Basic Math: Lesson 6, Part 6 Converting Fractional Expressions
Concepts covered: Step-by-Step directions on how to perform (by hand and using the calculator) many different type of conversions.
Step 1: Introduction
A Proper Fraction is a fraction whose numerator is smaller than the denominator (ignoring negative signs).
Step 2: Examples of Proper Fractions
Here's some examples of Proper Fractions. Here: One is smaller than two, three is smaller than seven, 157 is smaller than 1000. Again, I don't even consider the negative signs. Don't even look at those. So if the top is smaller than the bottom, they're Proper Fractions.
Step 3: Improper Fractions
An Improper Fraction is a fraction whose numerator is greater than or equal in value to the denominator (again, ignoring any negative signs). Here's some examples of Improper Fractions. As you can see, the numerators are all larger than the denominators. So, "seven" is larger than "two". "Five" is bigger than "four". Again we're not even considering the negative whatsoever. 90 is larger than 83. They're all Improper.
Step 4: Improper Fractions
Integers can be represented as Improper Fractions simply by placing a "one" in the denominator. So like this "four" can really be "four over one". This is now an Improper Fraction. "Negative three" can be done likewise. And see, according to our definition, the numerator doesn't have to be just larger, it can actually be equal. So this technically is an Improper Fraction. "One over one".
Step 5: Improper Fractions Example
Let's take a look at a pie graph of "nine-eighths". Nine slices is equal to eight slices plus one more. In a simulator we can represent this as pizza. Our "nine-eighths" is eight slices plus one more. But there's another way of seeing this. This is one entire whole pizza. So this is just "one". Plus we have one more slice. How would you represent that? An "eighth". So we have "one-eighth". Now we can actually combine this together and it will look like this. You would read this as "one and one-eighth".
Step 6: Mixed Numbers
And this brings us to our next definition. A Mixed Number is a compact form of expressing the sum of an integer and a proper (not improper) fraction. It is just the integer followed by the proper fraction, with no space or symbols between the numbers.
Step 7: Complex Fraction
When reading Mixed Numbers, always use the word "and" between the integer part and the fractional part. How would you read this? This is "two and three-fifths". Here we have "negative one and nine-tenths". A Complex Fraction is a fraction with other fractions within it. These mini-fractions can appear in the numerator, denominator or both. Now a Complex Fraction is also known as a Compound Fraction. But I don't want to use that term because it can be confused with Compound Fracture. But, according to some students, the fraction is actually more painful than the fracture. So let's stick with the Complex Fractions. Let's do some examples.
Step 8: Complex Fraction Example
Here's some examples of Complex Fractions. Notice on the first one here, I actually used the slanted fraction bar. In this case, we want to keep it nice and compact. So that looks kind of nice. And over here, we have a fraction divided by a fraction. And here we have an integer divided by a fraction. Again slanted. Fraction bar's okay, more compact. For example six, I'd like you to classify each of these fractions as either a Proper Fraction, Improper, Mixed Number or Complex Fraction. So I want you to give it a try first and then I'll give you the answer. Let's take a look at the first one. Well the numerator is smaller than the denominator so this is a Proper Fraction. Here, fraction over something is kind of complicated, well it's a Complex Fraction. Here we have an integer with a fraction; Proper Fraction. So this is actually a Mixed Number. Over here again, the numerator is smaller than the denominator. This is Proper. Over here we have a mess. But how would you classify this? Well it's a fraction or actually a Mixed Number over a fraction so, this is going to be a Complex Fraction. And here this is going to be, notice the numerator is bigger than the denominator, So this is going to be an Improper Fraction.
Step 9: Exercise
For Exercise number seven, I'd like you to write down the Improper and Mixed Number that's represented by the shaded regions right here. In this figure right here. So give that a try. Well let's see what's going on here. This is cut into four parts. One, two, three, four. So is this; one, two, three, four. We have four, five, six, seven. Seven shaded regions. So we have seven but it's really cut into four parts each. So our Improper Fraction is "seven-fourths". But now how would this be represented as a Mixed Number? Well this is one whole figure right? So it's going to be "one". And we have one, two, three out of the four. So we have "one and three-fourths". Or three quarters.Let's take a look at this one. I want you to try first.
Step 10: Conclusion
What's going to be the Improper Fraction and the Mixed Number? The representation of this scenario. Well we have everything cut into thirds. We have one, two, three, four, five, six, seven thirds. That's my Improper. But what else do we have? We have one whole, two wholes. We have two whole parts. And we have a little one left over. That's going to be one of three, so we do that. So it's going to be "two and one-third" is our Mixed Number. Now this is a little bit tougher. I want you to take a good look at this and decide well what's going to be the Improper and Mixed Number representation of this. Well let's see. Everything, you'll notice, is being cut in half. This is "one" so that's a half of that and there's half of there. So everything's in halves. We have one, two, three, four, five, six, seven halves. So that's going to be our Improper Fraction. What about the Mixed Number. Well we have one whole unit, two, three. And then we have half left over. So it can be "three and one-half".
