# Rate, Time Distance Problems With Solutions

The relationship between distance, rate (speed) and time

Distance = time � rate

is used to solve uniform motion problems. Detailed solutions to the problems and explanations are provided.

 Problem 1 Two cars started from the same point, at 5 am, traveling in opposite directions at 40 and 50 mph respectively. At what time will they be 450 miles apart? Solution to Problem 1: After t hours the distances D1 and D2, in miles per hour, travelled by the two cars are given by D1 = 40 t and D2 = 50 t After t hours the distance D separating the two cars is given by D = D1 + D2 = 40 t + 50 t = 90 t Distance D will be equal to 450 miles when D = 90 t = 450 miles To find the time t for D to be 450 miles, solve the above equation for t to obtain t = 5 hours. 5 am + 5 hours = 10 am Problem 2 At 9 am a car (A) began a journey from a point, traveling at 40 mph. At 10 am another car (B) started traveling from the same point at 60 mph in the same direction as car (A). At what time will car B pass car A? Solution to Problem 2: After t hours the distances D1 traveled by car A is given by D1 = 40 t Car B starts at 10 am and will therefore have spent one hour less than car A when it passes it. After (t - 1) hours, distance D2 traveled by car B is given by D2 = 60 (t-1) When car B passes car A, they are at the same distance from the starting point and therefore D1 = D2 which gives 40 t = 60 (t-1) Solve the above equation for t to find t = 3 hours Car B passes car A at 9 + 3 = 12 pm Problem 3 Two trains, traveling towards each other, left from two stations that are 900 miles apart, at 4 pm. If the rate of the first train is 72 mph and the rate of the second train is 78 mph, at whatt time will they pass each other? Solution to Problem 3: After t hours, the two trains will have traveled distances D1 and D2 (in miles) given by D1 = 72 t and D2 = 78 t After t hours total distance D traveled by the two trains is given by D = D1 + D2 = 72 t + 78 t = 150 t When distance D is equal to 900 miles, the two trains pass each other. 150 t = 900 Solve the above equation for t t = 6 hours. Problem 4 John left home and drove at the rate of 45 mph for 2 hours. He stopped for lunch then drove for another 3 hours at the rate of 55 mph to reach his destination. How many miles did John drive? Solution to Problem 4: The total distance D traveled by John is given by D = 45 � 2 + 3 � 55 = 255 miles. Problem 5 Linda left home and drove for 2 hours. She stopped for lunch then drove for another 3 hours at a rate that is 10 mph higher than the rate before she had lunch. If the total distance Linda traveled is 230 miles, what was the rate before lunch? Solution to Problem 5: If x is the rate at which Linda drove before lunch the rate after lunch is equal x + 10. The total distance D traveled by Linda is given by D = 2 x + 3(x + 10) and is equal to 230 miles. Hence 2 x + 3 (x + 10) = 230 Solve for x to obtain x = 40 miles / hour. Problem 6 Two cars left, at 8 am, from the same point, one traveling east at 50 mph and the other travelling south at 60 mph. At what time will they be 300 miles apart? Solution to Problem 6: A diagram is shown below to help you understand the problem. . The two cars are traveling in directions that are at right angle. Let x and y be the distances traveled by the two cars in t hours. Hence x = 50 t and y = 60 t Since the two directions are at right angle, Pythagora's theorem can used to find distance D between the two cars as follows: D = sqrt ( x 2 + y 2 ) We now find the time at which D = 300 miles by solving sqrt ( x 2 + y 2 ) = 300 Square both sides and substitute x and y by 50 t and 60 t respectively to obtain the equation (50 t) 2 + (60 t) 2 = 300 2 Solve the above equations to obtain t = 3.84 hours (rounded to two decimal places) or 3 hours and 51 minutes (to the nearest minute) The two cars will 300 miles apart at 8 + 3 h 51' = 11:51 am. Problem 7 By Car, John traveled from city A to city B in 3 hours. At a rate that was 20 mph higher than John's, Peter traveled the same distance in 2 hours. Find the distance between the two cities. Solution to Problem 7: Let x be John's rate in traveling between the two cities. The rate of Peter will be x + 10. We use the rate-time-distance formula to write the distance D traveled by John and Peter (same distance D) D = 3 x and D = 2(x + 20) The first equation can be solved for x to give x = D / 3 Substitute x by D / 3 into the second equation D = 2(D/3 +20) Solve for D to obtain D = 120 miles Problem 8 Gary started driving at 9:00 am from city A towards city B at a rate of 50 mph. At a rate that is 15 mph higher than Gary's, Thomas started driving at the same time as John from city B towards city A. If Gary and Thomas crossed each other at 11 am, what is the distance between the two cities? Solution to Problem 8: Let D be the distance between the two cities. When Gary and Thomas cross each other, they have covered all the distance between the two cities. Hence D1 = 2 * 50 = 100 miles , distance traveled by Gary D1 = 2 * (50 + 15) = 130 miles , distance traveled by Gary Distance D between the two cities is given by D = 100 miles + 130 miles = 230 miles Problem 9 Two cars started at the same time, from the same point, driving along the same road. The rate of the first car is 50 mph and the rate of the second car is 60 mph. How long will it take for the distance between the two cars to be 30 miles? Solution to Problem 9: Let D1 and D2 be the distances traveled by the two cars in t hours D1 = 50 t and D2 = 60 t The second has a higher speed and therefore the distance d between the two cars is given by d = 60 t - 50 t = 10 t For d to be 30 miles, we need to have 30 miles = 10 t Solve the above equation for t to obtain t = 3 hours. Problem 10 Two trains started at 10 pm, from the same point. The first train traveled North at the rate of 80 mph and the second train traveled South at the rate of 100 mph. At what time were they 450 miles apart? Solution to Problem 10: Let D1 and D2 be the distances traveled by the two trains in t hours. D1 = 80 t and D2 = 100 t Since the two trains are traveling in opposite directions, then total distance D between the two trains is given by D = D1 + D2 = 180 t For this distance to be 450 miles, we need to have 180 t = 450 Solve for t to obtain t = 2 hours 30 minutes. 10 pm + 2:30 = 12:30 am Problem 11 Two trains started from the same point. At 8:00 am the first train traveled East at the rate of 80 mph. At 9:00 am, the second train traveled West at the rate of 100 mph. At what time were they 530 miles apart? Solution to Problem 11: When the first train has traveled for t hours the second train will have traveled (t - 1) hours since it started 1 hour late. Hence if D1 and D2 are the distances traveled by the two trains, then D1 = 80 t and D2 = 100 (t - 1) Since the trains are traveling in opposite direction, the total distance D between the two trains is given by D = D1 + D2 = 180 t - 100 For D to be 530 miles, we need to have 180 t - 100 = 530 Solve for t t = 3 hours 30 minutes. 8 am + 3:30 = 11:30 am
More math problems with detailed solutions .