Expanding and Reducing Fractions
Two things you might be asked to do are expand and reduce (or simplify) fractions. Expanding a fraction means making it "bigger" while reducing a fraction means making it "smaller."
Expanding fractions means making them "bigger." For example, let's say you ate 3/8 of that pizza, but your friend wanted to know how many sixteenths you ate. You would have to expand 3/8 in order to give him an answer. Expanding a fraction does NOT change how much you have (or ate), it just changes the way you are telling the person what you ate. You would expand the fraction like this:
Look at the denominators. What can you multiply 8 by to get to 16? Think of your 8 times tables. 8 times what gives you 16? Well, we know that 8 x 2 = 16. That means that we are expanding the fraction by 2. You already have the denominator expanded, which means you still have to expand the top. To do this, you take the same number you used for the bottom multiplication (in this case, 2) and multiply that number by the top number (the numerator). It would say: 3 x 2 = ? What is 3 x 2? We know that the answer to this question is 6, so 6 is now our new numerator. When we put this altogether, we get 6/16, so we know that 3/8 expanded is 6/16. Then, you can tell your friend you ate 6/16 of the pizza. It's the same as what you had originally counted, 3/8, but it's a different way of telling him.
Let me show you one more time. This time, I'm going to give you the fraction, and the number we're expanding by, and you'll get the new fraction on your own.
Expand 3/4 by 5.
Think: expand the numerator (top number). 3 x 5 = ? We know that 3 x 5 = 15, so our new numerator is 15. Next, expand the denominator. 4 x 5 = ? We know that 4 x 5 = 20, so 20 is our new denominator. So, our entire expanded fraction is 15/20. 15/20 still means the same as 3/4, it's just a different way of writing it.
Reducing (Simplifying) Fractions
Now we'll talk about reducing fractions. Reducing fractions (also known as simplifying fractions) is the opposite of expanding them. Sometimes we see really big fractions that can be written (or said) in an easier way. For example, you wouldn't tell someone that you ate 75/200 of a pizza, you would want to say it more simply so that he or she can understand you. In order to do this, you would need to reduce (simplify) the fraction. There are two different ways to do this, I will show you both of them.
The first way is by trying to figure out the biggest number that divides into both the numerator and denominator of your fraction. I'm going to use 5/20 for my example. I want to think of a number that I can divide into both 5 and 20. I can put 5 into both 5 and 20. Then, I take the number I thought of (5) and divide the numerator and denominator by this number. In this example, 5 ÷ 5 = 1, and 20 ÷ 5 = 4. So now, my new numerator is 1 and my new denominator is 4, making my fraction 1/4.
It would look like this:
Now I'll show you the second way. For the second way, you need to list the prime factors of each number. In this case, the prime factors of 5 are 1 and 5. The prime factors of 20 are 2 x 2 x 5. You would put these where they belong on the parts of the fraction bar, so 1 x 5 would go on top (where the numerator goes) and 2 x 2 x 5 would go on the bottom (where the denominator goes). It would look like this:
Now you would look for any common factors between the two numbers. In this example, 5 is the common factor, so we would cross out the 5 on both the top and bottom.
Now you would look at the numbers you have left, 1 on the top, and 2 x 2 on the bottom. You would multiply the 2 x 2 back together to get 4. Then, you're done! Your answer would be 1/4.
Remember that when you're reducing, you want to get a prime number on the top (as the numerator). This means that you have reduced it "all the way;" you can't reduce it anymore.
Finally, whether you're expanding or reducing, keep in mind that you are not changing the AMOUNT that the fraction represents. Expanding and reducing fractions is just saying them in a different way. Normally, when we work with fractions, we want them to be in "simplest terms," or reduced all the way.