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# Ratio, Proportion, and Variation

## Identify variations from word problems

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Direct and Inverse Variation

Have you ever thought about how one variable can affect another? Take a look at this dilemma.

After a lot of debating and discussing, an eighth grade class decided to attend the Omni Theater. The presentation on the rainforest was fascinating and all of the students were glad that this was the decision that they had made. While the film did show the animals and insects of the rainforest, it also focused on ecology and on scientists and other people working to save the rainforest.

When they returned to school, Mr. Thomas had them talk about different parts that had interested them.

“I was fascinated on the bus ride that the scientists took to get to the rainforest. It was amazing to me that the bus could travel over all of those bumpy roads and not have an accident,” Mark commented.

“Yes, if you think about it, the bus ride was pretty cool to see from the Omni point of view. I mean I felt like I was going to fall over the edge some of the time,” Karen added.

Others smiled as well. Mr. Thomas seizing the opportunity wrote the following problem on the board.

A bus leaves Boston traveling at a constant speed of 60mph. You can make a table showing the distance, d\begin{align*}d\end{align*}, in miles that the train has traveled after h\begin{align*}h\end{align*} hours.

This is your task. After learning about slope, you will learn about direct and indirect variation. This Concepts will help you with Mr. Thomas’ dilemma.

### Guidance

Do you know that one variable can affect another variable? Did you know that this often happens in real-world situations?

Take a look at this situation.

A train leaves Boston traveling at a constant speed of 60mph. You can make a table showing the distance, d\begin{align*}d\end{align*}, in miles that the train has traveled after h\begin{align*}h\end{align*} hours.

h\begin{align*}h\end{align*} d\begin{align*}d\end{align*}
0 0
3 180
6 360
9 540
12 720

This relationship can be shown in the function d=60h\begin{align*}d = 60 h\end{align*}.

This type of function is called a direct variation.

It is a linear equation that can be written in the form y=kx\begin{align*}y=kx\end{align*}, where k0\begin{align*}k \neq 0\end{align*}.

This is a linear function whose x\begin{align*}x\end{align*}- and y\begin{align*}y\end{align*}-intercepts are always zero—it always passes through the origin. The variable k\begin{align*}k\end{align*} is called the constant of variation which is also the slope of the line. In the case above, the distance varies directly with the time because it increases in proportion to the time. That is, if the time doubles, the distance doubles, and so on.

In a direct variation, as one variable increases, the other increases, too. In a direct variation, for any ordered pair (x,y),k=yx\begin{align*}(x, y), k=\frac{y}{x}\end{align*},.

Good question. We call this an inverse variation.

An inverse variation can be written in the form y=kx\begin{align*}y=\frac{k}{x}\end{align*} where k\begin{align*}k\end{align*} is still the constant of variation as in direct variations but the product of the ordered pairs is k\begin{align*}k\end{align*}.

A plane flying is a situation describing an inverse variation. When the speed of the plane increases, the time that the plane is in the air actually decreases.

Look at each situation and determine if they represent an inverse variation or a direct variation.

#### Example A

A car racing on a track. The relationship between speed and time.

Solution: Inverse variation

#### Example B

The price of a plane ticket that increases year to year.

Solution: Direct variation

#### Example C

The number of miles traveled in relationship to distance.

Solution: Direct variation

Now let's go back to the dilemma from the beginning of the Concept.

Now here is the table and the graph to represent the data.

h\begin{align*}h\end{align*} d\begin{align*}d\end{align*}
0 0
3 180
6 360
9 540
12 720

This relationship can also be shown by the function d=60h\begin{align*}d = 60 h\end{align*}.

### Vocabulary

Direct Variation
a situation where as one variable increases, the other variable increases as well.
Inverse Variation
a situation where as one variable increases, the other variable decreases.

### Guided Practice

Here is one for you to try on your own.

The caloric intake of a hummingbird varies directly with the amount of nectar that it consumes. For each gram the hummingbird consumes, it takes in 5 calories. Write an equation that shows the direct variation. Identify k\begin{align*}k\end{align*}. Then create a t-table.

Solution

First, think about what is being affected. For each gram of nectar, a hummingbird takes in 5 calories. The 5 calories represents the k\begin{align*}k\end{align*} value. Here is an equation for this situation.

c=5g\begin{align*}c=5g\end{align*}

Where c\begin{align*}c\end{align*} is the calorie total and g\begin{align*}g\end{align*} represents grams.

Now we can create a t-table.

h\begin{align*}h\end{align*} d\begin{align*}d\end{align*}
0 0
1 5
2 10
3 15

This is our answer.

### Practice

Directions: State whether each situation described is an example of direct variation or inverse variation.

1. The train traveled at a speed of 80 miles per hour. The number of miles increased with every hour that the train was running.

2. The train traveled at a speed of 80 miles per hour, then it increased it's speed. The number of hours on the train decreased with the increase in speed.

3. Mary is training for a marathon. She runs many hours each week. After a few weeks of running, she noticed that her speed increased.

4. Kevin has been working for the same company for several years. He has received a raise each year that he has been with the company.

5. Joseph has been working for the same company too, but his salary has decreased each year.

6. Kelly spent more hours studying than she ever had before. She was surprised when she received a lower score on this test than she had on any previous tests.

7. Jeff is on a diet. He knows that the number of calories that he burns is directly connected with the number of hours that he exercises.

8. Seth and Sarah spent a lot of time eating while they were on vacation. After vacation was over, they both noticed that they had each gained five pounds.

9. Mary's training has increased. She has been keeping track of how long it takes her to run 1 mile. She noticed that the she trained, the faster her time became.

10. Over time, the price of a postage stamp has increased a few pennies each year.

Directions: Answer each question true or false.

11. In direct variation, one factor increases as the other factor increases.

12. In inverse variation, one factor increases, but the other factor decreases.

13. When you see the letter k\begin{align*}k\end{align*} in a situation, you can think of a constant variable.

14. If a plane ascends and then quickly descends, this is an example of direct variation.

15. If your exercise increases and you lose weight, then this is an example of direct variation.

### Vocabulary Language: English

Direct Variation

Direct Variation

When the dependent variable grows large or small as the independent variable does.
Inverse Variation

Inverse Variation

Inverse variation is a relationship between two variables in which the product of the two variables is equal to a constant. As one variable increases the second variable decreases proportionally.

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