Basics of Proportionality

Understand proportional relationships and learn how to find the constant of proportionality.

Proportional Relationships

Here are the results of an experiment conducted at a middle school to investigate the rate at which ice melts. They hypothesized that as the volume of ice increases, the speed at which it melts would also increase, so they examined the melting speed when changing the volume.

Experimental Procedure
1. Prepare cubic boxes with side lengths from 1cm to 7cm.
2. Fill these boxes with water and freeze for 24 hours.
3. Put the resulting ice cubes into a bowl of water
and measure the time it takes to melt.
4. Repeat steps 1-3 twice for each cube size.

Based on the experimental data, we summarized it as follows. Here, we've adjusted the values to be different from the actual experimental results to make the relationships easier to understand.

It seems that as the volume of the cube increases, the time it takes to melt also increases.

Equation of Proportion

Let's make a table of the above data, with $x\,cm$ as the side length of the cube and $y$ minutes as the time it takes to melt.

Looking at the relationship between $x$ and $y$ in this table, we can see that the value of $y$ is 5 times the value of $x$ . From this, we can express the relationship between $x$ and $y$ with an equation.

In this equation, $x$ and $y$ are called variables because their values change. On the other hand, the 5 in $y=5x$ is called a constant because it's a fixed value that doesn't change.

The proportional relationship $y=ax$ is sometimes also called the function $y=ax$ .

Constant of Proportionality

The constant of proportionality $a$ shows the ratio at which the two variables change.

In a proportional relationship, when $x$ becomes 2 times, 3 times, 4 times its original value, $y$ also becomes 2 times, 3 times, 4 times its original value. We can also say that the ratio of change in $x$ to the change in $y$ is constant.

Transforming the equation like this makes it easier to understand. Since $a$ is a constant, it's fixed. What's constant is $\frac{y}{x}$ , and if we view this as a ratio, it means the ratio of $y$ to $x$ is constant.

If we check with the values in the above table, we can indeed see that the ratio of $y$ to $x$ is always constant at 5.