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Operations with Proportions
In this video, we’re gonna talk about the mathematics of proportions. A proportion is an equation of the form, fraction equals fraction. In the ratio videos in the next module, you will learn about the many uses of proportion and problem solving. They’re actually a very powerful problem solving tool. In this video, we will simply discuss the rules, what we can do and what we can’t do with proportions.
You have to understand the basic mathematical rules of what you can do with a proportion before you can even begin to use it in problem solving. One big idea is cross-multiplication. When we have our proportion, we can immediately eliminate all fractions simply by cross-multiplying. So if we have a proportion, a over b equals c over d, that’s a proportion, we can simply cross-multiply.
And of course what we’ve done is, we’ve multiplied diagonally. We’ve multiplied the a and the d, we’ve multiplied the b and the c, and we have just set those two equal to each other. That’s cross-multiply. So for example, this is a very powerful problem solving tool. If I wanna solve for x, I just cross-multiply.
And then I get an ordinary algebra equation, I could just divide by 5 and find the value of x. For proportions with larger numbers, of course, we should cancel before we cross-multiply. But, be careful, what we can cancel in a proportion and what we can’t cancel in a proportion is one of the most frequent problem areas in mathematics.
So, we’re going to be very clear about this. What exactly are we allowed to cancel in the proportion a over b equals c over d? First of all let’s talk about legal cancellation. Obviously, we can cancel factors in the numerator and denominator of the same fraction, we can always do that. So for example, there’s a common factor in a and b, we can cancel that.
If there’s a common factor in c and d, we can cancel that. So I’m gonna call this vertical cancellation, cancellation up and down in the same fraction. We can cancel factors in the two numerators on the opposite sides. So for example if a and c have a common factor, we can cancel across that way. Essentially, what we’re doing is dividing both sides of the equation by the same number.
That’s always perfectly legal. Another type of horizontal cancellation, we can cancel factors on the two denominators on opposite sides. So b and d have a common factor, we can cancel there. So there, there’s vertical cancellation on each side and then two types of horizontal cancellation and all of these are legal.
So we can summarize, in a proportion we can cancel vertically, numerator and denominator of the same fraction or we can cancel horizontally, both numerators or both denominators on opposite sides. So all that is what is legal to do. Let’s talk about what’s illegal, now.
This is a very tricky topic, because there is a move that is 100% illegal and wrong and yet, it’s an extremely tempting mistake that some students are convinced is legal. In a proportion, diagonal cancellation is 100% incorrect and illegal. So for example, if we have a over b equals c over d, we absolutely can not cancel a common factor in a and d.
We absolutely cannot cancel the common factor in b and c. That is 100% illegal. In fact, if you think about it, cross-multiplying says you’re allowed to multiply those two things. Well, if you’re allowed to multiply any two things, you’re not allowed to divide them.
Dividing two things always gives you a different answer than multiplying them. Here’s where I think the confusion arises. When we multiply two fractions, cross cancellation is perfectly legal. So for example, here I have a over b times c over d. This is not a proportion, this is a product of two fractions. And in this, of course, we can cancel a common factor in a and d.
Cross cancellation is perfectly fine. But, in a proportion, this is completely illegal and incorrect. If I have a over b equals c over d, so this is a proportion now. If I cross cancel, that is illegal. I think what happens is people get very mechanical about the idea of cross cancellation, and they neglect the difference.
Is there a multiplication sign between the fractions? Is there an equal sign between the fractions? They neglect that detail and they carry over something that is legal in the multiplication case even though it’s illegal in the proportion case. So it’s very important to pay attention to this, because it’s a very easy trap in which to fall.
Now that we have all that straight, we are in a position to simplify proportions involving larger numbers. So suppose we have this proportion. There are some large numbers in this. The first thing I’m gonna do, I’ll notice that in those two numerators is a factor of 11, so I can cancel the common factor there.
And that become, goes down from 44 and 33 digits 4 and 3. Now notice also there’s a common factor of 5 in the denominator. So I’ll just divide the, the two denominators, divide them both by 5, and I get down to this. Now at this point, there are some people who are gonna look at those two 4s and say, wow, it really looks like we cancel there, we could cross cancel them.
And that is 100% illegal. We cannot cancel those two 4s. In fact, at this point, the only thing we can do is cross multiply. We cross multiply and divide by x to find the answer. Here are some proportions to practice on your own.
I recommend pausing the video now and practicing these. Here are the solutions. So in this video, we talked about the math with proportions. We learned about cross multiplication. And we talked about the importance of cancelling before you multiply, and then we spent a lot of time talking about what you can cancel on a proportion, vertical and horizontal cancellation, and what you can’t cancel that is to say diagonal cancellation.
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