Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College Solving application problems of rational expressions. The reciprocal of five plus the reciprocal of seven is the reciprocal of what number? In a story problem, we always want to start by defining our variable so X equal the unknown number. Then the reciprocal means one over. So 1 / 5 + 1 / 7 equal 1 / X. We're going to find the common denominator here, and the common denominator is going to be 35 X and we're going to get X not equal to 0. So if I take this entire equation and multiply each and every term by that 35X and then simplify in one step, we ought to get rid of all of our fractions. SO5 goes into 35 seven times, we get 7X7 goes into 35 five times, so 5X and then the last 1, the X is cancel, giving us 3512 X equal 35 SX equal 35 twelfths. So in a story problem, we define our unknown. We set up our equation, we solve our equation. Our next thing is we should check. When we check does 1/5 + 1/7 equal 1 / 35 twelfths while getting a common denominator, we'd have 730 fifths plus 530 fifths. And we take the reciprocal when it's one over equal 1230 fifths. And that's true. And when we're given an equation or a question in words, we should answer in words. So the unknown number is 35 twelfths the sum of a number, and 21 times its reciprocal is -10. So let X equal the unknown number, the sum of a number, and 21 times its reciprocal. Its reciprocal means 1 / X is -10. If we simplify this up, we could have X + 21 / X equal -10 because remember that 21 is over 1, so our common denominator is going to be X this time, and X can't equal 0. If we multiply each and every term by X in one step, we should be able to get rid of all the fractions. We get X ^2 + 21 equal -10 X. In this case, it's a quadratic, so we need to take all the terms to one side and then we're going to factor the two numbers that multiply to give us 21 but add to give us 10 are going to be 7 and three. So X equal -7, X equal -3 to do the check does -7 + 21 / -7 equal -10? Well, 21 / -7 is -3, so yes. The other one negative 3 + 21 / -3 21 / -3 is -7 so that one also checks. So the two numbers, the two possible numbers are -7 and -3. Looking at more examples, mail order. So an experienced shipping clerk can fill a certain order in five hours. Willie, a new clerk needs 9 hours to complete the same job. Working together, how long will it take them to fill the order? Well, there are a lot of ways to do this one, but the easiest way that I know is to think about a fraction as the number of jobs fulfilled per time. So jobs per time. So Zoe can do one job in five hours, Willie can do one job in nine hours working together. They can do one job in X hours. Where we're trying to find ** is the time working together. So here we're going to get our common denominator of 45 X. So we get 45 X times each of those terms. And in this one step we should be able to get rid of all of our fractions. And X of course, isn't going to equal 0 because we'd have zero in the denominator. 5 into 45 would give us 9X plus 5X equal 4514 X equal 45 X equal 45 fourteenths. So to check this we'd have 1/5 + 1 ninth equaling 1 / 45 fourteenths, getting common denominator 940 fifths plus 540 fifths equals 1440 fifths. And it does check, so it will take 45 fourteenths hours or 14 goes into 45 three hours three and three fourteenths hours 14 goes into 45 three whole times. That's 42 + 3 more 43 four 445 looking at the next one filling a tank, a tank can be filled in 18 hours. So once again, we're going to do the job per time. So a tank can be filled in 18 hours by pipe A alone, 22 hours by pipe B alone. How long will it take if the pipes are working together? So X equal pipes working together. So now we want to find our common denominator. This is 2 * 9 and this is 2 * 11. So we know that they have a 2IN common and a nine and 11 and an X have to all be in the common denominator. X can never equal 0. So in this first one, if I multiply by the common denominator, I'm going to see hopefully the the two and the 9 and the denominator are going to cancel with the two and the 9:00. So that would leave me 11X. If I multiply the next one by the common denominator, the two and the 11 are going to cancel, leaving us 9X. And if I multiply this last one by the common denominator, the XS are going to cancel, leaving us 198. So then 11X plus 9X is 20 X equals 198 X equal 198 twentieth. And that reduces those are both divisible by two. So if we divide by two, we're going to get 99 tenths or 9 9/10 hours. We need to do the check. 118th plus 122nd equal 1 / 99 tenths. The common denominator here is going to be 99, so we're going to get 1190 ninths plus 990 ninths. Oh wait, the common denominator is 2 * 9 * 11. OK, oops, 198 common denominator. So we're going to get 20 / 198, which does reduce to 1090 ninths. So it will take 9 9/10 hours working together, looking at the next one, cutting firewood. Jake can cut and split a court of firewood in six fewer hours than Skyler. So amount for Jake and amount for Skyler, Jake takes fewer hours. So we're going to subtract. So if we let Skyler be X, then Jake is X -, 6. He's 6 fewer hours. So J can do one job and X -, 6 time, and Skyler can do it in 1 / X. When they work together, they do it in four hours, so 1 / 4. The common denominator here is going to be X - 6 * X * 4, and we know that X can't equal six or zero. If we multiply everything through by the common denominator, we're going to have four X * X -, 6 times the first term, so those X -, 6 is canceled, leaving us just 4X4 X times X -, 6 times the second term. So the X is canceled, leaving 4 * X - 6, and then in the last one, the four is going to cancel, leaving us X * X - 6. We're going to distribute things out, and because we have an X ^2 term, we need to take everything to one side using the 0 property. So we're going to get X ^2 -, 14 X plus 24. So we need 2 numbers that are going to multiply to give me 24 but add to give me -14 so -12 and two X equal 12 X equal 2. Now this is a story problem, and we have more than one unknown that we're trying to solve for. We have X, which was the amount of Skylar, but we also have X - 6, which is the amount of Jake. If I have 1212 -, 6 is 6. If I had two, though, 2 - 6 is -4 That doesn't make any sense because we can't have -4 hours. So the one that's going to make our answer is the 12 and the 6th. If we check it, Jake did one job in six hours, Skyler did one job in 12 hours, Does that equal 1/4? Well, the common denominator here is going to be twelfths and we get 2 twelfths plus 1/12 equal 312. So it does check. Jake took six hours and Skyler took 12 hours. Claudia can paint a neighbors house 4 times as fast as Jan can. This year, the year they worked together it took them eight days. So we need Claudius time and we need Jan's time. And if Claudia can do it four times faster than Jan takes four times as long. So if Claudia was one hour, Jan would be 4 hours. So Claudia can do 1 / X and Jan can do 1 / 4 X and together they do 1 / 8. So our common denominator here is 8. ** can't ever equal 0. If we multiply everything 3 by 8X and simplify, the X is canceled and we get 88X and 4X reduced to two. The eights cancel and we get X. So X is going to equal 10. So 4X would equal 40. So when we check it does 1 / 10 + 1 / 40 equal 1 / 8 Common denominator of 40. So 4 / 40 + 1 / 40 equal 5 / 40. And yes, it does check. So Claudia takes 10 days and Jan takes 40 days. Kayaking. The speed of the current in Catamount Creek is 3 mph. Omens Kayak can travel 4 miles upstream. So we're going to use distance equal rate times time, and let's see 4 miles upstream and the same time it takes to go 10 miles downstream. We know the speed of the current, but we don't know how fast the boat's going in Stillwater. So if we're going upstream, we're going against or we're subtracting the current. If we're going downstream, we're going with the current. So we want to know what's the speed of the kayak in Stillwater. When we have these two equations, we could solve them each for time because the time is the same. So in the first equation, time equals 4 / X -, 3. The second equation time equal 10 / X + 3. Because the TS are the same, we're going to set them equal. The common denominator here is going to be X - 3 * X + 3. Going to say X can't equal 3 or -3. So we're going to end up with 4 * X + 3 equaling 10 * X - 3 or four X + 12 equal ten X - 36 X equal 42 X = 7. To check it does 4 / 7 - 3 equal 10 / 7 + 3. Well, 4 / 4 equal 10 / 10, so it does check. The speed in Stillwater is 7 mph. Looking at the next 1A paddleboat travels 2 kilometer an hour in Stillwater. So now we don't know the current, so we're going to say 2 -, C when we're going upstream and 2 + C when we're going downstream. The paddle boat. The boat is paddled 4 kilometers an hour downstream in the same time as it takes 1 kilometer to go upstream. We don't know the times, but we do know time would equal 1 / 2 - C and time would equal 4 / 2 + C If we set those equal and get our common denominator of 2 - C * 2 + C, we'd get 1 * 2 + C equaling 4 * 2 - C So 2 + C equal 8 - 4 C 5C equals 6 or C = 6 fifths, so 1 / 2 - 6 fifths. Does that equal 4 / 2 + 6 fifths? 2 here is really 10 fifths, so 10 fifths -6 fifths is 4/5, and 1 / 4 fifths would be 5 fourths over here for 10 fifths plus 6 fifths. So we'd get 4 / 16 fifths because it's division. We're going to say that's four times 5 sixteenths, the fours over 116 and 4 reduced to 5 fourths. So it does check. So the speed of the river is 6 fifths kilometers per hour, or one and one fifth kilometer per hour. Moped speed. Jamie's moped travels 8 kilometers of fat kmh faster in the Maria's. So Jamie's is faster. So X + 8 for Jamie, X for Mara. Jamie travels 69 in the same time that Mara travels 45. So this is the same kind of problem we're going to solve for T So we're going to get t = 69 / X + 8 and T equal 45 / X, so 69 / X + 8 equal 45 / X The common denominator X * X + 8. So 69 X equal 40, five X + 360. If we subtract all the X's to one side, we're going to get 24X equal 360. So X is 360 / 24, which would be 30 / 2 or 15. So if we check this to 69 / 15 + 8 equal 45 / 15, well 15 + 8 is 2369 / 23 is 3 and 45 / 15 is 3. So it does check. So the speed of each person X was Mara. So Mara's speed is 15 kilometers per hour and Jamie's speed. Remember it was eight mile kmh faster, so it's 23 kilometers per hour. Julius Boston Whaler cruised 45 miles upstream and 45 miles back in a total of 8 hours, so this one's a little different. The speed of the river is 3 mph, so we have 45 miles hour UPS or 45 miles upstream, and the speed of the river is 3, so we don't know the speed of the boat. So X + C is some time to go up river. Actually that would be downriver, wouldn't it? Because if we're adding the current, we're going faster, so it's down. 45 would equal X -, C times the time to go up river. So if we wanted to solve for those TS, we'd get 45 / X + C to equal the time of downriver and 45 / X -, C to equal the time of upriver. And now we know that the two times added together are a total of 8. So we have 45 / X + 3 because the current was 3 + 45 / X - 3. Because the speed of the river is 3 equal a total of eight. Our common denominator X + 3 * X - 3. If we multiply each term 3 by that common denominator, we'd get 45 * X - 3 + 45 * X + 3 equal 8 * X + 3 X -3. So we'd get 45 X -135 + 40 five X + 135 X + 3 X -3. Remember is difference of squares. So when I distribute the eight, I get eight X ^2 - 72 the -135 and +135 are going to cancel. So we're going to get everything to one side. Eight X ^2 - 90 X -72. Now without using a calculator, those are bigger numbers. So I'm going to divide everything through by two and see if that helps me a little bit, get a few smaller numbers. I'm going to figure out what this factors into. So four times the -36 four carry two 144. So I want 2 numbers that are going to add to give me -44 negative 45 but multiplied to give me -144 So when we list out our possibilities, I don't know this one off the top my head. 6 and 24. Let's see 36. And oh wait, 36 and 3 is not times 36. Three and 48, right? There's the one I want 3 and 48. So if we had X minus plus 3 / 4 and X -, 48 / 4, and we want to reduce those, once they can't reduce, we're going to take that denominator and put it out in front. So this is what that'll factor into -3 fourths doesn't make any sense because we're talking about a current, so 12 is the one that makes sense. When we check it, does 45 / 12 + 3 plus 45 / 12 - 3 equal 8? Well, 45 / 15 is 3 and 45 / 9 is 5 and 3 + 5 equal 8. So the boat speed is 12. What were it? What was the unit 12 mph Thank you and have a wonderful.