Back To CourseELM: CSU Math Study Guide
16 chapters | 140 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
We encounter fractions every day. Think about food. When you bake, you might need a half cup of sugar or a quarter teaspoon of vanilla. When there's pizza, you may hope for more than just 1/8 of the pie. When you're sneaking cookies, maybe you have 3 1/2, because 4 would just be gluttonous.
Fractions are also very common in sports. At the track, you might run 4 3/4 laps. When you watch a football game, sometimes it's the final quarter that's most exciting. On the golf course, you might celebrate making par on 5 out of 9 holes.
You can get through life without knowing how to solve problems with fractions, but what kind of life would that be? Half full? Half empty? You would never know.
Let's look at a few situations involving fractions and mixed numbers. Remember, a fraction is a part of a whole number, like 1/2 or 5/8. A mixed number is a whole number and a fraction, like 33 1/3.
Let's say you and your friend Ginger team up for a talent competition reality show. In order to win the final round, you need to get more votes than your competitors. Your flaming sword juggling routine gets 4/9 of the votes. The family that does trapeze tricks with a tiger gets 2/7 of the votes. The guy conducting the monkey string quartet gets 1/3 of the votes. Who wins?
This question is just asking us which fraction is largest. To do this, we need the least common denominator, or the smallest multiple of the denominators. If we started listing multiples, we'd find that 63 is what we want.
To convert 4/9 to 63rds, we multiply it by 7/7. That'll be 28/63. With 2/7, we multiply by 9/9 to get 18/63. With 1/3, we multiply by 21/21 to get 21/63. Now we just compare numerators - 28, 18, 21. You win!
Unfortunately, someone notices that 28 + 18 + 21 = 67, and 67/63 is more than 100% of votes, thereby invalidating your victory due to voter fraud. You and Ginger try to start fresh on a cupcake baking show. Together, you need to bake 1,000 cupcakes in one hour. If you make 1/2 of the cupcakes and Ginger makes 2/5, did you make enough?
We need to add 1/2 and 2/5. To do this, we again need the least common denominator. Here it's 10. To convert 1/2 to 10ths, we multiply it by 5/5. So it's 5/10. With 2/5, we multiply by 2/2, so it's 4/10. We then add the numerators, so 5/10 and 4/10 gives us 9/10. So 9/10 of our cupcakes are ready. Of course, we need 10/10. And now time is up!
To deal with your loss, you eat your way through many of those cupcakes. Then you and Ginger decide to go on a competitive weight loss show. One of the challenges is to run on a treadmill as far as you can go. You run 3 2/3 miles. Ginger ran 4 5/6 miles. How many more miles did Ginger run than you?
We want to know what 4 5/6 - 3 2/3 is. With a mixed number, we need to first convert it to an improper fraction, or a fraction with a flair for off-color jokes. Wait, no, it's a fraction with a larger numerator than denominator. To do this, just multiply the whole number by the denominator, then add it to the numerator. 4 5/6 becomes 29/6, and 3 2/3 becomes 11/3. Now we need a common denominator, which is 6. We multiply 11/3 by 2/2 to get 22/6. And then we just subtract numerators. 29 - 22 is 7. So Ginger ran 7/6, or 1 1/6, more miles than you.
Things are getting tense between you and Ginger and, after the weight loss show, you both go on a competitive dancing show. Now you're competing head-to-head. Of all the types of dances you do, Ginger is an expert at 5/6. And when you were getting along better, she taught you 2/3 of what she knows. How many dances can you do?
We need to multiply 5/6 * 2/3. To multiply fractions, simply multiply the numerators, then multiply the denominators. 5 * 2 is 10. And 6 * 3 is 18. So you know 10/18, or 5/9. That's a little better than 1/2, but definitely not as much as Ginger.
After Ginger beats you on the dance show with an amazing paso doble, you find out that she was behind that talent show voter fraud. And she sabotaged the cupcake competition. Plus, she cheated on the treadmill race. So you decide to take her down on a fashion design show. You've gotten so good at working with fractions that you think you have an edge here.
You tell her she has 6 2/3 yards of polka dot fabric. She needs to divide that into 1/4 yard segments for an awesome leisure suit pattern you gave her. How many segments can she get with that fabric? Okay, this is just 6 2/3 divided by 1/4. To divide fractions, just flip the second fraction, which gives you its reciprocal. But first, we need to convert 6 2/3 to an improper fraction. 6 * 3 is 18, and 18 + 2 is 20. So it's 20/3. 20/3 divided by 1/4 is 20/3 * 4/1, or 80/3. If we convert that back to a mixed number, we get 26 2/3. So she'll have 26 complete 1/4 yard segments and 2/3 of a segment. That's going to be one awesome leisure suit.
And, in fact, it was. She won the competition.
In summary, let's look at what we've learned here. First, Ginger is really not the friend you thought she was, is she? And it's not nice to seek fashion design vengeance. More important, fractions and mixed numbers are everywhere. And it only takes a few simple tricks to work with them.
Mixed numbers should first be converted to improper fractions. Then, with addition and subtraction, always find the least common denominator.
Multiplication is simpler - just multiply the numerators, then the denominators. With division, flip the second fraction to get its reciprocal, then multiply. And now your glass is half full!
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Back To CourseELM: CSU Math Study Guide
16 chapters | 140 lessons