quantities can change but the way they change is special. For instance another ratio is 30%. It does not seem like this is a ratio since it does not involve two quantities but think about what 30% means – 30 percent – 30 per cent (“per” can be translated to me “for each”) – 30 for each cent  (and “cent” is Latin for “Hundred”) – 30 for each hundred, or 30 out of each hundred or 30/100. Now 30/100 (or 30 out of 100) is the same as 3 out of 10 or 3/10, which is the same as 0.3/1 or 300/1000. All of these ratios are identical and that is because I changed the numbers “in proportion”, which means by the same amount. How can that be? When I changed  30/100 to 3/10 I changed the top number from 30 to 3 by 27 and the bottom number by 90 – that’s not the same you might say. But ratios don’t deal with differences, I changed 30 to 3 by dividing it by 10 and that’s exactly what I did when I changed 100 to 10. I changed them in proportion. In physics we will deal with a lot of quantities that are proportional and the reasoning that is involved in figuring out these proportions is one of the most important things we will do this year.

Before you go to check out the problems I provide in the power point and the workbook here is an example of proportional reasoning: Suppose you have $15 and know that you can buy 5 CDs with this money. What is the price of one CD?  Suppose you want to buy 12 CDs, what would this cost? How many CDs can you buy with $180? Since you have been dealing with money all your life this does not seem difficult. It wouldn’t be much more difficult if I changed the numbers around to say $176 instead of $180. However, whenever things other than money are involved, like mass and volume or distance and time this reasoning seams to become more difficult and often you will want to fall back onto the equation so the you have something “to plug in”. This is not good practice. Plugging in numbers into equation will very often not result in the right answer because it is done “blindly” and with no reasoning behind it. Instead always try to reason your way through a problem. Here is another example: If your car goes 60 mph (60 miles in one hour), how long will it take you to cover 10 miles? You know that you cover 60 miles in one hour. You want to find out how many hours does it take to cover 10 miles, therefore how many times does 60 fit into 10 – not once, but less than one. The answer is found by dividing 10 by 60 and is roughly 0.15. It takes 0.15 hours (which is not 15 minutes but rather about 9 minutes).

When you come across a problem that seems difficult because the numbers are small, or don’t work out like in the problem above, try to make the number easier to deal with. For instance in the example above, change 10 miles to 100 miles. How long does it take to cover 100 miles - 60 miles in one hour, so it will be more than an hour, but less than two hours. How do you find the answer? By dividing 100 by 60, which is roughly 1.5 hours. Now you know that you have to divide and you can go back to the original problem.

For other examples and the reasoning behind them, please look at the power point in my account.

 

Look at the different Rubrik's cubes below. It is not hard to conclude that of the volume increases the mass has to as well. But since all cubes are made from the same material the mass increaqses in the same increments as the volume. That means that if you double the volume the mass will ave doubled. The two quantities, mass and volume are said to be proportional - one of them increases as a result of the increase in the other one. Mass and volume actually form a linear relationship with one anothert. That menas that they increase (or decrease) in equal steps. Because of this, if you have the mass of one cube you can easilly find the mass of 10 or 5679 cubes.

 

 

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