Operations with Fractions
Transcript
In this video, we're simply gonna talk about how to add, subtract, multiply, and divide fractions. First of all, addition and subtraction. We can only perform addition or subtraction on two fractions when they have a common denominator. With a common denominator, we can just add or subtract across the numerators.
For example, 1/5 + 3/5, we just add to get 4/5. 9/13- 6/13, 9- 6 is 3, that would be 3/13. Obviously, the test will demand more complicated math than this. Most often, when we have to add or subtract fractions, the fractions given do not happen to have the same denominator. We're given something like this.
For example, 1/3 + 1/7. In this case, we need to find a common denominator. That is, we need to find equivalent fractions of each fraction such that these equivalent fractions have the same denominator. To get equivalent fractions with the same denominator, I will multiply the first fraction by 7/7, and the second fraction by 3/3.
So I'd start out with this, multiply the first by 7/7, the second by 3/3. Of course, in each case, I'm multiplying by 1, so I don't really change the value. I get these fraction, 7/21 is another way of writing 1/3, and 3/21 is another way of writing 1/7. But writing them this way, now they have the same denominator. Now we can just add, 7 + 3 is 10, we get 10/21, and that's the sum.
Another example, 3/5- 1/3, multiply the first one by 3/3, the second one by 5/5. We get 9/15- 5/15. 9- 5 is 4, that's 4/15. For small numbers, we can simply multiply the numerator and denominator of each fraction by the denominator of the other fraction.
So for example, if we're adding a/b + c/d, we can just multiply the first fraction by d/d/, the second one by b/b. The problem with using this as a default strategy, this runs into big numbers very quickly. So for example, if I'm adding this, if I multiply the first one by 24/24 and the second one by 12/12, I'm gonna get enormous numbers, much bigger than 100.
That's gonna be kind of cumbersome to do math with those numbers. We'll notice 24 is actually a multiple 12. So all I really have to do is multiply the first fraction by 2/2. If I multiply by 2/2, then immediately I get a common denominator, 24. I can add and simplify. Another example, 1/14- 1/21, well, if I multiply the first one by 21/21, the second one by 14/14, I'm gonna get a number way over 100.
As the general rule, if you're doing simple calculations and you wind up with a number way over 100, you're probably doing things the hard way. Here I could just notice, well, both of these are factors of the number 42. So if I multiplied the first one by 3/3 and the second one by 2/2, I would get a common denominator of 42. And then it's easy, 3- 2, 1/42.
In that last example, I quote, unquote noticed that 14 and 21 had a common multiple in 42. And this may be discouraging. You might think, gee, well, I have to notice these things. What if I don't notice them? Well, turns out there's a general procedure for finding the least common multiple of two numbers.
This is also known as the least common denominator. This is discussed in the Integer Property module. So once you get to the Integer Property module, you'll be able to do this procedure and find the least common denominator of any two numbers. Here's some practice problems. I recommend pausing the video and trying these on your own.
These are the solutions. Now, multiplication of fractions. Of the four operations, multiplication is the easiest. It is by far the easiest. We just multiply across in the numerator and the denominator, piece of cake. 2/7 x 2/3, that's just 2 x 2 in the numerator, 7 x 3 in the denominator, 4/21, very easy.
What's a little trickier about multiplication of fractions is what you can cancel. So if we're multiplying 5/14 x 7/15, we can actually cancel the common factor 5 between the 5 in the numerator of one fraction and the 15 in the denominator of the other. They go down to 1 and 3 respectively.
We can also cancel the 7 in the numerator with the 14 in the denominator, the other one. They go down to a 1 and a 2, and we just wind up with 1/2 x 1/3, which is 1/6. Much, much simpler. When multiplying two or more fractions, you can cancel any numerator with any denominator.
If there's a common factor between any numerator and denominator, you can cancel it. And I'll say right now, always cancel before you multiply. This is a huge mathematical strategy that people overlook. If you always cancel before you multiply, you will make your life so much easier. Here's some multiplications for practice.
You might pause the video here and practice these right now. Here are the results. Some folks are confused by multiplication between a fraction and a whole number. It can be important to remember here that we can write that whole number as a fraction by putting it over 1. And then it makes it very clear how the fraction multiplication works.
Finally, division of fractions. To divide by a fraction, we multiply by its reciprocal. So if we have 1/4 and we divide by 3/2, this is 1/4 times the reciprocal, 2/3. And of course we can cancel and we get 1/6. If we take the fraction 3/20 and divide by 6/5, of course, we can multiply by the reciprocal, 3/20 x 5/6.
We can do a little canceling, and once it's simplified, then we can get an answer. To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction. So 6 divided by 3/4 is the same as 6 over 4/3. Again, we're gonna write that 6 as 6/1, and then just do the fraction multiplication.
To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number, which will be in the form 1/n. So if I have 3/5 divided by 2, this would be 3/5 times the reciprocal of 2, which is 1/2. And of course, this is 3/10. Here's some practice division problems.
Pause the video and practice these on your own. Here are the solutions. In this video, we talked about adding and subtracting fractions, and the skill of finding a common denominator. We talked about multiplying fractions and cancellation. We had the proviso cancel before you multiply, one of the most valuable math strategies you can use in preparing for the test.
And we talked about division with fractions, including number divided by fraction and fraction divided by a number.
Read full transcriptFor example, 1/5 + 3/5, we just add to get 4/5. 9/13- 6/13, 9- 6 is 3, that would be 3/13. Obviously, the test will demand more complicated math than this. Most often, when we have to add or subtract fractions, the fractions given do not happen to have the same denominator. We're given something like this.
For example, 1/3 + 1/7. In this case, we need to find a common denominator. That is, we need to find equivalent fractions of each fraction such that these equivalent fractions have the same denominator. To get equivalent fractions with the same denominator, I will multiply the first fraction by 7/7, and the second fraction by 3/3.
So I'd start out with this, multiply the first by 7/7, the second by 3/3. Of course, in each case, I'm multiplying by 1, so I don't really change the value. I get these fraction, 7/21 is another way of writing 1/3, and 3/21 is another way of writing 1/7. But writing them this way, now they have the same denominator. Now we can just add, 7 + 3 is 10, we get 10/21, and that's the sum.
Another example, 3/5- 1/3, multiply the first one by 3/3, the second one by 5/5. We get 9/15- 5/15. 9- 5 is 4, that's 4/15. For small numbers, we can simply multiply the numerator and denominator of each fraction by the denominator of the other fraction.
So for example, if we're adding a/b + c/d, we can just multiply the first fraction by d/d/, the second one by b/b. The problem with using this as a default strategy, this runs into big numbers very quickly. So for example, if I'm adding this, if I multiply the first one by 24/24 and the second one by 12/12, I'm gonna get enormous numbers, much bigger than 100.
That's gonna be kind of cumbersome to do math with those numbers. We'll notice 24 is actually a multiple 12. So all I really have to do is multiply the first fraction by 2/2. If I multiply by 2/2, then immediately I get a common denominator, 24. I can add and simplify. Another example, 1/14- 1/21, well, if I multiply the first one by 21/21, the second one by 14/14, I'm gonna get a number way over 100.
As the general rule, if you're doing simple calculations and you wind up with a number way over 100, you're probably doing things the hard way. Here I could just notice, well, both of these are factors of the number 42. So if I multiplied the first one by 3/3 and the second one by 2/2, I would get a common denominator of 42. And then it's easy, 3- 2, 1/42.
In that last example, I quote, unquote noticed that 14 and 21 had a common multiple in 42. And this may be discouraging. You might think, gee, well, I have to notice these things. What if I don't notice them? Well, turns out there's a general procedure for finding the least common multiple of two numbers.
This is also known as the least common denominator. This is discussed in the Integer Property module. So once you get to the Integer Property module, you'll be able to do this procedure and find the least common denominator of any two numbers. Here's some practice problems. I recommend pausing the video and trying these on your own.
These are the solutions. Now, multiplication of fractions. Of the four operations, multiplication is the easiest. It is by far the easiest. We just multiply across in the numerator and the denominator, piece of cake. 2/7 x 2/3, that's just 2 x 2 in the numerator, 7 x 3 in the denominator, 4/21, very easy.
What's a little trickier about multiplication of fractions is what you can cancel. So if we're multiplying 5/14 x 7/15, we can actually cancel the common factor 5 between the 5 in the numerator of one fraction and the 15 in the denominator of the other. They go down to 1 and 3 respectively.
We can also cancel the 7 in the numerator with the 14 in the denominator, the other one. They go down to a 1 and a 2, and we just wind up with 1/2 x 1/3, which is 1/6. Much, much simpler. When multiplying two or more fractions, you can cancel any numerator with any denominator.
If there's a common factor between any numerator and denominator, you can cancel it. And I'll say right now, always cancel before you multiply. This is a huge mathematical strategy that people overlook. If you always cancel before you multiply, you will make your life so much easier. Here's some multiplications for practice.
You might pause the video here and practice these right now. Here are the results. Some folks are confused by multiplication between a fraction and a whole number. It can be important to remember here that we can write that whole number as a fraction by putting it over 1. And then it makes it very clear how the fraction multiplication works.
Finally, division of fractions. To divide by a fraction, we multiply by its reciprocal. So if we have 1/4 and we divide by 3/2, this is 1/4 times the reciprocal, 2/3. And of course we can cancel and we get 1/6. If we take the fraction 3/20 and divide by 6/5, of course, we can multiply by the reciprocal, 3/20 x 5/6.
We can do a little canceling, and once it's simplified, then we can get an answer. To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction. So 6 divided by 3/4 is the same as 6 over 4/3. Again, we're gonna write that 6 as 6/1, and then just do the fraction multiplication.
To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number, which will be in the form 1/n. So if I have 3/5 divided by 2, this would be 3/5 times the reciprocal of 2, which is 1/2. And of course, this is 3/10. Here's some practice division problems.
Pause the video and practice these on your own. Here are the solutions. In this video, we talked about adding and subtracting fractions, and the skill of finding a common denominator. We talked about multiplying fractions and cancellation. We had the proviso cancel before you multiply, one of the most valuable math strategies you can use in preparing for the test.
And we talked about division with fractions, including number divided by fraction and fraction divided by a number.
