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What are Direct Proportions

Direct proportionality, or direct variation, is a special relationship between two variables. When one variables changes, the other changes proportionally in the same direction.

For example, the hours you spend studying might vary directly with your test grade. If you double the hours you spend studying, then the score on your test will double. Similarly, if you study a third less than normal, your grade will be a third less than normal.

We use the following equation for direct variation. $$y=kx$$ Here, $k$ is the constant of proportionality. It will be a number that relates the two variables together.

$k$ Doesn't Change

When solving direct proportionality problems it's important to note that $k$ will never change in the problem.

If we rearrange the direct proportion equation, we see $k$ is a ratio of $y$ and $x$. $$k=\frac yx$$ Because $k$ is fixed (never changes), if we increase $y$, we then have to increase $x$ by some amount so that the ratio of the two still equals $k$.

Example: Is it Directly Proportional

Example 1$\\$ The equation for the area of a circle is $$A=\pi r^2$$ where $\pi$ (pi) is the constant $3.14\dots$ and $r$ is the radius of the circle.

Is this a directly proportional equation?

The answer is yes! There are two variables, $A$ and $r^2$, and they're related to each other through a constant, $\pi$.

Example 2$\\$ What about the following equation? $$y=\frac 4x$$ No. You cannot write this equation where $x$ and $y$ are a ratio $4$. IF you move $x$ over, you get $xy=4$, not $\frac yx=4$.

Example: Using Tables

Sometimes you'll be given a table and have to figure out if its directly proportional or not. $$\begin{array}{c|c} x&y\\ \hline 1&2\\ 4&8\\ 5&10\\ 13&26 \end{array}$$

To solve these, we find $k$ for each pair of coordinates. If $k$ is the same for each of them, then they're directly proportional. $$\begin{array}{c|c|c} x&y&\frac yx=k\\ \hline 1&2&\frac21=2\\ 4&8&\frac84=2\\ 5&10&\frac{10}5=2\\ 13&26&\frac{26}{13}=2 \end{array}$$ Since they all have a ratio of $2$, they're directly proportional.

Example: Word Problems

The general steps to solving these proportionality problems are:

1. Use the given information to find $k$.
2. Use $k$ to find the unknown information.

Let's look at an example:

Hooke's Law states that the distance $x$ which a spring can be stretched varies directly as the force $F$ exerted on it. If a force of $25\text{ lbs}$ stretches a spring $5\text{ inches}$, how much force does it take to stretch the spring $11\text{ inches}$?

We're told that the distance $x$ varies directly with the force $F$. This tells us to set these variables up in a direct proportion. Using the equation, we get: $$F=kx$$

Proportionality problems usually follow the same format. They give one piece of information which you use to find $k$, and then you use that to solve the last piece of information.

In this problem we're given that a force of $25\text{ lbs}$ stretches $5\text{ inches}$. Plug these into our equation and solve for $k$. $$\begin{align} F&=kx\\ 25&=5k\\ \frac{25}{\color{#8925ae}{5}}&=\frac{5k}{\color{#8925ae}{5}}\\ 5&=k \end{align}$$ We now have our constant of proportionality. This value doesn't change.

Because we know $k$, we can now find the unknown part of the problem. We want to find the force when the spring is stretched $11\text{ inches}$. $$\begin{align} F&=kx\\ F&=(5)(11)\\ F&=55\text{ lbs} \end{align}$$

Notice the ratio of both cases equals our constant of proportionality. $$\frac{25\text{ lbs}}{5\text{ in}}=5=\frac{55\text{ lbs}}{11\text{ in}}$$

Solution

We're told $y$ and $x$ are directly proportional. So, $$y=kx$$

Next, we use the given coordinate, $(5,9)$, to find $k$. $$\begin{align} y&=kx\\ 9&=k(5)\\ \frac 95&=k \end{align}$$

Finally, we use $k$ to find our unknown value. $$\begin{align} y&=\frac95x\\ y&=\frac95(25)\\ y&=45 \end{align}$$ So when $x_2=25$, $y_2=45$.

(Video) Variation

by mathman1024

This video by mathman1024 explains direct variation, also known as direct proportionality.

$\large{\text{Summary}}$

Variation or proportionality is a way to explain how things relate to one another. When you're doing word problems, phrases like "$y$ varies directly as $x$" or "$y$ is directly proportional to $x$" are clues that what you're working with might be direct proportionality.

Direct variation or proportionality are set up in the form: $$y=kx$$ where $k$ is the constant of variation or proportionality constant. This just means that when $x$ changes, $y$ changes directly with it.

We've actually seen things where $x$ and $y$ increase consistently relative to one another before. For example, the equation for the circumference of a circle is a direct variation: $$C=\pi d$$

This relates the circumference to the diameter. Here, the $\pi$ never changes so we know that it's our proportionality constant ($k$). We can also see that as we increase our diameter our circumference also increases. $$d\uparrow\text{ then }C\uparrow$$

The Constant of Proportionality

We can use the constant of proportionality, $k$, when solving direct variation problems. $k$ doesn't change in one equation.

$k$ signifies the relationship between two variables, typically $x$ and $y$. We can say that: $y$ varies directly as $x$ when we are demonstrating a direct relationship between two variables.

For example, say we are looking at the relationship between the number of bananas a monkey eats and the weather outside. We may notice that as the weather increases, so does the number of bananas that a money eats.

Say Georgia the money eats $5$ bananas when it's $75^oF$ outside. What would be the constant of proportionality for this relationship?

Step 1: Establish your variables. Let's say that: $$x=\text{bananas eaten}$$ $$y=\text{temperature outside}$$

Step 2: Plug the variables into the direct proportionality equation, and solve for $k$. $$y=kx$$ $$75=k(5)$$ $$15=k$$

Now that we've found $k$, we can use this value to solve further questions. What if we knew the temperature was $90$, but we didn't know the numbers of bananas eaten? Well, we can use our newly-solved $k$ value to figure that out!

Step 1: So, what are our variables? $$k=15$$ $$y=\text{temperature outside}=90$$

Step 2: Plug the numbers into the equation! $$y=kx$$ $$90=(15)x$$ $$6=x$$

So, now we know that if the temperature outside is $90^oF$, Georgia will eat $6$ bananas!