Rate-Time-Distance Problems

Algebra

Rate-Time-Distance Problems

• An object is in uniform motion when it moves without changing its speed, or rate. These problems fall into 3 categories: Motion in opposite directions, Motion in the same direction, & Round Trip. Each is solved using a chart, a sketch, and the distance formula.

T

DR

R

DTRTD or or

Motion in the Same DirectionA helicopter leaves Central Airport and flies north at 180 mi/h. Twenty minutes later a plane leaves the airport and follows the helicopter at 330 mi/h. How long does it take the plane to overtake the helicopter.

Rate Time Distance

Helicopter

Plane 330

180

t

t + ⅓

330t

180(t + ⅓)

Rate Time Distance

Helicopter

Plane

180

330

t + ⅓

t

180(t + ⅓)

330t

When the plane overtakes the helicopter, the two distances are equal.

3

1180330 tt

60180330 tt

60150 t

min. 24or h 5

2t

Officer Barbrady left his home at 2:15 PM and had driven 60 miles when he ran out of gas. He walked 2 miles to a gas station, where he arrived at 4:15 pm. If he drives 10 times as faster than he walks, how fast does he walk?

Distance Rate Time

Driving

Walking r

10r60

2r

2r10

60

Set to Equal

15):4 to15:(22h WalkingTime Driving Time

22

10

60

rr2

10

20

10

60rr

180102 rr4

Office Barbrady walked at a rate of 4/mph

1

2

10

80r

8020 r

Two cars leave the same town at the same time. One travels north at 60 mph and the other south at 45 mph. In how many hours will they be 420 miles apart?

Bob

Sherry

DistanceTimeRate

Car #1

Car #2

60

45

t

t

60t

45t

4204560 tt

Bob

Sherry

DistanceTimeRate

Car #1

Car #2

60

45

t

t

60t

45t

420105 t4t

In 4 hours the cars will be 420 miles apart.

Bicyclist Brent and Jane started at noon from 60 km apart and rode toward each other, meeting at 1:30 P.M. Brent’s speed was 4 km/h greater than Jane’s speed. Find their speeds.

Rate Time Distance

Brent

Jane r

r + 4 1.5

1.5

1.5(r + 4)

1.5r

Rate Time Distance

Brent

Jane

r + 4

r

1.5

1.5

1.5(r + 4)

1.5r

605.145.1 rr

605.165.1 rr

543 r

Speed sJane' 18r

Speed sBrent'

22418

Round TripA ski lift carried Maria up a slope at the rate of 6 km/h, and she skied back down parallel to the lift at 34 km/h. The round trip took 30 min. How far did she ski?

Rate Time Distance

Up

Down 34

6

t

0.5 - t

34 t

6(0.5 – t)

Rate Time Distance

Up

Down

6

34

0.5 - t

t

6(0.5 – t)

34 t

tt 5.0634

In round trip problems, the two distances are equal.

tt 6334 340 t

075.040

3t

km 55.2075.03434 t

Maria skied for 0.075h, or 4.5 min, for a distance of 2.55km.

Rate-Time-Distance Problems

Review Additional Problems Part B

Sherry and Bob like to jog in the park. Sherry can jog at 5 mph, while Bob can jog at 7 mph.If Sherry starts 30 minutes ahead of Bob, how long will it take Bob to catch up to Sherry?

Bob

Sherry

DistanceTimeRate

5

7 x

Let x = the time it takes Bob to catch Sherry

x + 1/2 5(x + 1/2)

7x

Sherry started 30 minutes or 1/2 hour before Bob, so her timemust reflect that amount. Rate is in miles per hour, time mustbe in hours so our units match.

When Bob catches-up with Sherry their distances will be equal. So the Equation will be sherry’s distance equals Bob’s distance.

It will take Bob 1.25 hrs to catch up with Sherry.

Bob

Sherry

DistanceTimeRate

5

7 x

x + 1/2 5(x + 1/2)

7x

xx 72

15

xx 75.25

x25.2 x25.1

Bernadette drove 120 miles. The first part of the trip she averaged 60 mph, but on the second part of the trip she ran into some congestion and averaged 48 mph. If the total driving time was 2.2 hours, how much time did she spend at 60 mph?

Rate Time Distance

Part 1

Part 2

60

48 2.2-t

t 60t

48(2.2-t)

Rate Time Distance

Part 1

Part 2

60

48 2.2-t

t 60t

48(2.2-t)

1202.24860 tt

120486.10560 tt

4.1412 t2.1t The time traveled at 60

mph was 1.2 hours.

A passenger train’s speed is 60 mi/h, and a freight train’s speed is 40 mi/h. The passenger train travels the same distance in 1.5 h less time than the freight train. How long does each train take to make the trip.

DistanceTimeRate

Freight train 4.5h.Passenger train 3h

Ali rode her bike to visit a friend. She traveled at 10 mi/h. While she was there, it began to rain. Her friend drove her home in a car traveling 25 mi/h. Ali took 1.5 hours longer to go to her friend’s than to return home. How many hours did it take Ali ti ride to her friend’s house?

DistanceTimeRate

2.5 h

Katie rides her bike the same distance as Jill walks. Katie rides her bike 10 km/h faster than Jill walks. If it takes Katie 1 h. and Jill 3 h to travel the same distance, how fast does each travel?

DistanceTimeRate

Katie 15 km/ hJill 5 km/h

At 10:00 A.M., a car leaves a house at a rate of 60 mi/h. At the same time, another car leaves the same house at a rate of 50 mi/h in the opposite direction. At what time will the car be 330 miles apart?

DistanceTimeRate

1:00 P.M.

Brittney begins walking ay 3 mi/h toward the library. Her friend meets her at the halfway point and drives her the rest of the way to the library. The distance to the library is 4 miles. How many hours did Marla walk?

DistanceTimeRate

2/3 h or 40 min.

Ryan begins walking towards John’s house at 3 mi/h. John leaves his house at the same time and walks towards Fred’s house on the same path at a rate of 2 mi/h. How long will it be before they meet if the distance between the houses is 4 miles?

DistanceTimeRate

4/5 h. or 48 min

A train leaves the station at 6:00 P.M. traveling west at 80 mi/h. On a parallel track, a second train leaves the station 3 hours later traveling west at 100 mi/h. At what time will the second train catch up with the first?

DistanceTimeRate

9:00 A.M.

It takes 1 hour longer to fly to St. Paul at 200 mi/h than it does to return at 250 mi/h. How far way is St. Paul.

DistanceTimeRate

1000 mi