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Pre-Algebra: Fraction & Mixed Number Arithmetic

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About this Lesson

  • Type: Video Tutorial
  • Length: 6:11
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 66 MB
  • Posted: 01/28/2009

This lesson is part of the following series:

Pre-Algebra Review (31 lessons, $61.38)

Adding or subtracting fractions or mixed numbers can become tricky if the denomintators are not the same. Professor Burger shows us how to find common denominators by finding common multiples of the denominators. First, he shows us that we can use the product of the two denominators as a common denominator. To do this, we must also multiply the numerators by the same numbers by which we are multiplying the denominators. (Effectively, we will be multiplying each fraction by X/X, which is equal to one, in which X is equal to a common multiple of both denominators). Then, we will rewrite both fractions with the common denominators and add them together, simplifying if needed. This process can also be done by finding the Least Common Multiple or Least Common Denominator (LCD), which is the least common multiple of the two denominators. By using the LCD, you will often have less work to do simplifying your answer once the math operation is performed.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Pre Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/prealgebra. The full course covers whole numbers, integers, fractions and decimals, variables, expressions, equations and a variety of other pre algebra topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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ADDING AND SUBTRACTING FRACTIONS AND MIXED NUMBERS

Suppose we want to now add or subtract two fractions, when the denominators aren’t the same. This is tricky, because we only know how to add and subtract fractions when we have the same denominator. How do we proceed? We convert our question into a question that we can answer. In this case, we convert the fractions into equivalent fractions that have the same denominator. We call that the common denominator. We get a common denominator by finding some kind of common multiple that each of these numbers in the denominator have. Let’s take a look at an example here. Let’s take one-sixth and add to it two-sevenths. Notice the denominators are not the same. So I can’t just add the numerators. I’ve got to convert these denominators to be the same. I need a common multiple. Remember a common multiple is a number that has each of these two numbers as factors. So, an easy way of doing it is just to pick the common multiple to be the product of the two denominators. In this case, 6 times 7. Let’s think about that. How would I make that fly? I can’t change the rational numbers numerically, but I can write them as equivalent fractions. That means if I multiply the denominator by something, I have to multiply the numerator by the exact same something. If I want a common denominator to be 6 times 7, which is 42, then I have to multiply the top and the bottom of the first fraction by 7. 7 on top; 7 on the bottom. It’s an equivalent fraction, so it hasn’t changed anything. To make this fraction 42 on the bottom, I have to multiply the top by 6 and the bottom by 6. It doesn’t change anything. Now, check out what I’ve got. I’ve got 7 times 1, which is 7, over 42, plus 12 over 42. Voila! Magically, these equivalent fractions now have the same denominator. What does that mean? It means that I can just add the numerators at my whim. 7 plus 12 is 19, over 42. The sum of those 2 fractions (with different denominators) actually equals 19 over 42. What was the method? The method was to generate a common denominator. In this case, the simplest way was just to get the common denominator to be the product of the two denominators.

Now, let’s look at this question. Here I have 4 and four-sevenths. I’m adding to it negative 4 and a half. I see two mixed numbers, and the first thing I should do is convert them into improper fractions. That’s going to allow me to add them up. Here we go. Converting to improper fractions is not a big deal. I take this 4 (the whole number part) and multiply it by the denominator, 7, and I add to it the 4 here. What do I see? I see 32, and I write that over the same bottom, 7. I’m going to do the same process here. What I want to do here is hold off that negative sign. That negative sign is there, but that’s in front of everything. So we take the 4, multiply it by the 2 (which is 8), then I add the 1 (that’s 9), and then I add the negative sign. That’s still over 2. See that—the negative sign we’re going to hold off on until I’ve converted that to an improper fraction, and then stick that negative always on the numerator. Great, now I’ve got two fractions, but unfortunately the denominators aren’t the same. Now I need to multiply the top and the bottom of this fraction by some number. Then multiply the top and the bottom of this fraction by some number, so the denominators will be the same.

Again, I’m looking for a common multiple. We could look for the least common multiple, which is the smallest, positive number that has both 7 and 2 as a factor. In this case, the least common multiple is 14. What do I have to multiply 7 by to make it 14? 2. If I multiply the bottom by 2, I have to multiply the top by 2. What do I have to multiply the denominator here by to make it 14? I have to multiply it by 7. That forces me to multiply the numerator by 7, too. I’m converting this, but I’m converting this to an equivalent system of two fractions. 2 times 32 is 64 over 14 plus negative 9 times 7 (negative 63) over 14. Look—the denominators are the same (by design). We can just add the tops. 64 plus negative 63 is simply 1, and that’s over the common bottom 14. So this equals 1/14. That’s the answer. Neat. There’s no problem. Even when we have mixed numbers, we first just convert them. We convert them to improper fractions, then we find a common denominator by finding a common multiple (maybe the least common multiple if you want). The product of the two denominators will always be a common multiple that will work. Neat stuff. Tricky, but not hard.

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