Word Problems - Travel

Richard and Grace planned to meet at a restaurant. Since it was a lovely day outside they decided to walk there. Richard started at point A, located 820 feet away from point B, which is where Grace began her walk. They both walked in the same direction and arrived at the same time. If Richard walks 6 times faster than Grace what is the ratio between the respective distances covered?

6:1 This is because you are told that Richard and Grace walk in the same direction. Furthermore, Richard walks six times faster than Grace. Finally, you are told they arrive at the same time. Therefore, the ratio of walking speed between Richard and Grace must be the same as the ratio of distance walked. Therefore, Richard walks six times the distance that Grace walked: X Richard : X Grace = 6v*t : v*t = 6 : 1 Hence, the ratio is 6:1.

Jonathan lives in town A and works in town B. Since he is not a morning person he usually ends up leaving the house later than he intended and has to drive to work at 110 m/h. On his way back home he likes to enjoy the lovely scenery around him and drives at 42.5 m/h. The journey back home takes him 84 minutes longer than his drive to work in the morning. What is the distance in miles between town A and town B?

96.9 miles Explanation The correct answer is 96.9 m.In order to solve this type of questions you need to be familiar with the basic formula:x = v*tThis formula represents: distance = velocity*time (the time it takes to cover the given distance).We can use the information provided to form a system of two equations- one for the way from A to B and one for the way back from B to A. Distance: The distance from A-> B is equal to the distance from B->A and will be indicated by "x". Velocity: The velocity from A->B: VA,B = 110 m/h The velocity from B->A: VB,A = 42.5 m/h Time: The time it took Jonathan to get from A->B will be indicated by "t", thus: tA,B = t hours. The time it took Jonathan to get from B->A: tB,A = t + 84 min. Since the velocities and the answers are given in units of hours and not minutes we shall convert the 84 minutes to 1.4 hours. Thus: tB,A = t + 1.4 hours. Inserting the above data in the formula:(i) x = 110*t(ii) x = 42.5*(t + 1.4)Since we are only interested in finding x we will isolate t from the first equation:(i) t = x/110And insert it into the second equation (the purpose was to remain with "x" as the only variable):(ii) x = 42.5*(x/110 + 1.4)=> x = 0.386x + 59.5=> 0.614x = 59.5=> x = 96.9 m Another way to approach this question would be to insert options 1-4 into the formula and find which one aligns with the given information. Note, it is important to make sure you use the same units throughout your calculations and that the units of options 1-4 match those you are using. Sometimes it is best to convert prior to calculations (like in this case), but note that sometimes it may be easier to do the calculations first and convert the answer at the end.

Robert and William are running towards each other from point A and point B, respectively, which are 60 km apart. When they meet, they turn around and run back to their starting points. If they both started running at 08:00, Robert at an average speed of 25 km/hour and William at a speed of 15 km/hour, at what time will William be back at point B?

The best answer is 11:00. When Robert and William run towards each other, add their speeds to find when they will meet: t = d/v = 60/40 = 1.5 , They will meet after 1.5 hours. After they meet, each of them runs back to his starting point. The distance that William made from point B to the meeting point is 1.5 x 15=22.5 km. He has to run back the same distance at the same speed, it will also take him 1.5 hours, So, William will be back in point be 3 hours after 08:00, at 11:00.

On her way to work, Joann walked 12 minutes from her house to the bus station and waited 5 minutes for the bus. The bus ride was 25 kilometers long and took 15 minutes, and after another 8 minutes of walking, Joann arrived at her work place. If Joann walks at a speed of 50 meters per minute, what was Joann's average speed for the whole trip?

The best answer is 39 km/h. The total distance Joann walked is (12+8) x 50 = 1000 meters, which is 1 km. Together with the bus ride, Joann made a total distance of 26 km. The total time is 12+5+15+8=40 minutes, which is 2 thirds of an hour. Therefore, the average speed of Joann in terms of kilometers per hour is:

Car A and car B start driving at the same time from the same place. If car A is driving at a speed of 85 mph, and car B is driving at a speed of 105 mph, and they are both driving on the same road, what will be the distance between them after 4.5 hours?

The best answer is 90 miles. When both cars are driving in the same direction, subtract their speeds to find the difference between them: (105-85) x 4.5 = 90 miles.

Steve drove his car to work at 100.8 km/h in an area where the speed limit is 90 km/h. He passed a police officer, who noticing that Steve was over the speed limit, decided to chase him on his motorbike. By the time the officer started out, Steve's car had travelled 250 meters. If the officer pursued him at 136.8 km/h, what is the distance Steve drove before the police officer caught up with him (in meters)?

he correct answer is 950m.We know from the previous question that the police officer caught up with Steve 25 seconds after the officer set out in pursuit. In order to find out where they meet, we can insert: t = 25 sec in one of the equations from the previous question; (1) Xs = 28*t(2) Xp = -250 + 38*t => X = 28*25 = 700mSteve drove through these 700m distance + an initial 250 meters. Therefore, the meeting point, compared to Steve's starting point, is: 250m + 700m = 950m.

A delivery guy is working 5 hours a day, delivering packages from the central post office to the local post office and back, at a speed of 70 miles per hour. What can be the number of times that the delivery guy returns to the central post office, if the distance between both offices is 14 miles?

The best answer is A. The total distance that can be made by the delivery guy in 5 hours is 5x70=350 miles. Thus, he can make the distance between both offices 350/14=25 times. From this follows that he can drive 13 times to one direction and 12 times to the other direction. Since he starts from the central post office, he will arrive at the local post office 13 times and at the central post office only 12 times.

Richard and Grace planned to meet at a restaurant. Since it was a lovely day outside they decided to walk there. Richard started at point A, located 820 feet away from point B, which is where Grace began her walk. If Richard walks 6 times faster than Grace what will be the distance he'll cover before catching up with her?

The correct answer is (984 feet). Grace walks a certain distance from point B to the restaurant, denoted x. Richard walks the same distance that was denoted x and 820 meters that separate point A from point B. Since Richard walks six times faster than grace, he covers six times the distance that Grace walks in that same time frame. Finally, you know that they walk in the same direction; therefore, you can conclude the following equation: Therefore, Richard walks a total of 820 + 164 = 984 meters

John bikes for exercise. If John bikes for 100 minutes and travels 20 miles, what was John's speed in mph?

The correct answer is 12.If John bikes for 100 minutes and travels 20 miles, then John bikes 20 miles / 100 minutes = 0.2 miles per minute. However, the question asks for John's speed in miles per HOUR. Since there are 60 minutes in an hour, we need to multiply 0.2 by 60.Therefore, John's speed is 0.2*60 = 12 mph.

One morning a truck left the village of Poe and headed on towards the village of Dunk at a speed of 76 km/h. 42 minutes later, another truck left Dunk and traveled to Poe at a speed of 87 km/h. If the two villages are 232.5 km apart, how far from Poe will the trucks meet?

The correct answer is 136.8km.The truck from Poe, travelling 76 km/h, leaves 42 minutes before the truck from Dunk, travelling at 87 km/h. Therefore, if the truck from Poe travels for T hours until meeting the truck from Dunk, then the truck from Dunk travels for T - 42/60 = T - 0.7 hours. Since the total travel distance of the two trucks is 232.5 km, we can solve for T as follows:76 * T + 87 * (T - 0.7) = 232.576T + 87T - 60.9 = 232.5163T = 293.4T = 1.8Therefore, the total distance traveled by the truck from Poe, which is how far from Poe the two trucks will meet, is 76*1.8 = 136.8 km

A ship can travel 25 mph when the water is calm but has to slow down to 18 mph when the water gets stormy. If the ship makes a 315-mile journey in 14.5 hours, how many more hours were spent travelling through calm water?

The correct answer is 13⁄14. Let x represent the number of hours the ship spent travelling through calm water and 14.5-x the number of hours the ship spent travelling through stormy water. Since Distance = Velocity * Time and the Distance is 315 miles, we can formulate the following equation in order to find the value of x:315 = 25x + 18(14.5 - x)315 = 25x + 261 - 18x315 - 261 = 25x - 18x7x = 54x = 54⁄7This means that 54⁄7 hours were spent travelling in calm waters and hours were spent travelling in stormy waters. The question asks how much MORE time was spent travelling through calm water so the answer is hours.

An airplane has enough gas to fly for 18.2 hours with an average speed of 934 km/h. What is the greatest distance the airplane can travel before it needs to be refueled?

The correct answer is 16,998 km. Since Distance = Velocity * Time, the greatest distance the plane can fly before it needs to be refueled is 934*18.2 = 16,998.8 km. Since the plane is capable of travelling 16,998 km but does not quite have enough fuel to travel 16,999 km, the correct answer is 16,998 km.

A ball rolls down a 1-metre ramp at a speed of 2.7 km per minute. How many seconds does it take the ball to roll down the ramp?

The correct answer is 1⁄45. If the ball rolls at an average speed of 2.7 km per minute, and there are 1000 metres in 1 km, then its speed is 2,700 metres per minute. Therefore, the ball will need 1/2,700 minutes to roll down the 1-metre ramp. However, the question is asking for the number of seconds. Since there are 60 seconds in one minute, the correct answer is: seconds.

Jasmine walks every Sunday for 5 hours. Half of the way she walks at A mph and the remainder she covers at 3A mph. How long will it take her to complete her entire walk if she proceeds at a constant speed of 3A mph?

The correct answer is 2.5 hours. When tackling this type of question you will need to represent the given information in terms of the basic formula: x=v*t. In some cases, it is best to first rearrange the formula so it will be easier to work with. For example, in this case, it will be easier to work with: t=x/v. Now, you can form a system of two equations whereby one is aimed directly at what you are looking for (equation ii). (i) 5 = (x/2)/A + (x/2)/3A(ii) ? = x/3ARearranging the first equation:(i) 5 = (3x/2)/3A + (x/2)/3A /multiplying both the nominator and denominator of a number maintains the value of the fraction and enables you to create a common denominator that's easy to work with.=> 5 = (4x/2)/3A=> 5 = 2x/3Ax/3A = 5/2 = 2.5 hours

Paul and Emma set out to meet in London. They travelled on different trains which approached one another from opposite directions. The distance between the stations of departure was 100 km. Paul's train departed at 8 am travelling at a velocity of 100 km/h, whilst Emma's train departed 15 minutes later travelling at 80 km/h. When will the distance between the two trains be 60 km?

The correct answer is 20 minutes. In this question we first need to determine which point in time will be point zero for our calculations and which direction will be considered as the "positive" direction. For simplicity, we will consider Paul's train (the first one to depart) as our point zero (t=0) and its direction as the "positive" one. Inserting the information in the basic formula: X = V*t :(1) Xp = 100*t Emma's train departs 15 minutes later travelling in the opposite direction, which we will refer to as the "negative" direction. In addition, since we have decided that Paul's train is the reference point, we need to take into account the initial 100 km difference between the two stations. Inserting the information in the basic formula: X = X0 + V*t :(2) Xe = 100 - 80*(t - 1/4)Note that we have converted 15 minutes to 1/4 h in order to maintain corresponding units throughout the equation. We are interested in the moment when the distance between the trains is 60 km, i.e. Xe - Xp = 60 km Thus:[ 100 - 80*(t - 1/4) ] - 100*t = 60=> 100 - 80*t + 20 - 100*t = 60=> 60 = 180*t=> t = 1/3 h = 1/3 * 60 = 20 min The distance between the trains will be 60 km, 20 minutes after departure.

Steve drove his car to work at 100.8 km/h in an area where the speed limit is 90 km/h. He passed a police officer, who noticing that Steve was over the speed limit, decided to chase him on his motorbike. By the time the officer started out, Steve's car had travelled 250 meters. If the officer pursued him at 136.8 km/h, when would he catch up with Steve (in seconds)?

The correct answer is 25 seconds. In this question we first need to determine which point in time will be the starting point for our calculations. In order to simplify calculations we recommend choosing the moment when the police officer set off in pursuit of Steve as: t=0.All parameters regarding Steve will be indicated with an "s" subscript while those regarding the police officer will be indicated with "p" subscript. Before beginning calculations it is advisable to make sure that all the parameters are expressed in the same units and also the distracters given in the question. Basic conversions you should be familiar with:1 km = 1000 m1 h = 60 min = 3600 sec Thus: Vs = 100.8 km/h = 100.8*(1000/3600) = 28 m/sVp = 136.8 km/h = 136.8*(1000/3600) = 38 m/s Now we can insert the given information in the basic formula X = X0 + V*t :(1) Xs = 28*t(2) Xp = -250 + 38*t Notice that the police officer's starting point was 250 meters behind Steve's position at that moment. When the police officer finally caught up with Steve, Xs = Xp, therefore:28*t = -250 + 38*t / +250 - 28t250 = 10t=> t = 25 sec The police officer caught up with Steve after 25 seconds.

A courier needs to deliver three packages to three different addresses. He travels from the main office to the first address at a speed of 25 mph, taking him 15 minutes. Then he travels to the second address at a speed of 33 mph, taking him 21 minutes. Finally, he travels to the third address at a speed of 28 mph, taking him 8 minutes. If the courier travels back to the main office at a speed of 35 mph and takes 25 minutes to arrive, what is the courier's average speed for the entire journey?

The correct answer is 31.4 mph. In order to solve the question we need to calculate the sum of distances the courier passed and divide it by the total time it took him: Although all the times are measured in minutes while the velocities are measured in miles per hour, we do not need to convert all the times to hours. The reason for this is that the times appear both in the numerator and in the denominator, which means that performing the same operations on both would not change the final result. Therefore, the average speed for the entire journey was 31.4 mph.

Dylan drove 66 miles in 55 minutes. How many minutes longer would he have had to drive if his speed had been 5 mph less?

The correct answer is 4.1.We know that Dylan drives a distance of 66 miles in 55 minutes (=55/60 hours). Using the basic formula: Distance = Velocity * Time, we can find Dylan's velocity: Therefore, Dylan's velocity is 72 mph. Had it been 5 mph slower, it would have equal 72 - 5 = 67 mph. In that case, the time it would have taken him to drive 66 miles is: Therefore, it would have taken Dylan 59.1 - 55 = 4.1 minutes longer.

Charles jogs for 30 minutes, and then walks back at the end of his jog, every morning. Charles's speed when walking is 3 mph. If Charles's morning jog and walk back takes a total of 75 minutes, how fast does Charles jog every morning?

The correct answer is 4.5 mph. If Charles's morning jog and walk takes a total of 75 minutes, and Charles spends 30 minutes jogging, then the walk back takes 75 - 30 = 45 minutes. If Charles walks back at a speed of 3 mph, then Charles walks a distance of: Since the distance Charles jogs must be equal to the distance Charles walks back, the distance Charles jogs is also 2.25 miles. If Charles jogs for 30 minutes (=1/2 hour), then Charles jogs at a speed of .

A turtle and a rabbit argued as to who would win in a long distance running competition, each claiming the throne. A crow that happened to be in the area said it depends upon the length of the track. If the turtle ran at a speed of 10 m/h and the rabbit ran at a speed of 30 m/h but had to rest every 10 miles for 43 minutes, which of the following answers could be the length of the track for both of them to cross the finish line at the same time?

The correct answer is 43 miles. First, we need to understand that for both animals to reach the finishing line at the same time, the time element (t) in our basic formula: x=v*t has to be equal ('x' represents the length of the track - the variable we are interested in).Next, we need to represent each animal's time using 'x' and then compare the two. Turtle: t = x/10Rabbit: There are two components to the rabbit's time: running + resting. The time the rabbit ran = x/30The time the rabbit rested is determined based on the number of rests he took, which we will indicate by 'y'. We know that each time the rabbit rested for 43 minutes, thus the total time he spent on resting is y*43 min. However, since the velocities and distractors are given in units of m/h we recommend converting the minutes to hours. Thus, the time the rabbit rested = y*43/60 hours. Now we can form the following equation: The rabbit The turtle Resting Runningy*43/60 + x/30 = x/10 /Multiply both side by 6043y + 2x = 6x43y = 4x10.75y = x This equation represents the pattern according to which the rabbit and the turtle are meeting, whilst running (Remember: 'y' represents the number of rests the rabbit took, and 'x' represents the total distance they passed).The coefficient 10.75 tell us that the rabbit and the turtle meet every 10.75 miles. Thus, we need to look for an optional track's length that can be divided by 10.75.The only option that adheres to this rule is answer choice 1: 43 m/10.75 = 4 miles.

Alice and Bob start driving towards each other at the same time. Alice drives at 35 mph. How fast was Bob driving if they started 320 miles apart and it took them 4 hours to meet each other?

The correct answer is 45 mph. Since Alice and Bob are driving towards each other, we can add their speeds together. If we let x represent Bob's speed, then their joint speed is 35 + x. We know that they drive for 4 hours and pass a total of 320 miles together. Therefore, we can formulate the following equation based on the Distance = Velocity * Time formula:320 = (35 + x) * 4 / Divide both sides by 480 = 35 + xx = 45

Richard and Grace planned to meet at a restaurant. Since it was a lovely day outside they decided to walk there. Richard started at point A, located 820 feet away from point B, which is where Grace began her walk. If Richard walks 6 times faster than Grace, how much time in minutes passes before they meet?

The correct answer is none of the above. The equations we have formed previously (in question 3) are virtually the same as what is required here: Let's mark v as velocity and t as time(i) 984 = 6v*t / reduce by 6=(ii) 164 = v*t This is a case of 2 variables and only one equation. Thus, we have no means of extracting the absolute value of t. We can, however, represent it as: t = 164/v, which doesn't, however, appear in any of the given options.

Catharine drove 250 miles to Hertfordshire in order to spend New Year's Eve with her family. She drove at an average speed 12 miles per hour faster than her usual average speed. If she completed the trip in 1 hour less than usual, what is her usual driving speed, in miles per hour?

The correct answer is 49.1 mph. There are two ways to approach this type of question:1. Organize the given information in a table. When there is more than one variable we recommend representing them with reference to the one we are interested in (in this case, represent time using the velocity variable).Since we are told that the current trip is 1 hour less than the usual trip we can easily form an equation using the information we have organized:250/(v+12) = 250/v - 1 / Multiply by (v+12) and also by v=>250v = 250*(v+12) - v*(v+12) / Open brackets=> 250v = 250v + 3000 - v^2 - 12v=> V^2 + 12v - 3000 = 0In order to find the value of "v" we need to use the "Quadratic formula":=> V1,2 = (-12 +- √[12^2 - 4*1*{-3000}])/(2*1)=> V1,2 = (-12 +- √12144)/2 = (12 +- 110.2)/2 = 49.1 Or -61.1Catherine could not have been driving in a negative velocity and therefore: V = 49.1 mph2. You can solve this problem using the basic formula: x = v*t and the given information in order to form a system of two equations-Usual trip:(i) 250 = v*t Current trip:(ii) 250 = (v+12)*(t-1)We are interested in Catherine's velocity and therefore we will isolate "t" from the first equation:(i) t = 250/v And insert it into the second equation (the purpose is to remain with the velocity variable only):(ii) 250 = (v+12)*(250/v -1)The rest of the calculation is as shown in the first method.*Please note that you can check yourself by inserting the velocity into the original equations and see if there is indeed a 1 hour difference between the usual and current trips. We recommend checking yourself only if you can spare the extra time.

A person ran a marathon (26.22 miles) in less than five hours but in more than three hours. Which of the following could not be the speed that the person ran the marathon?

The correct answer is 5.24 mph .In order for a person to run a marathon in more than three hours, he must run at a speed slower than 26.22 miles / 3 hours = 8.74 mph. In order for him to run a marathon in less than five hours, he must run at a speed faster than 26.22 miles / 5 hours = 5.244 mph. Since the person ran the marathon in more than three, but less than five, hours, his speed must be between 5.244 mph and 8.74 mph. All of the answer options but the first one are included in this range. Therefore, 5.24 is the correct answer.

A train travelling at 72.5 mph enters a tunnel that is 5.8 miles long. The train is 1.2 mile long. How many minutes does it take for the entire train to pass though the tunnel?

The correct answer is 5.79 minutes. The total distance the train has to pass is the length of the tunnel + the length of the train itself. Thus: X total = 5.8 + 1.2 = 7 miles Insert the information in the basic formula: X = V*t :7 = 72.5*t=> t = 0.09655 h=> t = 0.09655 * 60 = 5.79 min.

Mary and Neal are moving and need to drive 1,000 miles to reach their new home. Mary drives first for three hours. Then, Neal drives for another three hours. Mary and Neal keep switching the role of driver every three hours until they reach their destination. If the entire journey takes exactly 15 hours, and Neal's driving speed is 10% greater than Mary's driving speed, for how many miles was Mary driving?

The correct answer is 577 miles. If Mary drives x miles per hour, then Neal drives 1.1x miles per hour. If Mary drives the first three hours of the 15-hour journey, then she drives for hours 1-3, 7-9, and 13-15, while Neal drives for hours 4-6 and 10-12.Therefore, Mary drives for 9 hours at x miles per hour, and Neal drives for 6 hours at 1.1x miles per hour. Since Distance = Velocity * Time, Mary drives a total distance of 9x miles and Neal drives a total distance of 6.6x miles. Since the total distance driven by both Mary and Neal is 1,000 miles, we can formulate the following equation: 9x + 6.6x = 1,00015.6x = 1000x ≈ 64.1 miles The total distance driven by Mary is therefore 9 * 64.1 = 576.9 miles ≈ 577 miles.

A person driving on the highway, travelling at a speed which is approximately 20% faster than the posted speed limit, travels 45 miles in only 36 minutes. What is the posted speed limit on the highway?

The correct answer is 62.5 mph. If a person travels 45 miles in 36 minutes (=36/60=0.6 hours), then his speed is 45/0.6 = 75 mph. According to the question, this speed is approximately 20% faster than the posted speed limit. Therefore, the posted speed limit is 75/1.2 = 62.5 mph.

Amy and Ashley bike along the same trail. Amy bikes the entire trail at a speed of 20 mph and Ashley bikes it at a speed of 15 mph. If Amy finishes biking the entire trail 8 minutes before Ashley, how many miles long is the bike trail?

The correct answer is 8.According to the question, Amy's speed is 20 mph and Ashley's speed is 15 mph. If Amy finishes the trail 8 minutes before Ashley, and we let t represent the amount of time it takes Amy to finish the trail, then Ashley will take t + 8 minutes to finish it. However, since the speeds are all in miles per hour, we must convert the 8 minutes into 8/60 = 2/15 hours. Therefore, Ashley's actual time is (t + 2/15).Since the distance they both bike is the same, we can now formulate the following equation comparing their times:20t = 15(t + 2/15)20t = 15t + 25t = 2t = 0.4We are asked to find the trail's length, i.e. the distance they both passed. Plugging the value for t back into the distance Amy has passed, we arrive at: 20 * 0.4 = 8 miles.

Tanya likes to run five laps around a 1000m track. She runs each successive lap 10% faster than the previous lap. If Tanya runs the second lap in 110 seconds, in how much less time would she run all five laps if she ran all of them at her fastest speed?

The correct answer is 91.35 seconds. We can see from the formula: Distance = Velocity * Time, that Velocity and Time are inversely related - when one increases by a certain amount the other decreases by the same amount. So if Tanya runs each successive lap 10% faster, the time it takes her to run each lap decreases by 10%. Therefore, if Tanya runs the second lap in 110 seconds, she ran the first lap in 110 * 1.1 = 121 seconds. The times for the other laps can be calculated as follows: Lap #3: 110/1.1 = 100 seconds Lap #4: 100/1.1 ≈ 90.91 seconds Lap #5: 90.91/1.1 ≈ 82.64 seconds The total time spent running all five laps is 121 + 110 + 100 + 90.91 + 82.64 = 504.55 seconds. If all five laps had been run at Tanya's fastest speed (= the least amount of time per lap), each lap would have been completed in 82.64 seconds for a total time of 82.64 * 5 = 413.2 seconds. Therefore, the difference between Tanya's actual running time and her running time if she had ran each lap at her fastest speed is 504.55 - 413.2 = 91.35 seconds.