A connection between the Root and Ratio Tests for Series in Calculus Courses
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So based on this, it looks like that we have a proportional relationship between y and x. So this one right over here is proportional. So given that, what's an example of relationships that are not proportional. Well those are fairly easy to construct.
Ratios with tape diagrams
So let's say we had-- I'll do it with two different variables. So let's say we have a and b. And let's say when a is one, b is three. And when a is two, b is six. And when a is 10, b is So here-- you might say look, look when a is one, b is three so the ratio b to a-- you could say b to a-- you could say well when b is three, a is one.
Or when a is one, b is three.
Intro to proportional relationships
So three to one. And that's also the case when b is six, a is two.
Or when a is two, b is six. So it's six to two. So these ratios seem to be the same. But then all of sudden the ratio is different right over here. This is not equal to 35 over So this is not a proportional relationship. In order to be proportional the ratio between the two variables always has to be the same. So this right over here-- This is not proportional. So the key in identifying a proportional relationship is look at the different values that the variables take on when one variable is one value, and then what is the other variable become?
So for every three blue squares or that seem a similar color, for every three blue squares, we are going to have one, two, three, four, five green squares.
So the ratio of blue to green is three to five, and so in quilt A, she has 21 blue squares.
So she has 21 blue squares. How many green squares would she have? Well, to go from three to 21, you have to multiply by seven, and so you would take five and then multiply that by seven. So you'd multiply five times seven to get to As long as you multiply both of these by the same number or divide them by the same number, you're going to get an equivalent ratio.
Intro to ratios (video) | Khan Academy
So 21 to 35 is the same thing as three to five. Now we have a situation in quilt B. They've given us the number of green squares, so that's Well, how do we get 20 from five?
Well, we would multiply by four.Ratio Test
So if you multiply the number of green squares by four, then you would do the same thing for the number of blue squares. Three times four is going to be equal to Let's do another example. Here, we are told the following diagram describes the number of cups of blue and red paint in a mixture. What is the ratio of blue paint to red paint in the mixture? So try to work it out. All right, so let's just see. We have one, two, three, four, five, six, seven, eight, nine, So this would be a legitimate ratio, a ratio of 10 cups of blue paint for every six cups of red paint, but this isn't in I guess you could say lowest terms or most simplified terms because we can actually divide both of these numbers by two, so if you divide 10 by two, you get five.