Direct Proportion Animation

The heights of these two bars vary in When one gets large, the other gets large; when one gets
small, the other gets small -
When the blue bar grows so that it
its size.doubles (times 2)
If the green bar shrinks so that its height is cut to
its
height.one third (times 1/3)
The height of both bars Further explanations follow. |

Quantities change.

When a quantity gets larger or smaller, we say that it changes.

Changes can be related through factor 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

*Twice the soup costs twice the money.*

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

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

Definition of Direct Proportion

When quantities are related this way we say that they are in ** direct proportion**.
That is, when two quantities both

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
A changes, B changes by the factor.same factor |

Half the volume of a liquid has half the weight.

Let us present another example of a direct proportion. The ** weight** of a liquid
is

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

Symbol for directly proportional is alpha.

Here is a shorthand way to say that the quantities A and B are directly proportional:

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

The form of the equation for a direct proportion looks like ** y = mx**.

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:

The above would read ** 'A equals k times B.'** The quantity k is a

Note that the above equation has the same form as ** y = mx**,
a simple
linear equation from mathematics that graphs a straight line
through the origin with a slope of m.

The graph of a direct proportion is a straight line through the origin.

If ** A=kB**, then a graph of A vs. B will yield a straight line through the origin
with k as the slope:

So, ** if you graph data for two related quantities**, and that graph yields a

Note again the similarity to the equation ** y = mx**, where a
line passes through the origin at a slope of m.

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.

Related:

- Factor changes
- The graph of a direct proportion
- Linear functions
- Inverse proportions
- The graph of an inverse proportion