Direct Proportion
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When a quantity gets larger or smaller, we say that it changes.
Sometimes a change in one quantity causes a change, or is linked to a change, in another quantity. If these changes are related through equal factors, then the quantities are said to be in direct proportion. Or one might say that the two quantities are directly proportional.
For example, suppose that you are buying cans of soup at the store. Let us imagine that they cost 50 cents, or $0.50, each.
Case #1:
Suppose that you buy 4 cans. You would pay $2.00.
Case #2:
Suppose that you buy 8 cans. You would pay $4.00.
So, changing the number of cans that you buy will change the amount of money that you pay.
Notice that the number of cans changed by a factor of 2, since 4 cans times 2 is 8 cans.
Also, notice that the amount of money that you must pay also changed by a factor of 2, since $2.00 times 2 is $4.00.
Both the number of cans and the cost changed by the same factor, 2.
When quantities are related this way we say that they are in direct proportion. That is, when two quantities both change by the same factor, they are in direct proportion.
In the above example the number of soup cans is in direct proportion to the cost of the soup cans. The number of soup cans is directly proportional to the cost of the soup cans.
The formal definition of direct proportion:
| Two quantities, A and B, are in direct proportion if by whatever factor A changes, B changes by the same factor. |
Let us present another example of a direct proportion. The weight of a liquid is directly proportional to its volume. Suppose that you had a container holding 6 quarts of a liquid, and that liquid weighed 3 pounds. If we poured out half of the liquid so that only 3 quarts remained, that liquid would now weigh 1.5 pounds. So, the volume of the liquid changed by a factor of 1/2, since it went from 6 to 3 quarts, and since 6 quarts times 1/2 equals 3 quarts. The weight of the liquid also changed by a factor of 1/2 since it went from 3 to 1/5 pounds, and since 3 pounds times 1/2 equals 1.5 pounds. Both the volume and the weight changed by the same factor, 1/2. So, in this example the weight and volume of the liquid are in direct proportion.
Here is a shorthand way to say that the quantities A and B are directly proportional:
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The Greek letter between the A and the B is called alpha. It is here written in lower case script. In this context it is shorthand for the phrase "is directly proportional to." So, the above statement reads "A is directly proportional to B."
Whenever you have a direct proportion as stated above you can change it into an equation by using a proportionality constant. Here is how the direct proportion would look as an equation:
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The above would read "A equals k times B." The quantity k is a proportionality constant. If two quantities, A and B, are directly proportional, then there is a proportionality constant, k, such that k times B will equal A.
If A=kB, then a graph of A vs. B will yield a straight line through the origin with k as the slope:
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So, if you graph data for two related quantities, and that graph yields a straight line through the origin, then you know that the two quantities are in direct proportion.
Examples of direct proportions abound in physics. For example, Newton's second law of motion states that the acceleration of an object is in direct proportion to the force on the object. So, if you triple the force on an object, then the acceleration of that object will also triple. Of course, when you triple a quantity you are changing it by a factor of 3.