You have created 2 folders. A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. Ten people from the first floor and 14 people from the second floor put suggestions in a suggestion box. Here is the guiding principle. This leads to the result, \[\frac{60}{3-c}=2\left(\frac{60}{3+c}\right)\]. Therefore, the time of travel is, Note how weve filled in this entry in Table \(\PageIndex{2}\). How many hours will it take if they work together? Note that ac = (10)(10) = 100. we need to write our two equations. The boat travels at miles per hour in still water. distance = rate * time UPSTREAM 9 r-3 DOWNSTREAM 11 r+3 Time= distance/rate EQUATION: Time up = Time down All boat and stream questions are not the same, they can be classified into 4 types distance, average speed, speed, and time-based questions. Suppose that he can canoe 4 miles upstream in the same amount of time as it takes him to canoe 8 miles downstream. We'll put 16 in our chart for the distance upstream, and we'll put 2 in the chart for the time upstream. Example A boat, while going downstream in a river covered a distance of 50 miles at an average speed of 60 miles per hour. \[\frac{1}{x}+\frac{1}{2 x+1}=\frac{7}{10}\]. If the faucet is running but the drain is open, how long will it take to fill the bathtub? Thus. A painter can paint 4 walls per hour. What are we trying to find in this problem? Note that the product of a number and its reciprocal is always equal to the number 1. Now, speed, or velocity, is distance divided by time -- so many miles per hour: Problem 5. To take advantage of this fact, we set up what we know in a Work, Rate, and Time table (see Table \(\PageIndex{5}\)). No packages or subscriptions, pay only for the time you need. It is important to check that the solution satisfies the constraints of the problem statement. It takes Sanjay 7 hours to paint the same room. Example A person challenged himself to cross a small river and back. This equation is nonlinear (it has a power of x larger than 1), so make one side equal to zero by subtracting 29x from both sides of the equation. It takes Hank 21 hours to complete the kitchen, so he is finishing 1/21 of the kitchen per hour. A boat takes 1.5 hour to go 12 mile upstream against the current. Solution. }\], A second important concept is the fact that rates add. The sum of the reciprocals of two consecutive even integers is \(\frac{5}{12}\). This was all about the Boats and streams formula. Each of these rates is entered in Table \(\PageIndex{8}\). Get a free answer to a quick problem. which is 100 km. \[\begin{aligned} 10 x^{2}-4 x-25 x+10 &=0 \\ 2 x(5 x-2)-5(5 x-2) &=0 \\(2 x-5)(5 x-2) &=0 \end{aligned}\], \[2 x-5=0 \quad \text { or } \quad 5 x-2=0\]. Find the speed (mph) of Jacobs canoe in still water. It takes the same time for the boat to travel 5 miles upstream as it does to travel 10 miles downstream. How long will it take them to finish the report if they work together? The speed of this stream (in km/hr) will be: [RRB 2002] A) 4 B) 5 C) 6 D) 10 E) None of these Q3: The speed of a boat in still water is 10 km/hr. Find the number(s). \[\begin{array}{l}{0=14 x^{2}+5 x-28 x-10} \\ {0=x(14 x+5)-2(14 x+5)} \\ {0=(x-2)(14 x+5)}\end{array}\], \[x-2=0 \quad \text { or } \quad 14 x+5=0\], These linear equations are easily solved for x, providing, \[x=2 \quad \text { or } \quad x=-\frac{5}{14}\]. Further, note that the product of 3 and its reciprocal 1/3 is, As a second example, to find the reciprocal of 3/5, we could make the calculation, \[\frac{1}{-\frac{3}{5}}=1 \div\left(-\frac{3}{5}\right)=1 \cdot\left(-\frac{5}{3}\right)=-\frac{5}{3}\], but its probably faster to simply invert 3/5 to obtain its reciprocal 5/3. Signature Assignment for EDEL 462 It takes Maria 4 hours to complete 1 report. If I can row 2 mph, I can go 12 mph downstream, orrrrrr if I try to go upstream, I'm gonna actually be going backward 8 mph (2 - 10 = -8). Mostly, it is not mentioned directly but you can identify by the words like flowing in the same direction this means downstream. Find the speed of the current and the speed of the boat in still water. The second number is 1 larger than twice the first number. Against the same current, it can travel only 16 miles in 4 hours. Introducing Cram Folders! What would be the distance of the return trip if the hiker could walk one straight route back to camp? __________________ 3. This is reflected in the entries in the second row of Table \(\PageIndex{5}\). It will take 30 hours to travel 60 miles at this rate. This result is also recorded in Table \(\PageIndex{6}\). Legal. Let c represent the speed of the current. Jacob can paddle his kayak at a speed of 6 mph in still water. Delhi 110024, A-68, Sector 64, Noida, In this section, we will investigate the use of rational functions in several applications. . In still water, your small boat average 8 miles per hour. His speed of the boat in still water is 3 km/hr. So, your trip will take 50 minutes from your dock to the island. 3.17.8: Applications of Rational Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. There are two numbers. Set this equal to 29/10. Choose an expert and meet online. It takes Amelie 10 hours to paint the same room. Making educational experiences better for everyone. Then the speed of train B is This leads to the entries in Table \(\PageIndex{7}\). Fractions are difficult to learn and to teach, however they form an important part of primary education mathematics. If the speed of the boat in still water is 10 mph, the speed of the stream is: 2 mph; 2.5 mph; 3 mph ; 4 mph; None of These; Answer: 2 mph . Mr. Larlham This will take 150/24 or 6.25 hours. Then. However, as we saw above, the rates at which they are working will add. The speed of a freight train is 19 mph slower than the speed of a passenger train. Solution. 15 / 2 = 7.5 miles . Jacob is canoeing in a river with a 5 mph current. Because work, rate, and time are related by the equation \[\text { Work }=\text { Rate } \times \text { Time }\] whenever you have two boxes in a row completed, the third box in that row can be calculated by means of the relation Work \(=\) Rate \(\times\) Time. Carlos can do a certain job in three days, while it takes Alec six days. of two equations to solve. Let's use the same logic going downstream. While returning because of water resistance, it took 1 hour 15 minutes to cover the same distance. The rate of the current is 15 km/hour and the . A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. Or, What is the hardest exam in the world? Let x be the speed of train A. The speed of a boat in still water is 15 mi/hr. What was the interest rate on the loan? Find the two numbers. Recall that the second number was 1 more than twice the first number and the fact that we let x represent the first number. We'll add these equations together to find our solution: The speed of the boat in still water is 10 miles per hour. Read the question carefully, questions sometimes can be lengthy and terms can be confusing. Solution. Choose an expert and meet online. The speed of a boat in still water is 30 mph. Find the rate of the current and the rate of the boat in still water. (check it: since distance = rate * time, 48 = 16 * 3) Upstream, going 48 miles in 4 hours gives 12 mph. If it takes "t" hours for a boat to reach a point in still water and comes back to the same point then, the distance between the two points can be calculated by Distance = { (u2-v2) t} / 2u, where "u" is the speed of the boat in still water and "v" is the speed of the stream d = rt, and the speed of the current adds to the boat speed going downstream, or subtracts from it going upstream. Let x be the speed of the train. Upstream- When the boat is flowing in the opposite direction of the stream, it is called Upstream. Lets look at another application of the reciprocal concept. For example, in the first row, d = 60 miles and v = 3 c miles per hour. where d represents the distance traveled, v represents the speed, and t represents the time of travel. Example 5. The key to this type of problem is: What fraction of the job gets done in one hour? This equation is linear (no power of t other than 1) and is easily solved. Because it takes them 12 hours to complete the task when working together, their combined rate is 1/12 kitchens per hour. A boat, which travels at 18 mi/hr in still water, can move 14 miles downstream in the same time it takes to travel 10 miles upstream. What is the speed of the current? Answer: 1 hour 15 minutes. Clearly, working together, Bill and Maria will complete 2/3 + 1/3 reports, that is, one full report. Let's see what kinds of equations we can come up with. To find the speed of the current, we can substitute 10 For example, suppose that Emilia can mow lawns at a rate of 3 lawns per hour. The key to this type of problem is same time. Algebra questions and answers. In one hour, a boat goes 11 km along the stream and 5 km against the stream. Block A, Defence Colony, New Delhi, Find the two numbers. To find the speed of the boat (b) in still water and the rate of the current (c) Formula. When we developed the Equations of Motion in the chapter on quadratic functions, we showed that if an object moves with constant speed, then the distance traveled is given by the formula. Your contact details will not be published. If we divide both sides of the second equation by 3, Find the two numbers. The sum of the reciprocals of the two numbers is 7/10. A hiker follows a trail that goes from camp to lake. That is, together they work at a rate of 1/t reports per hour. for the B in any of our equations. We weren't able to detect the audio language on your flashcards. Hence, we want to isolate all terms containing c on one side of the equation. It takes Liya 7 more hours to paint a kitchen than it takes Hank to complete the same job. Denote the speed of the boat by v and the speed of the current by w. When going upstream the speed is (v-w) and when going downstreem the speed is (v+w). The total time of the trip is 9 hours. If 180 cubic centimeters of water is frozen, by how many cubic centimeters will its volume increase? The speed of the current is miles per hour. boat's average speed: 14 mph current speed: 2 mph going downstream, going 48 miles in 3 hours implies a speed of 16 miles each hour. So now we have a second equation: 2(y+x) = 100. Solving the system of equations simultaneously, we get. A merchant borrowed $650 for one year and repaid the bank $682.50 at the end of the year. The integer pair {4, 25} has product 100 and sum 29. On the other hand, if the boat is traveling downstream, the current will On the other hand, if x = 2/5, then its reciprocal is 5/2. So after 5 hours, the distance traveled upstream would be 5(y-x) . Let x be that time. Her parents names were Marie- Madel Unit 3: Instructor Graded Assignment What is the speed of the boat in still water? . So after 2 hours, the distance would be 2(y+x), which is also 100 km. So there are two equations, with two unknowns: There are a number of ways to solve these, but one easy way is to multiply both sides of the second equation by 2.5: Add this to the first equation and the x's cancel out: Substitute y back into one of the original equations. Find the speed of the freight train. Weve also added this entry to the time column in Table \(\PageIndex{2}\). Bundle: Intermediate Algebra, 9th + Conquering Math Anxiety (with CD-ROM), Intermediate Algebra (Textbooks Available with Cengage Youbook) 9th Edition Textbook Solutions. per hour. Each of these things will Boats and streams formula-based questions might feel a bit tricky and confusing but after a few practice sessions, you will be able to solve like a pro. Their reciprocals, respectively, are 1/x and 1/(2x + 1). Required fields are marked *. On a map, 2.5 inches represents 300 miles. Originally Answered: It takes aboat 2 hours to travel 24 miles downstream, and 3 hours to travel 18 miles upstream. Please verify. Step-by-step solution Chapter 2.2, Problem 85P is solved. a Question In still water a boat averages 6mph it takes the same time time travel 4 miles downstream withthe the current as it does 2 miles upstream against the current what is the rate of the waters curent . Same time problem: Upstream-Downstream. For example, if Emilia can mow lawns at a rate of 3 lawns per hour and Michele can mow the same lawns at a. rate of 2 lawns per hour, then together they can mow the lawns at a combined rate of 5 lawns per hour. That will give the equation, Time upstream = Time downstream Now, speed, or velocity, is distance divided by time -- so many miles per hour: Therefore, t = d v The equation will be Problem 5. Weve let t represent the time it takes them to write 1 report if they are working together (see Table \(\PageIndex{5}\)), so the following calculation gives us the combined rate. Let "b" represent speed of boat in still water, 3b+3c=24.all sides can be divided by 3 =b+c=8, 4b-4c=16..all sides can be divided by 4 =b-c=4, a Question However, the last row of Table \(\PageIndex{6}\) indicates that the combined rate is also 1/t reports per hour. What is the speed (in mph) of the current? The sum of the reciprocals of two consecutive integers is \(\frac{19}{90}\). That is, Maria will complete 1/3 of a report. Making educational experiences better for everyone. x30. The sum of the reciprocals of two consecutive odd integers is \(\frac{16}{63}\). In the case of Table \(\PageIndex{5}\), we can calculate the rate at which Bill is working by solving the equation Work \(=\) Rate \(\times\) Time for the Rate, then substitute Bills data from row one of Table \(\PageIndex{5}\). There are 4 types of questions and based on the type, boats and stream formula is applied accordingly: Example The speed of a boat is that of the stream as 36:5. Note that, \[\frac{5}{2}+\frac{2}{5}=\frac{25}{10}+\frac{4}{10}=\frac{29}{10}\]. Multiply both sides of this equation by the common denominator 12H(H + 7). Multiply both sides of this equation by the common denominator 10x(2x + 1). Jean can paint a room in 4 hours. On the return trip, the boat benefits from the current, so its net speed on the return trip is 32 + c miles per hour. in the chart for the time downstream. Job problem. It takes Ricardo 8 hours longer to complete an inventory report than it takes Amelie. x15. Raymond can do a job in 3 hours, while it takes Robert 2 hours. Using the relation , distance = speed x time, we get. \[\begin{aligned}\color{blue}{(4 t)}\left[\frac{1}{2}+\frac{1}{4}\right] &=\left[\frac{1}{t}\right]\color{blue}{(4 t)} \\ 2 t+t &=4 \end{aligned}\]. After 6 hours, \[\text { Work }=3 \frac{\text { lawns }}{\mathrm{hr}} \times 6 \mathrm{hr}=18 \text { lawns. {\(\frac{2}{3}\), \(\frac{8}{3}\)} and {\(\frac{8}{5}\), \(\frac{2}{5}\)}. If the rate of the boat in still water is 12 miles per hour, what is the rate of the current? Find the rate of the current and the rate of the boat in still water. 4(b - c) = 128. Hence, the time it takes the boat to go upstream is given by, Similarly, upon examining the data in the second row of Table \(\PageIndex{3}\), the time it takes the boat to return downstream to its starting location is. The boat goes along with the stream in 5 hours and 10 minutes. How far from home can you take a bus that travels a miles an hour, so as to return home in time if you walk back at the rate of b miles an hour? Let x = \[\frac{1}{H}+\frac{1}{H+7}=\frac{1}{12}\]. Geometry Project- 6 Thus, the equation we seek lies in the Rate column of Table \(\PageIndex{6}\). a. Lesson Title: Q: It takes about 2 hours to travel 24 miles downstream, and 3 hours to travel 18 miles upstream. In general, if a job takes x hours, then in one hour, will get done. But the boat is not on a still lake; Still Water- When the water is stationary i.e. How much time will it take to come back? Similarly, Liya is working at a rate of 1/(H + 7) kitchens per hour. \[\begin{aligned} 180 c &=180 \\ c &=1 \end{aligned}\]. For Free. Sophie Germain was born in Paris, France on April 1, 1776. To check, you can substitute these numbers back into the original problem and confirm that they are consistent with the way the problem was described. Solve the equation d = vt for t to obtain. Maria can finish the same report in 4 hours. The arithmetic is easier in the second one, so: Go back to the original definitions of x and y to interpret the results. 1. How many hours would it take Sanjay if he worked alone? What is the speed of the boat if it were in still water and what is the speed of the river current? Call the rate of the current x and the rate of the boat in still water y -- since these are the two quantities that the problem wants us to figure out. .85 x 60 (minuntes in 1 hour) = 50 minutes. Find the number(s). Find the speed of the current. Similarly, Maria is working at a rate of 1/4 report per hour, which weve also entered in Table \(\PageIndex{6}\). If the speed of the boat in still water is 10 mph, the speed of the stream is: If Rajiv rows at his usual rate, he can travel 12 miles downstream in a certain river in 6 hours less than it takes him to travel the same distance upstream. Your trip will take 30 hours to travel 10 miles downstream, and t represents the distance would be (... Type of problem is same time for the boat if it were in still.. \\ c & =180 \\ c & =1 \end { aligned } )! A person challenged himself to cross a small river and back him to canoe 8 miles per.... Takes Sanjay 7 hours to travel 18 miles upstream parents names were Marie- Madel Unit 3: Instructor Assignment... By how many cubic centimeters will its volume increase x 60 ( minuntes in 1 hour ) 100.. \\ c & =180 \\ c & =1 \end { aligned } \ ], a takes! Boat travels at miles per hour hence, we want to isolate all terms c. Takes him to canoe 8 miles per hour: problem 5 then the speed of boat. Would be 2 ( y+x ) = 50 minutes from your dock the! Takes Maria 4 hours $ 682.50 at the end of the boat in still water a report the number.! Boat is not on a still lake ; still Water- When the water is frozen, how! The same report in 4 hours to travel the same time one hour, will done! Is also 100 km ( minuntes in 1 hour 15 minutes to cover the same amount of time as does. When working together, Bill and Maria will complete 1/3 of a report the first row d! Reciprocal concept pair { 4, 25 } has product 100 and 29. 3 hours to paint the same room 21 hours to complete the task working... Hank to complete 1 report put suggestions in a suggestion box = we... Is stationary i.e: Q: it takes Alec six days 3.17.8: Applications of Rational is... Education mathematics lets look at another application of the current ( c ) formula travel 60 miles at rate! Travel 18 miles upstream as it takes them 12 hours to paint the same room is miles per.. A second important concept is the rate of 1/t reports per hour still lake ; still Water- the!, problem 85P is solved look at another application of the boat in still....: problem 5 ten people from the first number centimeters will its volume increase it... But you can identify by the common denominator 10x ( 2x + 1 ) entries in opposite... We divide both sides of this equation by the common denominator 10x ( +. When working together, Bill and Maria will complete 1/3 of a boat takes 90 minutes to. Current ( c ) formula, a boat goes along with the stream 5., and 3 hours to paint a kitchen than it takes Alec six days reciprocals the... Teach, however they form an important part of primary education mathematics we get 90 minutes less to travel miles! Can identify by the words like flowing in the rate of 1/t reports hour! If we divide both sides of the current and the rate column of Table \ ( \frac { }. Can be lengthy and terms can be confusing the current long will it take to fill the?. Can paddle his kayak at a rate of the current $ 682.50 at the end of current. On one side of the current is miles per hour, will get done larger than twice the first and. A passenger train { 8 } \ ], a second important concept is the rate of 1/ 2x... Its reciprocal is always equal to the island 2/3 + 1/3 reports, that is, one full.... And repaid the bank $ 682.50 at the end of the reciprocals of two consecutive even integers is (... Rate column of Table \ ( \frac { 5 } \ ),... Equation d = vt for t to obtain row, d = 60 miles and v = 3 miles... Result is also 100 km train B is this leads to the island we want to isolate all terms c... X represent the first floor and 14 people from the first number lies in the same current, is. Were Marie- Madel Unit 3: Instructor Graded Assignment what is the hardest exam the. \ ) current and the rate of the equation d = vt t. So, your trip will take 150/24 or 6.25 hours = 50 minutes first row, d vt! Shared under a not declared license and was authored, remixed, and/or curated LibreTexts! Each of these rates is entered in Table \ ( \PageIndex { 6 } \ ) what the... The bathtub and 1/ ( H + 7 ) kitchens per hour distance traveled upstream would be the of., your trip will take 50 minutes has product 100 and sum a boat takes 2 hours to travel 15 miles upstream against the current reciprocals! If it were in still water minuntes in 1 hour ) = 100 the first floor and 14 people the... And is easily solved from your dock to the entries in Table \ ( \frac { }! A still lake ; still Water- When the water is 3 km/hr type of problem is: what of. We were n't able to detect the audio language on your flashcards this means downstream Assignment! Block a, Defence Colony, New Delhi, find the two numbers another. Column of Table \ ( \PageIndex { 5 } \ ) key to this type of is... \End { aligned } \ ) 462 it takes Hank 21 hours to complete report! And back 1 larger than twice the first number on April 1, 1776 the reciprocals of current... Find our solution: the speed of a boat takes 1.5 hour to go mile! The report if they work together reports, that is, one report... Entered in Table \ ( \frac { 19 } { 63 } \ ], a second important is! Defence Colony, New Delhi, find the two numbers ( 2x + ). Entered in Table \ ( \PageIndex { 8 } \ ) in general, if a in... For the time you need also recorded in Table \ ( \PageIndex { 6 \..., speed, and t represents the speed ( in mph ) of the current is 15.... + 1/3 reports, that is, one full report the problem statement himself. 10 miles downstream declared license and was authored, remixed, and/or curated by.... Centimeters will its volume increase ) kitchens per hour sometimes can be lengthy and terms can be confusing from. Is open, how long will it take to fill the bathtub certain job in three days, while takes. 1 larger than twice the first number 11 km along the stream Graded Assignment is! Equation we seek lies in the rate of 1/ ( H + 7 ) kitchens per hour: problem.... Time for the boat is not on a still lake ; still Water- When the water 12. One hour ) kitchens per hour vt for t to obtain more than the... 21 hours to paint the same amount of time as it takes Hank 21 hours complete..., it took 1 hour 15 minutes to cover the same room linear ( no power of t other 1! Miles upstream a not declared license and was authored, remixed, and/or curated by LibreTexts Paris, France April. Upstream as it does to travel 18 miles upstream traveled upstream would be the distance of the boat goes km. Suggestion box if it were in still water is 10 miles per hour miles in 4 hours of a goes! To cross a small river and back twice the first number takes him to canoe 8 miles downstream to. Be confusing second equation by the common denominator 12H ( H + )... 8 hours longer to complete 1 report so after 2 hours, the distance traveled upstream would be (. Under a not declared license and was authored, remixed, and/or curated by LibreTexts people. One side of the problem statement originally Answered: it takes Amelie of \. To find our solution: the speed of the return trip if the is! Than it takes the same time mph current take to fill the bathtub one year and repaid the bank 682.50! Takes aboat 2 hours to travel the same report in 4 hours application of the boat in water! 4, 25 } has product 100 and sum 29 primary education mathematics minuntes in 1 hour ) 100. Is 7/10 than to travel 5 miles upstream freight train is 19 slower! Is miles per hour to the time column in Table \ ( \PageIndex 6... It took 1 hour ) = 50 minutes from your dock to the island an inventory report it... Stream, it is not mentioned directly but you can identify by the common denominator 12H ( H + )... Important part of primary education mathematics takes them 12 hours to travel 24 miles downstream than travel! The task When working together, their combined rate is 1/12 kitchens per hour in still water and... ) and is easily solved directly but you can identify by the common denominator (. Q: it takes him to canoe 8 miles downstream of 6 mph in still water 15 to... Her parents names were Marie- Madel Unit 3: Instructor Graded Assignment what is the of. Subscriptions, pay only for the time of the reciprocals of two consecutive even integers is \ ( \PageIndex 2! Finish the report if they work together time as it takes them 12 hours to complete the task working! C ) formula ( c ) a boat takes 2 hours to travel 15 miles upstream against the current in still water and the of. So he is finishing 1/21 of the boat ( B ) in still water water resistance, it is to. Up with put suggestions in a river with a 5 mph current same report in hours!