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Proportions - Concept
We can find unknown quantities when we know similar ratios for comparison, using proportions. We find proportions often in word problems, for example those involving baking ingredients, and in comparing similar figures in Geometry. We can represent an unknown quantity in a proportion with a variable, and then solve it using one-step, two-step, or multi-step equation methods.
A proportion is two equal ratios, you can also think of it as two equal fractions. And you guys already know a lot about fractions for example lets say I were to write five sixths is equal to ten twelfths, you guys know those are equivalent fractions and incase you forgot let me just remind you that I multiply 5 by 2 in order to get 10 and I multiply 6 by 2 in order to get twelve that's how I know those are the same quantity the same fraction. Sometimes with proportions you see two equal ratios written like this, a lot of times you're going to have a variable where instead of telling that was 12 they would ask you to find out what it is. That's something you're going to run into a lot. So when you're given a problem like this there's a couple of different ways to solve. The first way is the way I just described to you where you think about how 5 times 2 gave you the answer 10, so 6 times 2 would have given me that 12. That's one way to think about it, is to look for what number you're multiplying by.
Another way to look at it is to do what's called "cross multiplying," and "cross multiplying" is where you multiply the diagonal quantities and set the products equal to each other, let me say that one more time you multiply the diagonal quantities and set the products equal to each other. So here we go 5 times x that's my first product, it's going to be equal to 6 times 10 and you can kind of do it in your head how this would give you the answer x equals 12. So cross multiplying is another good technique for solving proportions, the last method I want to show you is the one that makes most sense to me mathematically. We write that same problem, what I would want to do is try to undo what's happening to x just like I would with solve or [IB] solving an equation. So right now I have 10 divided by x, if I wanted to get x out of the denominator I would multiply both sides of the equation by x so that those cancel out. And then I'd be working with the problem 5x divided by 6 equals 10, and from there you would go through any the multiply by six fifths which is the reciprocal of this or you can multiply in two different steps multiply by six over one, so you have 5x=60. And then finally divide both sides by 5to see x equals 12. Tthis way is the longest but it's the most mathematically precise.
A lot of students jump right to cross multiplying cause they think it's pretty easy and you're right it is probably the easiest method but you have to make sure that you understand cross multiplying is really doing this process in fewer steps. So when you guys come with a [IB] come across a proportion which is two equal fractions think about which method you want to use, whether it's looking for what's the number you're multiplying by to get from one fraction to the next, whether you choose to cross multiply or to do it out using solving techniques like this.