My College Options - Ratios, Proportions & Variations
In this lesson, you'll learn how to approach questions about direct and inverse variation with a simple explanation of what the terms mean and how Elizabeth has been involved with tutoring since high school and has a B.A. in Classics. In this An example of this is relationship between age and height. In an inverse relationship when one thing increases, the other decreases, and by the to know if the answer is needed by a fifth grader, or by a college student. Science is all about describing relationships between different variables, and direct and inverse relationships are two of the most important.
So if x is equal to 1, then y is 2 times 1, or is 2. If x is equal to 2, then y is 2 times 2, which is going to be equal to 4.
So when we doubled x, when we went from 1 to so we doubled x-- the same thing happened to y. So that's what it means when something varies directly. If we scale x up by a certain amount, we're going to scale up y by the same amount.
If we scale down x by some amount, we would scale down y by the same amount. And just to show you it works with all of these, let's try the situation with y is equal to negative 2x. I'll do it in magenta. Let's try y is equal to negative 3x.
So once again, let me do my x and my y. When x is equal to 1, y is equal to negative 3 times 1, which is negative 3. When x is equal to 2, so negative 3 times 2 is negative 6. So notice, we multiplied. So if we scaled-- let me do that in that same green color. If we scale up x by it's a different green color, but it serves the purpose-- we're also scaling up y by 2. To go from 1 to 2, you multiply it by 2. To go from negative 3 to negative 6, you're also multiplying by 2.
So we grew by the same scaling factor. To go from negative 3 to negative 1, we also divide by 3. We also scale down by a factor of 3.
So whatever direction you scale x in, you're going to have the same scaling direction as y.
- Intro to direct & inverse variation
- SAT Math Skill Review: Ratios, Proportions & Variations
That's what it means to vary directly. Now, it's not always so clear. Sometimes it will be obfuscated. So let's take this example right over here.
And I'm saving this real estate for inverse variation in a second. You could write it like this, or you could algebraically manipulate it. Or maybe you divide both sides by x, and then you divide both sides by y. These three statements, these three equations, are all saying the same thing. So sometimes the direct variation isn't quite in your face. But if you do this, what I did right here with any of these, you will get the exact same result.
Or you could just try to manipulate it back to this form over here. And there's other ways we could do it. We could divide both sides of this equation by negative 3.
And now, this is kind of an interesting case here because here, this is x varies directly with y. Or we could say x is equal to some k times y. Direct Relationships A direct relationship is proportional in the sense that when one variable increases, so does the other.
Using the example from the last section, the higher from which you drop a ball, the higher it bounces back up. A circle with a bigger diameter will have a bigger circumference.
If you increase the independent variable x, such as the diameter of the circle or the height of the ball dropthe dependent variable increases too and vice-versa. Sciencing Video Vault A direct relationship is linear. Pi is always the same, so if you double the value of D, the value of C doubles too. The gradient of the graph tells you the value of the constant. Inverse Relationships Inverse relationships work differently. If you increase x, the value of y decreases.
Recognize direct & inverse variation (practice) | Khan Academy
For example, if you move more quickly to your destination, your journey time will decrease. In this example, x is your speed and y is the journey time. Problems involving direct variation can be solved using proportions. If it takes 3 gallons of paint to cover square feet, how many gallons of paint will be needed to cover square feet?Direct and inverse variation - Rational expressions - Algebra II - Khan Academy
The problem gives us the ratio of 3 gallons of paint to cover square feet. We can use this information to set up our proportion: When we say "y varies directly as x," we could also write: In the paint example, the number of gallons of paint varies directly with the square footage that will be covered.
You would then have: Inverse Variation When two variables or quantities change in opposite directions, you have inverse variation. The time it takes to paint a house varies with the number of people doing the work.
In this example, the time required to paint the house varies inversely with the number of people painting. This means the more people painting the house, the less total time it will take to paint.
Direct, Inverse, Joint and Combined Variation
When we say "y varies inversely with x," we can express this as: A particular hotel has a custodial staff of 12 employees, and they can typically clean all of the hotel rooms in 6 hours. If four members of the custodial staff are not at work today, how long will it take the remaining custodians to clean all of the hotel rooms? In this example, the total time taken to complete the job is inversely proportional to the number of workers.
We want to know the numbers of hours it will take the remaining custodians to do the job. We can thus solve the problem as detailed below: