When working with fractions, you sometimes need to change your fraction to make it easier to work with. Watch this video lesson to learn how you can raise and reduce fractions to make your problem solving easier.
Raising and Reducing Fractions
When you are working with fractions, sometimes you need to change your fractions so you can easily solve your problem. The times that you need to change your fractions are when you are adding or subtracting your fractions. When you add or subtract fractions, your denominator (the bottom number) needs to be the same before you can complete your operation. Why is this? Well, think of adding 1/4 to 1/2. When you first look at your problem, you might think, how in the world am I supposed to add those? But, if you stop and think about it in terms of slices of your favorite pie, things might make a little more sense.
You know that 1/4 means that your pie is sliced into 4 slices where you have 1 piece; 1/2 means that your pie is sliced into 2 slices where you have 1 piece. You can't say that you have 2 slices because that answer doesn't explain that you have 2 different-sized slices. So, is there a way to cut your pies so that your pie slices are the same for both fractions? Yes, there is!
If you change your 1/2 fraction so that we have 4 slices instead of 2, we can take 2 of our 4 slices so that we still end up with 1/2 of our pie. So, our new fraction here would be 2/4. Since the pie is sliced the same as our 1/4 fraction, we can easily see that adding 1/4 to 1/2 would give us 3/4, or 3 slices out of a pie that is sliced into 4.
What we have just done is raise our fraction. Mathematically, raising a fraction involves multiplying the numerator and denominator by the same number. Reducing a fraction involves dividing the numerator and denominator by the same number. We reduce fractions when we have a fraction where we can divide both the numerator and denominator by the same number, such as 6/8, which can be divided by 2 on both top and bottom. Now, let's see how to raise and reduce fractions mathematically.
How to Raise a Fraction
We begin with raising fractions. If we have a problem such as 2/5 + 1/3, we see that we need to raise our fractions so that they have the same denominator, and we can answer our problem. We look at each of our fractions and compare it with the other. If I think about my pie slices, I ask myself if there is a way to the cut the pies so that the slices are the same size in both pies. I look at the denominator in both. I have a 5 and a 3. Well, I can't change my 3 to a 5 that easily. But I can change both to 15 easily. All I would have to do is multiply my 3 by 5 and my 5 by 3.
What did I just do? I just found my least common multiple, the closest number that both numbers can multiply to. So, I need to raise 2/5 by multiplying both the numerator and denominator by 3. Doing that, I get 6/15. I also need to raise 1/3 by multiplying both the numerator and denominator by 5. Doing that, I get 5/15. So, now my problem is 6/15 + 5/15. Because both fractions now have the same denominator, I can go ahead and add my numerators together. My answer is 11/15. Thinking about my pies, if both pies are cut the same, I just need to count the number of slices I have from both pies. If I have 6 slices from one and 5 slices from the other, I have 11 total slices from a pie that is sliced into 15. So, my answer is 11/15.
How to Reduce a Fraction
If our answer was something like 10/30, we would have to reduce our fraction. Why? We would have to reduce our fraction because both 10 and 30 can be divided by the same number. Both can be divided by 10. It's like cutting a pie for 3 people. We could cut it into 3 slices so each person gets 1, or we can cut it into 30 slices so each person gets 10 little slices. It's the same, but it is much easier to just slice the pie into 3 slices and be done with it. That is why we reduce fractions when we can. So, for the fraction 10/30, because both 10 and 30 can be divided by 10, we go ahead and divide both the numerator and denominator by 10 to get 1/3. So, our fraction 10/30 reduces to 1/3.
Let's try another example. See if you can follow along on your own. We are going to add 1/8 and 3/8. We see that both have the same denominator, so we can go ahead and simply add the numerators together. So, 1/8 + 3/8 = 4/8. Hmm. It looks like my answer can be reduced since both 4 and 8 can be divided by the same number: 4. Dividing both 4 and 8 by 4, I reduce 4/8 to 1/2. Oh wow, so 1/8 + 3/8 = 1/2. I get half a pie!
Now, let's review what we've learned. We learned that to raise a fraction means to multiply the numerator and denominator by the same number, and to reduce a fraction means to divide the numerator and denominator by the same number. We raise fractions when we have to add or subtract and our denominators are not matching. We raise either one or all the fractions so that they all end up with the same denominator. We reduce fractions whenever we have an answer where both the numerator and denominator can be divided by the same number.
This lesson can help you to accomplish the following objectives:
- Understand what it means to raise or reduce a fraction
- Determine when to raise and reduce fractions
- Follow the steps necessary to raise and reduce fractions