The Power of Algebra; No. 009; Fractions

<v Watson>[music playing] [water rushing] Oh by George, Holmes. What a romantic sight. <v Watson>Niagara Falls by moonlight. <v Holmes>Yes, indeed, quite breathtaking. <v Watson>Well what the ?deuce? is that? <v Holmes>Just some ducks quacking Watson. Relax and enjoy the view. <v Watson>Uh yes, but I say, isn't it a bit too dangerous standing right on the edge of the falls <v Watson>in the middle of the night? I mean, what if Moriarty- <v Holmes>Don't worry about that bird brain, Watson. <v Holmes>I'm much too clever to fall for any of his tricks. <v Moriarty>[boat engine whirring] [laughing] Those two flat foots are going to fall a lot further <v Moriarty>than they think. [music plays] <v Watson>?inaudible?. Oh, oh, oh good grief! <v Watson>It's Moriarty. <v Moriarty>I knew I'd triumph in the end. <v Moriarty>If you two are so fond of Niagara Falls, now's your chance to get a really good <v Moriarty>close up view! <v Holmes>But you can't push us over the falls just like that. <v Holmes>It's it's simply not cricket. <v Watson>No, it's just not done, old chap. <v Moriarty>Isn't it? Well, how should I push you over then? <v Moriarty>[music playing]

<v Holmes>In a barrel, of course. That's the proper way. <v Moriarty>Oh, good idea. Oh what am I saying! <v Moriarty>I left my barrel 40 kilometers upstream. <v Moriarty>Oh I know! I'll tie you two up against this tree, then I'll canoe up the <v Moriarty>river to fetch the barrel. After all, there's plenty of time. <v Moriarty>I've got all night. <v Holmes>I wouldn't be so sure Moriarty. The current in this river runs at 6 <v Holmes>kilometers per hour. It's now just about midnight and the park ranger comes <v Holmes>on duty shortly after 5 A.M. so he'll rescue us before you can get back. <v Moriarty>Curses. Could Holmes be right? Let me see. <v Moriarty>I have 5 hours to go 40 kilometers against the current and 40 kilometers <v Moriarty>with the current. So how fast must I go? <v Moriarty>Ah... ah yes, I must average 8 kilometers an hour <v Moriarty>over the whole trip. But that's exactly the speed of my motorized canoe! <v Moriarty>[laughs] I'll be back before the park ranger <v Moriarty>gets here. No problem. <v Moriarty>Then you two will really be for the high jump.

<v Moriarty>[water rushing] [laughter] <v Watson>Are you quite sure Moriarty can't make it back in 5 hours Holmes? <v Holmes>Absolutely certain. Instead it'll take him nearly 23 hours to do the roundtrip. <v Watson>You're certain? <v Holmes>Yes. In fact, and, here comes the clue Watson, I'll bet you a <v Holmes>bill [water rushing] of any denomination that I'm right. <v Man 1>Welcome to the 9th program in the Power of Algebra series. <v Man 1>[music playing] In the previous program, we continue the exploration of factoring <v Man 1>and saw how what appears to be a quite complicated equation can <v Man 1>be simplified by looking for the common factor, which in this case is 4. <v Man 1>In this program, we're going to see how to simplify algebraic expressions that involve <v Man 1>fractions. Algebraic fractions are no more difficult to do than any other <v Man 1>fractions. Exactly the same rules apply whether you're talking about three eighths <v Man 1>or X over Y.

<v Man 1>You still have a numerator on top and a denominator on the bottom. <v Man 1>But beware of unknowns in the denominator. <v Man 1>You can't divide by zero. <v Man 1>You can reduce a more complicated looking arithmetic fraction to its lowest terms <v Man 1>by looking for the factor common to the numerator and the denominator. <v Man 1>Here it's 7. <v Man 1>When you have a more complicated looking algebraic fraction, you can use the same <v Man 1>technique. In this case, the common factors are 3, X, <v Man 1>and Y. So the fraction can be reduced to 2 X over 3. <v Man 1>Here's one that looks even trickier but works in exactly the same way. <v Man 1>Only here, in order to see what the common factor is, we have to first factor <v Man 1>the top and bottom into binomials. <v Man 1>X minus 1 square becomes X minus 1 times X <v Man 1>minus 1 and X square minus 9 X plus <v Man 1>8 becomes X minus 1 times X minus 8. <v Man 1>Now it's easy to see that the common factor is X minus 1, which allows us

<v Man 1>to reduce the original fraction to something much simpler. <v Man 1>If factoring works the same way for fractions and algebra as it does for fractions and <v Man 1>arithmetic, so do all the basic operations. <v Man 1>When you multiply three eighths by five sevenths, you first multiply the two numerators <v Man 1>to get 15 and then multiply the two denominators to get 56. <v Man 1>So when you multiply A over B by C over D, <v Man 1>you first multiply A times C, then B times D to get <v Man 1>AC over BD. <v Man 1>The same goes for division. If you have to divide three eighths by five sevenths, <v Man 1>you replace the divisor, five sevenths, by its reciprocal. <v Man 1>In other words, you turn it upside down and multiply by seven fifths to get <v Man 1>twenty one fortieths. And that's exactly what you'd do in algebra. <v Man 1>If you have to divide A over B by C over D, you replace the divisor <v Man 1>by its reciprocal and multiply by D over C to get <v Man 1>AD over BC.

<v Man 1>[water flowing] [birds chirping] That's easy for multiplication and division. <v Man 1>Unfortunately, it's not quite so straightforward for addition and subtraction. <v Man 1>I need to get this straight because I've got to figure out how long it would take me to <v Man 1>fill this pool if I didn't have a fill pipe and instead I had to just do <v Man 1>it with this garden hose. [music plays] <v Orville Baker>My name is Orville Baker. I'm a technician here at WLPB. <v Orville Baker>I just received my B.S. degree in physics here at Southern University in Baton Rouge. <v Orville Baker>Algebra is important to all mathematics. <v Orville Baker>And regardless of what type of mathematics that you do, algebra is always <v Orville Baker>?inaudible? in order to get the end result you're trying to find in any mathematical <v Orville Baker>endeavor. And in physics in particularly no matter how complex the equation is, <v Orville Baker>you always have to use algebra to find the solution. <v Orville Baker>Mathematics and particularly algebra, the language of science. <v Orville Baker>In fact, it would be s-almost safe to say that any physicist that is competent <v Orville Baker>is also a very competent mathematician.

<v Orville Baker>If you 'xpect to survive in the next century, you gon' have to have a background in <v Orville Baker>mathematics. Because our technology, our job force, our job descriptions <v Orville Baker>are changing more and more to mathematically inclined understanding. <v Orville Baker>If you're going to deal with the computers, if you're going to deal even with car <v Orville Baker>manufacturing, you have to understand the difference between metric and English <v Orville Baker>system and be able to make those conversions usin' mathematics, usin' algebra. <v Orville Baker>Those will survive the next century. Those who have the jobs and those who have a better <v Orville Baker>standard living were those who have- who are not afraid to look at math <v Orville Baker>and tackle it. <v Man 1>[water rushing] [music playing] I said that adding and subtracting algebraic fractions <v Man 1>isn't quite as straightforward as multiplying and dividing them. <v Man 1>The big difference is that when you're multiplying any fractions, you can multiply both <v Man 1>the numerators and the denominators. <v Man 1>When it comes to addition, you can only add the numerators if both of the denominators <v Man 1>are the same. In other words, you can only add fractions if they have a common <v Man 1>denominator. Here's a very simple example.

<v Man 1>Add two ninths and five ninths. <v Man 1>You add the numerators and keep the same denominator to get two plus five over <v Man 1>nine or seven ninths. <v Man 1>It's the same when adding algebraic fractions. <v Man 1>When you add A over C and B over C, you get A plus B <v Man 1>over C. <v Man 1>The identical process takes place with subtraction. <v Man 1>[birds chirping] But you can only do that if the two fractions that you're adding or <v Man 1>subtracting have got a common denominator. <v Man 1>What if they don't? What if you're faced with adding, say, oh, three sevenths and two <v Man 1>fifths? How do you take those two different denominators and turn them into a common <v Man 1>denominator? Well, we look for the least common denominator, the smallest <v Man 1>multiple of 7 and 5, which is 35. <v Man 1>And to change these two denominators into 35, we multiply both terms in the first <v Man 1>fraction by 5 and both terms and the second fraction by 7, which gives us <v Man 1>15 35ths plus 14 35ths. <v Man 1>Now we can add these fractions to get 29 35ths.

<v Man 1>We use precisely the same technique to add algebraic fractions with unlike denominators. <v Man 1>Hmm as here, the least common denominator, LCD, in <v Man 1>this case is 12 X. To change the two unlike denominators into <v Man 1>12 X, we multiply both terms in the first fraction by 4 <v Man 1>and both terms in the second fraction by 3, which gives us 20 <v Man 1>over 12 X plus 27 over 12 X. <v Man 1>Then we add the two fractions to give us 47 over 12 X. <v Man 1>[water rushing] So once you understand how the least common denominator process works for <v Man 1>adding and subtracting fractions and arithmetic, you'll also know how it works in <v Man 1>algebra. Which brings me back to the swimming pool problem. <v Man 1>How long would it take to fill this pool if I didn't have that fill pipe? <v Man 1>[bubbles popping] And instead I had to do it with just a garden hose. <v Man 1>Well, with the fill pipe running alone, takes it 5 <v Man 1>hours to fill the pool. With both the fill pipe and the hose running, it only takes

<v Man 1>3 hours. So I take the formula for a rate of work, which is rate quals <v Man 1>work divided by time. <v Man 1>Now the work to be done is to fill one pool. <v Man 1>So work equals one. <v Man 1>And since the fill pipe takes 5 hours to fill the pool, we can say that its rate is <v Man 1>1 over 5. But since the hose's rate is unknown, we can only <v Man 1>say it's one over X hours. <v Man 1>In other words, the total rate is 1 over 5 plus 1 <v Man 1>over X. And since this combined rate fills the pool in 3 hours, <v Man 1>our time here is 3. <v Man 1>Now, if we rearrange all this into the equation for work, which is work equals <v Man 1>rate multiplied by time we get this. <v Man 1>And if we multiply that out, we get 1 equals 3 over 5 plus 3 <v Man 1>over X. The least common denominator here is 5 X. <v Man 1>Since we're working with an equation, we keep it balanced by multiplying both sides <v Man 1>by 5 X and we end up with 5 X equals 3 X

<v Man 1>plus 15 or 2 X equals 15 or <v Man 1>X equals 7.5. <v Man 1>[water running] So, I've been able to figure out that should this fill up ever be out of <v Man 1>action [bubbles popping], I'll be able to fill the pool with this garden <v Man 1>hose in 7 and a half hours. <v Man 1>All thanks to knowing about the least common denominator. <v Man 1>Holmes and Watson should thank their lucky stars that Moriarity didn't have access to the <v Man 1>same information. <v Watson>[music playing] I wish we knew the time Holmes, but I can't get my pocket watch. <v Holmes>Look at the sun man, it's just coming over the horizon. <v Holmes>[water rushing] The sun rises just after 5:00 A.M. <v Holmes>today. <v Watson>And still no sign of Moriarty. <v Holmes>They're right on cue, here comes the park ranger to set us free! <v Holmes>[grunts] Thank you, Ranger. <v Holmes>Now we can take a leisurely stroll along the river. <v Holmes>Moriarty won't make it back until just before 11 o'clock tonight. <v Watson>Oh what an ?intellect?! But what was the clue this time?

<v Watson>Eh something to do with uh denomination? <v Holmes>Least common denominator. <v Holmes>Do I have to spell everything out for you? <v Watson>Well ?inaudible? [sighs] But what exactly did Moriarty do wrong? <v Watson>[music plays] <v Holmes>Well what did he want to calculate? <v Watson>How fast his canoe should go to get from the top of the falls, eh point <v Watson>A uh 40 kilometers upstream to <v Watson>where his barrel was at point B. <v Watson>Ran back again. <v Holmes>Facing a current as he goes upstream of 6 kilometers per hour. <v Holmes>Don't forget that. <v Watson>Uh yes and having to compete the round trip between midnight and <v Watson>5 A.M., a total time of 5 hours. <v Holmes>So the equation we want here is time equals distance divided by <v Holmes>rate. <v Watson>Right. The time is 5 hours, the distance <v Watson>is 40 kilometers upstream, divided by the rate which is our unknown <v Watson>X minus the current, which is 6 kilometers an hour, because

<v Watson>the current would be going against Moriarty. <v Holmes>And then we add to that the distance downstream, which is also 40 kilometers <v Holmes>divided by X, plus the current of 6 kilometers an hour because <v Holmes>now Moriarty would have the current going with him. <v Watson>But I say, you can't add those two fractions because they have unlike denominators. <v Watson>Oh, I see it all now. We have to look for the least common denominator. <v Holmes>Which is very simple here. The smallest multiple of X minus 6 <v Holmes>and X plus 6 is X minus 6 times <v Holmes>X plus 6. <v Watson>So we multiply everything in the equation by that. <v Holmes>So the whole thing multiplies out to this. <v Holmes>And then this. And then this. <v Watson>Which we change into the standard for import a quadratic equation. <v Watson>Then simplify by factoring out the five. <v Watson>And then factor that into two binomials.

<v Holmes>Revealing that the two possible values of X are negative two, which is <v Holmes>impossible or positive 18. <v Holmes>Which tells us that the speed at which Moriarty should have gone to make the round trip <v Holmes>was 18 kilometers per hour, not 8 kilometers per hour. <v Watson>Well, then how did Moriarty arrive at that incorrect answer? <v Holmes>Because instead of looking for the least common denominator, he not only added <v Holmes>the numerators, but he also tried to add the unlike denominators. <v Holmes>Then he simplified that to get 40 over X, which gives X <v Holmes>a value of 8. <v Watson>No wonder you said the round trip would take nearly 23 hours. <v Moriarty>[geese honking] [music] Huh. Call me a bird brain, would they? I'll give them birds on the brain! <v Holmes>[geese honking] Hey! Get away you nasty things. Shoo. Shoo! Hey I say! [water splashing] <v Man 2>In 1964, China exploded its first nuclear bomb.

<v Man 2>In 1974, India followed suit. <v Man 2>Once the exclusive servant of the superpowers, the nuclear genie now had many <v Man 2>masters. Today, the anxiety and the arsenals continue to <v Man 2>grow. The politics and dangers of nuclear proliferation. <v Man 2>Next time on War and Peace in the Nuclear Age. <v Man 3>Cut 4 National Geographic, special number 1403, 30 seconds. <v Richard Kiley>A dog's work is never done.

<v Richard Kiley>I'm Richard Kiley. Join me when a new National Geographic special <v Richard Kiley>covers canines in action. <v Richard Kiley>Guiding, herding, performing, racing with their human partners. <v Richard Kiley>Don't miss those wonderful dogs. <v Richard Kiley>A new look at man's best friend. <v Man 3>Cut six. Nature number 715. <v Man 3>30 seconds. <v Man 4>When it's springtime in Antarctica, temperatures can soar to a balmy zero degrees. <v Man 4>Perfect weather for penguins to have fun. <v Man 4>Ice sliding, egg rolling and boxing are very popular. <v Man 4>So is posing with the kids for a family portrait. <v Man 4>And of course, there's always time for romance.

<v Man 4>You'd make time, too, if you only made it once a year. <v Man 4>Join us for a formal affair with the best dressed bird in town. <v Man 4>The Iceberg. Next time on nature. <v Woman 1>Cut seven core promo number 891. <v Woman 1>30 seconds. <v Man 5>[SImon and Garfunkel singing] It was the concert of the decade. <v Man 5>A legendary reunion after 11 years. <v Man 5>Two old friends singing the songs that evoked an era. <v Man 5>Join half a million people for Simon and Garfunkel, the concert in Central

<v Man 5>Park. <v Woman 1>Cut 27, the Power of Excellence with Tom Peters, the Forgotten Customer. <v Woman 1>30 seconds. <v Tom Peters>Our jobs are at stake and more significantly, in my mind, our way of <v Tom Peters>life is at stake. <v Tom Peters>We are losing economically- <v Narrator 1>[Tom still speaking] Tom Peter's Power of Excellence shocks us with remarkable examples <v Narrator 1>of service in America. <v Tom Peters>I've seen the good news examples. I know it can be done. <v Tom Peters>They include no magic, but the terror is that we're not movin' fast enough. <v Man 6>[music playing] They are one of the most endangered species on Earth.

<v Man 6>And because breeding is so difficult, the birth of a baby panda is cause for <v Man 6>celebration. <v Man 6>This is a celebration of a tiny miracle and a rare intimate look at <v Man 6>the first 8 months of Chu Lin, the baby panda.