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Fraction Properties - II
In this video, we’re gonna talk about four mathematical properties for fractions. The first topic is just more on the topic of Canceling and Order of Operations. Big idea number 1, multiplication and division are at the same level of priority in GEMDAS and can be done in any order. GEMDAS is the acronym I use for order of operations. If this is unfamiliar to you.
You can watch the order of operations video. But the big idea here is is that multiplication and division are at the same level of priority and can be done at any order. Big idea number 2 is that canceling itself is a form of division. It is division of the same factor of the numerator and the denominator. The big idea number 3 is always cancel before you multiply.
Always make the numbers smaller before you make them bigger. For example if we have A times B over C times D. We could write this as A over C times B over D or we could write this as A over D times B over C. In other words, we could reorganize it as long as what’s in the numerator stays in the numerator and what’s in the denominator stays in the denominator.
The order doesn’t matter. Remember, when multiplying fractions, you can always cancel a factor in any part of the numerator with a factor in any part of the denominator. If it helps you to see what to cancel, you certainly can split the fractions into several fractions, but you don’t have to. So for example, suppose we have to perform this.
This would be a horrible thing if we just multiplied out the numerator separately, multiplied out the denominator. This would be incredibly ugly. Instead. We’re gonna do some canceling. Now we could write these as separate fractions.
But we don’t have to do that, we can just leave them as is and cancel in place. As almost to remembering we can cancel any part of the numerator, with any part of the denominator. So I’m going to cancel a factor of 8 between the 56 and the 64, they’re going to go down to 7 and 8. Then I’m going to cancel a factor of 9 between the 54 and the 45, they’re going to go down to 6 and 5.
Then I’m gonna cancel a factor of 12 between the 72 and the 60. They’re gonna go down to 6 and 5. Then, this is tricky, the 8 has several factors of 2 in it, so I’m gonna cancel a factor of 2 in each one of the 6s. That means each 6 goes down to 3, and I’ve cancelled two factors of 2, so the 8 goes, gets divided by 4, and goes down to 2.
Well now I can just multiply across, and I get 63 over 50, and that is the simplified version of this product. These same ideas can be used to cancel factors in algebraic fractions. For example, in this algebraic fraction, I’ll start out by canceling a factor of 3, between the 27 and the 6. So that goes down to 9 and 2 then I’m going to factor it 2 out from the parenthesis so instead of writing 2y minus 2 that’s gonna be 2 times Y minus one which gives me the same factor.
In the numerator denominator those can cancel directly and I’m just left with 9 times y plus 5. There will be more of this in the algebra module. Here I’m just showing the fraction principles that will apply. The second idea that we’ll talk about is the addition or subtraction in the numerator or denominator.
We can separate a fraction into two fractions by addition or subtraction in the numerator. So if we have A plus B over C we could write that as A over C plus B over C. If we have D minus E over F we can write this as D over F minus E over F. We cannot separate a fraction into. Into, we can, into two fractions by addition or subtraction in the denominator.
So, for example, here people want to do this separation. Separate it out by the addition or subtraction of the denominator and those are illegal moves. This is a very tempting mistake to make. It’s very important to recognize this and realize this is something you can’t do. If we have addition and subtraction book the numerator and the denominator we can split up the numerator, but the denominator has to stay unchanged so, for example, A plus B over C plus D, we can not write it like this.
We have to write it like this. The denominator stays unchanged. Most of the applications of this will also be in the algebra module. One application with pure numbers is changing from an improper fraction to a mixed numeral. For example, suppose I start with the improper fraction 17 over 3.
Well I’m gonna write that as a sum, that 17 as the sum of the largest multiple of 3, less than 17 plus something else. So the largest multiple of 3 less than 17 is 15, so I’m gonna write the 17 as 15 plus 2. Separate this into two fractions. The 15 over 3 becomes just ordinary 5, and then 5 plus two-thirds, conventionally this is written as 5 and two-thirds, the mixed numeral.
The third idea we’ll talk about is multiplying a fraction by its denominator. And the big idea is. Fraction times its denominator equals the numerator. So for example, 4/7th times 7 just gives us 4. This can be very useful in solving equations. So for example, if I have to solve for x.
All I have to do is multiply both sides by 5, and it will cancel on the left side and of course I’ll get x equals 15. Simplifying complex fractions. What’s a complex fraction? A complex fraction is a big fraction that has smaller fractions either in the numerator or the denominator or both.
A complex fraction purely of numbers can just be turned into ordinary fraction division. Complex fractions become a bit trickier when a bit of algebra is involved. So for example, suppose we have this ugly complex fraction. The strategy would be to multiply the numerator and the denominator of the big fraction by each denominator of the inner fractions.
So I’m gonna start with that 15, gonna multiply the numerator and denominator by 15 in the denominator it will just cancel and I’ll get the numerator which is x plus 6, the numerator of that little fraction. I’ve eliminated the fraction in the denominator at this point. I’ll distribute the 15 in the numerator.
So I get 15x and then I get 1 6th times 15 and that simplifies to 5 halves. Well now I still have a fraction in the numerator. The fraction in the numerator has a denominator of 2 so I’m gonna multiply the entire fraction by 2 over 2. And when I multiply through, I get this reduced version. So this is a simplified version, equal to the original, but no longer a complex fraction.
Now this is just a simple fraction with a single numerator and a single denominator. We discussed patterns of cancellation available in fraction multiplication in this video. Then talked about its relationship to the order of operations. We can split fractions by addition and subtraction in the numerator, but not in the denominator.
That becomes important in Algebra. We talked about the shortcut of multiplying a fraction by its denominator and we talked about how to simplify complex fractions.
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