Lesson 70:  More About Fractions

In this lesson, we'll review some of the things we learned about fractions, to prepare us for the next lesson where we'll be adding and subtracting them. 

First, review all of the previous lessons on fractions, as well as the lesson on LCM, and the lesson on GCF. Make sure that you fully understand all of the material in those lessons.  Recall that to compare fractions, we need for them to have a common denominator.  This is because if they don't, then we are not comparing parts of the same whole. 

For easy and everyday fractions, we might not have to do this.  For example, we know that 1/2 of a pizza is bigger than 1/4 of a pizza.  But if the fractions were not as common, the answer might not be so obvious.  For example, we might need to compare 4/7 and 5/9.  We would first need to get these two fractions to have a common denominator, and then compare the numerators to see which was larger.

Remember that we said that it's always OK to multiply the numerator and the denominator of a fraction by the same number.  Let's look at the reason for this.  Let's say we multiply the numerator and denominator of a fraction by 3.  What we are really doing is multiplying the fraction by 3/3 (three-thirds).  What is 3/3?  It's really one whole, or the number 1.  If we cut a pizza into 3 pieces, and eat all 3, then we ate the whole pie.

So we're multiplying the fraction by 1.  What do we know about multiplying any number by 1?  It doesn't change it at all.  In the case of a fraction, we're changing its numerator and denominator, but we're converting the fraction to one that is equivalent.

For practice, let's convert 2/5 and 1/2 so that they have a common denominator.  This is where the lesson on LCM comes in.  The best thing for us to do is find the lowest common multiple of the denominators.  That is what we will convert each fraction to.  What is the LCM of 2 and 5?  According to the lesson on LCM, the answer is 10.  We can multiply the numerator and denominator of the first fraction by 2, to get 4/10.  For the second fraction, we can multiply each by 5, to get 5/10.  Now the fractions have the same denominator.  It is easy to compare them, and as we'll see in the next lesson, it will be easy to add them.

Very often we are asked to reduce a fraction to lowest terms.  This is sometimes referred to as simplifying a fraction.  Sometimes we need to do this in order to help us add or subtract fractions, and sometimes we just do it to present our answer in a simpler way.  For example, 1/2 is just simpler than 400/800, although they are actually the same fraction.

What we do is similar to what we did above, except we must first find the GCF of the numerator and denominator.  For example, let's reduce 24/40 to lowest terms.  Referring to the lesson on GCF, we can see that the GCF of the numerator and denominator is 8.  We then divide both numerator and denominator by 8 to get 3/5.  Note that what we really did was divide the fraction by 8/8, which is 1.  We know that dividing a number by 1 doesn't change it, so that is why we are allowed to do this.  

Note that in problems such as the ones in this lesson, it will usually be easy to find the LCM or GCF as needed, using the methods shown in those lessons.  If you are given a fraction with very large numbers, it can be trickier to find the LCM or GCF.  Please contact me if you have such a problem, and I can show you a special technique that can be used.

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