Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Rational equation applications. The sum of a number and 21 times its reciprocal is -10 find the number. Traditionally we have 5 steps in a story problem in mathematics. The first is to define our unknown. So let X equal the number. The second step is to set up an equation so the sum of X and 21 times its reciprocal. The reciprocal is one over, so 1 / X is -10. Typically -10, or the word is in front of a number frequently means equals. So if we simplify this, we get X + 21 / X equal -10. So that's our equation. To solve it, we need our common denominator, which is going to be X. And we're going to know that X can never equal 0 or we'd have zero in the denominator. So if I multiply everything through by the XI get X ^2 + 21 equaling -10 X. To solve this, we're going to take everything to one side, and then we're going to factor. So when I solve, I get X + 3 * X + 7 because 3 * 7 is 21 and 3 + 7 is 10. So X equal -3 and X equal -7. That's the third step in a story problem is to actually solve the equation. Then we're going to check it. So does -3 + 21 / -3 really give us -10? And the answer is yes, because 21 / -3 is -7 What about -7 + 21 / -7? Does that really equal -10? And the answer, once again, is yes. So then the last step of the five steps is to actually answer. So the number is -3 or -7 The next example, mail order Zo. An experienced shipping clerk can fill a certain order in five hours. Willie, a new clerk needs 9 hours to complete the same order. Working together. How long will it take them to fill the order? So let's let X be working together. Time working together on work problems. I like to think of a ratio as jobs done per time that it takes. Now, there are a lot of different ways to do this, but I personally find this the easiest. So Zoe can do one job in five hours, Willie can do one job in nine hours. If they're working together, we would add those rates together. So 1 / 5 + 1 / 9, and that's going to equal 1 over. We want to know how long it takes one job with them working together or 1 / X. Now, sometimes you'll see it as X / 5 + X / 9 equaling just one hole, and that works too. But I like the ratio jobs per time because then it's consistent throughout the entire equation. Jobs per time, jobs per time, jobs per time. So common denominator is going to be 45. X&X is never going to equal 0. So we just did our first two steps in the story problem. We defined our variable. We set up the equation. Now we're going to do Step 3, which is solve. So if I multiply everything through by 45, XI would get 9X plus 5X equaling 4514 X equal 45 S X = 45 fourteenths. And if I wanted that as a whole number with the fractional portion, it'd be three and three fourteenths. So if we checked it 1/5 + 1 ninth equal 1 over, I'm going to leave that as 45 fourteenths in the check. So here we'd get 940 fifths plus 540 fifths and one over. Something just means to take the reciprocal, so 1440 fifths. So we can see that that is indeed true. So working together takes three and three fourteenths hours. Our next example is on cutting firewood. Jake can cut and split a cord of firewood in six few hours, then few fewer hours than Skylar can. So let's let Jake and Skylar. Jake takes 6 fewer hours than Skyler. So if we let X be Skyler, than Jake is X -, 6 because he's taking fewer. So job one job per time and one job per time equaling one job per time. And the time for them working together is 4. So the common denominator is four X * X - 6. So multiplying 3 we get four X + 4 * X - 6 equaling X ^2 - 6 X distribute 4X plus four X - 24 equal X ^2 - 6 X. Taken everything to one side, we get X ^2 - 14, X plus 24. Factoring that, we're going to get X - 2, X -12 So X equal 2 or X equal 12. But we also have X - 6. If X equal 2, that would mean it took -4 hours for Jake, and that doesn't make any sense. So the two can't work. But X - 6 when X is 12 would give me 6 to 1 / 6 + 1 / 12 really equal 1 / 4. Common denominator would be twelves SO2 twelves plus one twelves does equal 3 twelves so it does work. And it says how long for each of them. So Jake will take six hours and Skyler 12 hours. Now, if either of these numbers came out to be less than four, hopefully, logically we'd realize that couldn't happen because if it takes 4 to do it together, each one separately would have to take more than the 4A paddable travels 2 kilometers per hour in Stillwater. The boat is paddled 4 kilometers downstream in the same time it takes to go 1 kilometer upstream. What's the speed of the river? So we're going to have downstream and we're going to have upstream, and the current going downstream is going to make us go faster. So Stillwater plus current is going to go downstream and going upstream it's going to be the Stillwater minus the current, because in upstream it makes us go slower. If distance equals rate times time, we know that time is really just distance divided by rate. So the distance to go downstream is 4 and upstream is 1. The rate in Stillwater was 2, so we're going to have 2 + X and 2 - X. And the time in this case are going to be the same, so let's just call them T. So in the first equation we get 4 equaling 2 + X * t or T is just 4 / 2 + X and the second one one equal to minus X * t or T is just 1 / 2 - X. The two TS are equal, so by transitive property we can set the two things that are equal to as an equation common denominator 2 + X * 2 - X. So we get 4 * 2 - X equaling 1 * 2 + X or 8 - 4 X equal 2 + X five X = 6. So X is 6 fifths or one and 1/5 hour. So to check it we would have 4 / 2 + 6 fifths. Does that equal 1 / 2 -, 6 fifths? We can think of this two as being 10 fifths plus 6 fifths. So that would give us 16 fifths. And when we multiply by its reciprocal, we get 20 / 16. OK, so now this one we're going to think of as 10 fifths -6 fifths or 1 / 4/5, and it's reciprocal would be 5 fourths. 2016 is indeed reduced to 5 fourths, so it does check. So the speed of the river is 01 and 1/5 kmh. I actually thought we were saving for time back here. So that wasn't hours. This is kilometers per hour. Let's look at one more example before we give you a few to try. Julius Boston Whaler cruised 45 miles upstream and 45 miles downstream and a total of 8 hours. The speed for the river is 3, so X -, 3 and X + 3. Let's call this time to go up and time to go down for just a second. So we know that 45 / X -, 3 is the time to go up and 45 / X + 3 is the time to go down. But the total time. So the time up and the time down is 8. So if we took the time to go up plus the time to go down, we get total time. Common denominator X - 3 * X + 3. So I'd get 45 * X + 3 + 45 * X - 3 equaling 8 * X - 3 X +3. So we'd have 40 five X + 135 + 45 X -135 equaling eight X ^2 - 72. These last two terms is X ^2 - 9. And then I distributed the 8 so the -135 positive 135 we're going to cancel. So we'd get 0 equaling eight X ^2 -, 90 X -72. I don't know that one off the top of my head. So I think I'm going to divide everything through by two to get an equivalent equation that's a little smaller. Four X ^2 -, 45 X -36 So I need 2 numbers that are going to multiply to give me -36 * 4. So -144 So one one 44272 three 48436. And I want them to add to give me -45 S actually that 3 and 48's going to be what I want. So I'm going to get 0 equaling 4X. I'm just going to come down here for a minute because I'm at a room up there 4X and X, and we're going to have -12 and +3 because there's my -48 X and there's my positive 3X and that adds to give me my -45 S. Here X would be negative 3/4, which doesn't make any sense. And here X is 12. So the rate of the boat in Stillwater is going to be 12 and we could check that 45 / 12 - 3 + 45 / 12 + 3. Does that equal 812 - 3 is 945 / 9 is 545 / 15 is 3/5 plus 3 is 8. It does check so speed about in Stillwater 12 mph. Let you try the remaining problems. Thank you and have a wonderful day.