Basic Math: Lesson 6, Part 7 Arithmetic Of Fractions: Multiplication
In this vide, the concepts covered are Multiplication of Improper and Proper Fractions, and also Mixed Numbers. So watch and learn now with VideoJug user Video Math Tutor
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Step 1:
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The Problem
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Let's say I have half a pizza, and I want to know what is half of a half. Then we just slice it right there, and take it off and have it for lunch. Half of a half is a quarter.
Now I have to find out what is a third of a third. Here's a third of a pizza, cut this in three parts right here, so we have to grab this, that's one of the thirds, that's another third right here, so what is one of these slices? It turns out, one third of a third is one ninth.
Now I'd like to find out what is two-thirds of a quarter, right here. So if I slice it like that, notice if you extend the slices through this whole pizza here, you're really cutting it into twelfths. Two-thirds of this quarter, that right there, is really two twelfths, which is of course one-sixth.
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Step 2:
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Replace Of With Multiply
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Now let's replace the word "of", which we said previously, with the times sign.
A half times a half is a quarter, and what happens is I'm just multiplying through the numerators and the denominators.
Let's do the same for the others. A third times a third is a ninth, and two thirds times a quarter, I get two twelfths, which is a sixth.
So it looks like you can just multiply straight through on your numerators and get the answer, and multiply straight through on your denominators and get the answer.
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Step 3:
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The Rule
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Let's make a rule out of this:
To multiply proper or improper fractions: Multiply the numerators together to get the new numerator.
Multiply the denominators together to get the new denominator.
Reduce the fraction to lowest terms, if possible.
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Step 4:
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Example 26
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For this example, I'm going to multiply these two fractions together.
What we really need to do is multiply the numerators together, and you get 24. Multiply the denominators together, and you get 36.
Oh wait, I can reduce that and it becomes two thirds.
Now there's actually a slicker way of doing this, and let's try that next.
Another thing we can do is actually cancel out common factors. You have an 8 and a 4 here, well there's a 4 in each. I have 1 and I have 2 up there. These 3's and 9's have 3's in common. I can zap out 3's, and I have 1 and 3. And I just multiply the red numbers together. 1 times 2 is 2, 1 times 3 is 3, and there's our answer.
Notice how much easier that was to do.
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Step 5:
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The Improved Version Of Multiplication
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Now we have an improved version of multiplication:
To multiply proper or improper fractions (improved version):
Cancel any common factors by "cross-cancelling."Multiply the numerators. Multiply the denominators. Simplify, if possible.
Now that last step isn't really needed if you actually did cancel out all common factors. But sometimes you might forget a common factor. So, when you're done with your fraction, take a second look at it, and go, "hmm, is there anything I can maybe cancel out of this thing? Oh yeah, maybe something else." So you always try to simplify if you can.
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Step 6:
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Example 27
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Now I'd like to evaluate this expression right here.
Well, you can multiply straight through, but you're going to get some pretty nasty numbers, and then you need to simplify that. So it's much easier to see if you can cross-cancel stuff.
Alright, here's a 6 and a 2, they have 2's in common. A 3 and a 9, I can zap things out there. There's 5's right here, I'm left with 11 and 7. Right here, 7 and 7. So we're left with 1 times 11 times 1, which is 11. 1 times 3 times 1, is 3. See how easy that was?
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Step 7:
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On The Calculator
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Evaluate the following expression:
(6/35) * (55/9) * (7/2)
Our first step is to clear. Now, let's input our three fractions.
Parentheses, 6 divided by 35, closed parentheses. Open parentheses again, 55 divided by 9, closed parentheses. Open parentheses again, 7 divided by 2, and closed parentheses.
Let's press enter, and it outputs as a fraction. Math, enter, enter again. So our answer is 11/3.
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Step 8:
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Multiplying Mixed Numbers
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To multiply mixed numbers, convert them to improper fractions first, then proceed as before.
Change the result back into a mixed number.
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Step 9:
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Example 28
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Now I'd like to multiply these two mixed numbers.
What's your first step?
Change them to improper fractions.
So to do that, you multiply these together, then add that, then multiply these together, and add.
So we end up getting 12/5 for this guy and over here we get 40/9. Let's do our little cancellation technique. There's 5's here, 5's there, we get 8 and 1. Then we have 3's in common, so we get that. Now we multiply those together, and we get 32/3.
And always change your improper fraction to a mixed number. Using long division, you're going to get 10 and 2/3.
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Step 10:
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On The Calculator
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Now we're going to do the problem we just did on the calculator.
What do we do?
The first thing we have to do is make sure you enclose your mixed numbers in a set of parentheses. And of course, put a plus sign between the whole part and the fraction part.
Now it's ready to put into the calculator. Multiply 2 2/5 X 4 4/9
Press clear. Open parentheses, 2 plus 2 divided by 5, closed parentheses.
Open them again, 4 plus 4 divided by 9, closed parentheses. Enter. So let's just write down the 10. Subtract 10, and let's just make this a fraction. Now let's just tack on the 2/3. Two thirds, and we're done.
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Step 11:
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Exercise 34
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For exercise 34, I'd like to multiply the following fractions together.
For the first one, it doesn't look like anything can be zapped out here. Just multiply straight across. You're going to get 6 over 35.
Now over here, what's happening? There's 3's I can knock off here, so I'm left with 3 and 5 up here. There's 8's in both. And you just multiply those straight through. So you're going to get 5/6. Let's do some others.
Now I want you to multiply these together. The first step is to put our whole number over 1, and we need to change our improper fractions. We get 3/1, times, 2 goes below, 2 times 2 is 4 plus 1 is 5. Five stays below, 5 times 1 is 5 plus 1 is 6. Can we zap things out of here? Well, the 5's cancel nicely. The 2 and the 6 can be zapped out. That's it. So I'm left with 9 over 1, which just becomes 9.
Finally, try multiplying these three numbers together.
The first step is to put 4 over 1, right? And, let's change my mixed number. So what do we get? Four over 1, times, that just stays the same, and over here I'm going to get 10/9. Okay, let's zap some stuff here. The 4 and the 8, 2 into 10, this can go away, these are common here, and I'm left with this. And now just multiply straight through. Five over 3, and we're done.