Respect Responsibility Readiness
1. A freight train starts from Los Angeles and heads for Chicago at 40 mph. Two hours later a passenger train leaves the same station for Chicago traveling at 60 mph. How long will it be before the passenger train overtakes the freight train?
First draw a picture of the movement
Notice that the distances are equal.
Second make a diagram to put in information:
|Freight train||x + 2||40|
|Freight train||x + 2||40||40(x + 2)|
Every time rate and distance has some kind of relationship between the distances. This one had the distances equal. That is, the trains traveled the same distance because they started at the same place and traveled until one caught up with the other. This fact was not stated. you have to watch for the relationship. We have two distances in the table above. 40(x + 2) represents the distance for the freight train. 60x represents the distance for the passenger train. Set these two distances equal for your equation:
40(x + 2) = 60x
40x + 80 = 60x
-20x = -80
x = 4 hours
It takes 4 hours for the passenger train to overtake the freight train?
2. A car leaves San Francisco
for Los Angeles traveling an average of 70 mph. At the same time, another car
leaves Los Angeles for San Francisco traveling 60 mph. If it is 520 miles
between San Francisco and Los Angeles, how long before the two cars meet,
assuming that each maintains its average speed?
1. Find out if you know both speeds or both times. In this problem we find that both speeds are given.
2. What is the unknown? The question is, "How long before the two cars meet?" So their time is unknown. But which shall we let x equal? They leave at the same time and meet and meet at the same time, so the traveling times must be equal if neither one stops. Watch for the times being equal when two objects start at the same time and meet at the same time.
3. What fact is known about distance? The 520 miles is the total distance, so here is equation information. Remember, don’t put distance information in the distance box unless you have to (for example, when the exact distance each object travels is given)
First, draw a sketch of movement:
Second, make a diagram and fill in all the information you have determined. And remember, time multiplied by rate equals distance in the table.
Let x = time in hours for the two cars to meet
|Car from Los Angeles||x||60||60x|
|Car from San Francisco||x||70||70x|
You know that the total distance is 520 miles, so add the two distances in the diagram and set them equal to 520 miles for the equation.
60x + 70x = 520
130x = 52O
x = 4 hours
60(4) + 70(4) = 520520=520
3. Two planes leave New York
at 10 A.M., one heading for Europe at 600 mph and one heading in the opposite
direction at 150 mph. At what time will they be 900 miles apart? How far has
1. You know both speeds.
2. The times must be unknowns
and they are equal (or the same) because the planes leave at the same time and
travel until a certain time (when they are 900 miles apart).
3. The total distance is 900
Warning! Don’t let x = clock time, that is, the time of day. It stands for traveling time in hours! The rate is in miles per hour, so time must be in hours in all time, rate, and distance problems.
Let x = time in hours for planes to fly 900 miles apart. Remember, x equals the time for the planes, since their times are the same in this problem.
600x + 150x = 900
750x = 900
x = 900/750
x = 1 1/5 hours
is that what the problem asked
for? Always check the question, especially on time, rate, and distance problems.
The problem asked what time they would be 900 miles apart. Since the problem
says they left at 10 a.m., add 1 1/5 hours to that time. So the answer is 11:12
The problem also asked far each
would travel. Since we know that rate multiplied by time equals distance,
600x = distance for fast plane
and substituting the value of x,
600(1 1/5) = 600(6/5) = 720
150x = distance for slow plane
150(1 1/5) = 150(6/5) = 180
4. Moe Tell makes a business trip from his house to Logan in 2 hours. One hour later, he returns home in traffic at a rate 20 mph less than his rate going. If Moe is gone a total of 6 hours, how fast did he travel on each leg of the trip?
1. This problem doesn’t give
us the rates, but it does give us the time both ways. It gives the time to
Loganville as 2 hours. The total time is 6 hours, but he didn’t travel during
one of those hours. Deducting this hour leaves only 5 hours of total traveling
time. That means that if it took 2 hours to travel to Loganville, it must have
taken 3 hours to travel back.
2. The rates are both unknown. We can let x equal either one. Just be careful that the faster rate goes w/the shorter time. If x equals the rate going, then x – 20 equals the rate returning. If x equals the rate returning, then x+20 equals the rate going. Either way is correct.
Let x = rate in mph going to
X - 20 = rate in mph returning home.
|Toward Home||3||x - 20||3(x - 20)|
The distance to Loganville
equals the distance back home.
2x = 3(x - 20)
-x = -60
x = 60 mph (rate going)
x – 20 = 40 mph (rate
2(60) = 3(60 - 20)
120 = 3(40)
5. Tom and Jerry went on a camping trip with their motorcycles. One day Jerry left camp on his motorcycle to go to the village. Ten minutes later Tom decided to go too. If Jerry was traveling 30 mph and Tom traveled 35 mph, how long was it before Tom caught up with Jerry?
We know both speeds.
2. We know Tom traveled for 10 minutes less than
Jerry. But minutes cannot be used
in time, rate, and distance problems because rate is in miles per hour. So we
must change 10 minutes into 10/60 or one-sixth of an hour.
3. They travel the same distances so the distances are equal.
Let x = time in hours for Tom to catch Jerry.
|Jerry||x + 1/6||30||30(x + 1/6)|
= 30(x + 1/6)
= 30x + 5
5x = 5
x =1 hour (Tom's time)
|Tom||x - 1/6||35||35(x + 1/6)|
you use the alternate solution, you should be good at fractions!
be sure to answer the question and solve for x - 1/6, Tom's time.
6. Two cars headed for Las Vegas. One is 50 miles ahead of the other on the same road. The one in front is traveling 60 mph while the second car is traveling 70 mph. How long will it be before the second car overtakes the first car?
speeds are given.
The times are the unknowns and are equal.
The distances are not equal; this inequality is the tricky variation we
mentioned above. But if we add 50 miles to the distance of one, it will equal
the distance of the other.
x = time in hours for the second car to overtake the first.
+ 50 = 70x
= 5 hours
one car is 50 miles ahead of the other at the same time, it will take the cars
the same length of time to reach the spot where they are together (or where one
overtakes the other). So each takes x hours. The difficulty is that the two cars
do not start from the same point, so their distances are not equal. The sketch
shows that you have to add 50 miles to the distance the car first travels to
make it equal to the distance the second car travels. Of course, the equation
Rate and distance problem involving moving air (wind) or moving water (current)
Some more difficult problem have planes flying in a wind or boats traveling in moving water. The only problems of this type which we can solve are those where the objects move directly with or against the wind or water. The plane must have a direct headwind or tailwind and the boat must be going upstream or downstream. In this type of problem the plane's speed in still air would be increased by a tailwind or decreased by a headwind to determine how fast it actually covers the ground. For example, a plane flies 200 mph in still air. This is called airspeed. If there is a 20-mph headwind blowing, it would decrease the speed over the ground by 20 mph. so the ground speed of the plane would be 200-20 or 180 mph. The ground speed is the rate in time, rate, and distance problems. A headwind reduces the speed of the pane by the velocity of the wind. A tail wind increases the speed of the plane over the ground by the velocity of the wind. A plane with an airspeed of 400 mph with a 30-mph tailwind actually travels over the ground (ground speed) at 430 mph. A current affects a boat the same way.
7. The Red Baron takes 5 hours to fly from Los Angeles to Honolulu and 4 1/11 hours to return from Honolulu to Los Angeles. If the wind velocity is 50 mph from the west on both trips, what is the airspeed of the plane? (Airspeed is the speed of the plane in still air)
The 2 times are given
You are asked to find the speed of the plane in still air
Going to Honolulu you have headwind so subtract the velocity of the wind.
Returning to Los Angeles, you have tailwind, so add the velocity to airspeed
The distances are equal.
x = speed of plane in airspeed
- 50 = speed of plane over the ground to trip from Los Angeles to Honolulu
+ 50 = speed of plane over ground on the trip from Honolulu to Los Angeles
Ground speed determines how long it takes the plane to travel from one place to another.
|To Honolulu against the wind||5||x - 50||5(x - 50)|
|To Los Angeles with the wind||4 1/11||x + 50||4 1/11(x + 50)|
5(x - 50) = 4 1/11(x + 50)
5(x - 50) = 45/11(x + 50)
5x - 250 = (45x + 2250)/11
both sides by 11 to clear fractions.
- 2750 = 45x + 2250
= 500mph (airspeed)
- 50) = 45/11(500 + 50)
The plane will fly at the speed at the same airspeed regardless of the wind velocity. The speed at which it actually covers the ground is the airspeed plus or minus the wind velocity assuming a direct tailwind or headwind.
8. In his motorboat, Don Stream can go downstream in 1 hour less time than he can go upstream the same distance. If the current is 5 mph, how fast can Don travel in still water if it takes him 2 hours to travel upstream the given distance?
1. The times are 2 hours upstream and 1 hour downstream.
2. The rates are unknown. If you let x equal his rate of traveling in still water, his rate upstream will be x - 5 and his rate downstream will be x + 5. You subtract or add the rate of the current.
3. The distance upstream equals the distance downstream.
Let x = rate of boat in mph in still water
x - 5 = rate of boat in mph upstream
x + 5 = rate of boat in mph downstream
|Upstream||2||x - 5||2(x - 5)|
|Downstream||1||x + 5||1(x + 5)|
2(x - 5) = 1(x + 5)
2x - 10 = x + 5
x = 15 mph (rate in still water)
9. Kay Oss leaves Seattle for New York in her car, averaging 80 mph across open country. One hour later a plane leaves Seattle for New York following the same route and flying 400 mph. How long will it be before the plane overtakes the car?
Let x = time in hours for plane to travel same distance as car.
|Car||x + 1||80||80(x + 1)|
The plane leaves 1 hour later, so the car travels 1 hour longer.
80 (x + 1) = 400x
80x + 80 = 400
-320x = -80
x = 80/320
x = 1/4 hour
80(1 + 1/4) =
80 (5/4) = 400
100 = 100
Let x = time in
hours car travels before plane overtakes it.
|Plane||x - 1||400||400(x - 1)|
The plane takes 1
hour less time than the car.
80x = 400(x + 1)
80x = 400x – 400
-320x = -400
x = 5/4 hours
x – 1 = 1/4 hour
10. A train averaging 50 mph leaves San Francisco at I P.M. for Los Angeles 440 miles away. At the same time a second train leaves Los Angeles headed for San Francisco on the same track and traveling at an average rate of 60 mph. At what times does the accident occur?
Let x = time in hours for trains to meet.
The train leaves at the same time and arrive at the meeting point at the same time, so the times are the same.
Distance traveled for the first train +
distance traveled for the second train = total distance.
50x + 60x = 440
x = 4 hours
Since the trains left at 1
they meet 4 hours later at 5 p.m.
50(4) + 60(4) = 440
11. Doug Upp rides his bike at 6 mph to the bus station. He then rides the bus to work, averaging 30 mph. If he spends 20 minutes less time on the bus than on the bike, and the distance from his house to work is 26 miles, what is the distance from his house to the bus station?
Let x = time in
hours from home to bus station
|Bus||x - 1/3||30||30(x - 1/3)|
Twenty minutes is 1/3 hour.
6x + 30 (x –1/3)
6x + 30x - 10 = 26
x = 1 hour
Since the problem
asked the distance from the house to bus station, we multiply 6 times x,
6x = 6 miles
6(1) + 30(1 – 1/3) = 26
6 + 20 = 26
12. Two planes leave Kansas City at I P.M. Plane A heads east at 450 mph and plane B heads due west at 600 mph. How long will it be before the planes are 2100 miles apart?
x = time in hours each plane travels
+ 600x = 2100
x = 2 hours
13. Doomtown is 200 miles due west of Sagebrush, and Joshua is due west of Doomtown. At 9 A.M. Mr. White leaves Sagebrush for Joshua. At I P.M. Mr. Earp leaves Doomtown for Joshua. If Mr. Earp travels at an average speed 20 mph faster than Mr. White and they each reach Joshua at 4 P.M., how fast is each traveling?
Let x = Mr. White's speed in mph
x + 20 = Mr. Earp's speed in mph
|Earp||3||x + 20||3(x + 20)|
Mr. White travels 200 miles farther than Mr. Earp.
7x = 3(x + 20) + 200
7x = 3x + 60 + 200
4x = 260
x = 65 mph (Mr. White's speed)
x + 20 = 85 mph (Mr. Earp's speed)
7(65) = 3(35) + 200
455 = 255 + 200
455 = 455
14. Meg A. Bucks left Rome at 8 A.M. and drove her Ferrari at 80 mph from Rome to Sorrento. She then took the boat to Capri for the day, returning to Sorrento 5 hours later. On the return trip from Sorrento to Rome she averaged 60 mph and arrived in Rome at 8 P.M. How far is it from Rome to Sorrento?
Let x = time in hours from Rome to
|Return||7 - x||60||60(7 - x)|
The traveling time
is the total time (12 hours) less 5 hours out for Capri, or 7 hours. The time on
the return trip is the total traveling time minus the time on the trip to
80x = 60( 7 - x)
80x = 420 - 60x
x = 3 hours
Since the problem asked for the distance from Rome to Sorrento, we
multiply 80 times x,
80x = 80(3) = 240 miles
80(3) = 60(7 - 3)
240 = 60(4)
15. Superman flies to San Francisco from Santa Barbara in 3 hours. He flies back in 2 hours. If the wind was blowing from the north at a velocity of 40 mph going, but changed to 20 mph from the north returning, what was the airspeed of superman (his speed in still air)?
Let x = speed of plane still air (airspeed)
40 = velocity of wind from north (headwind) going
x - 40 = ground speed of plane going to San Francisco (SF) against wind
20 = velocity of wind from north (tailwind) returning
x + 20 = ground speed returning to Santa Barbara (SB)
|SB to SF||3||x - 40||3(x - 40)|
|SF to SB||2||x + 20||2(x + 20)|
Distance going equals distance returning.
3(x - 40) = 2(x = 20)
3x - 120 = 2x + 40
x = 160 mph (airspeed) answer
16. Phil T. Rich leaves home for Fresno 400 miles away. After 2 hours, he has to reduce his speed by 20 mph due to rain. If he takes I hour for lunch and gas and reaches Fresno 9 hours after he left home, what was his initial speed?
Let x = Phil T. Rich's original speed in mph.
|Second||6||x - 20||6(x - 20)|
First distance + second distance = total distance.
2x + 6(x - 20) = 400
8x - 120 = 400
8x = 520
x = 65 mph
17. A highway patrolman spots a speeding car. He clocks it at 70 mph and takes off after it 0.5 mile behind. If the patrolman travels at an average rate of 90 mph, how long will it be before he overtakes the car?
Let x = time in hours for patrol car to overtake car.
The patrol car travels 0.5 mile
farther than the other car.
90x = 70x + 0.5
20x = 0.5
Multiply by 10 to clear the
200x = 5
x = 1/40 of an hour
If you wish, change the 1/40
hour to minutes, 1/40 x 60 = 1 1/2 minutes.
90(1/40) = 70(1/40) + 1/2
90/40 = 70/40 + 20/40
90/40 = 90/40
18. Boy Scouts hiking in the mountains divide into two groups to hike around Lake Sahara. They leave the same place at 10 A.M., and one group of younger boys hikes east around the lake and the older group hikes west. If the younger boys hike at a rate of 3 mph and the older boys hike at a rate of 5 mph, how long before they meet on the other side of the lake if the trail around the lake is 8 miles long? At what time do they meet?
Let x = time boys take to meet.
The distance the younger Scouts traveled plus the distance the
older Scouts traveled equals the total
distance around the lake.
3x + 5x = 8
8x = 8
x = 1 hour
Since the problem also asks what time they meet,
a.m. + 1 hour = 11 a.m.
3x + 5x = 8
3(1) + 5(1) = 8
3 + 5 = 8
19. A boat has a speed of I 5 mph in still water. It travels downstream from Greentown to Bluetown in two-fifths of an hour. It then goes back upstream from Bluetown to Redtown, which is 2 miles downstream from Greentown, in three-fifths of an hour. Find the rate of the current.
Let x = rate of current in mph
15 + x = rate of boat downstream in mph
15 - x = rate of boat upstream in mph
|Downstream||2/5||15 + x||2/5(15 + x)|
|Upsteam||3/5||15 - x||3/5(15 - x)|
The distance from Greentown to Bluetown is equal to the distance from Bluetown to Redtown plus 2 miles.
2/5(15 + x) = 3/5(15 - x) + 2
(30 + 2x)/5 = (45 - 3x)/5 + 2
Multiply each term by the LCD, 5, to clear fractions.
30 + 2x = 45 - 3x + 5(2)
30 + 2x = 45 - 3x + 10
5x = 25
x = 5 mph
20. Calvin can run a mile in 6 minutes. Hobbes can run a mile in 8 minutes. If Hobbes goes out to practice and starts I minute before Calvin starts on his practice run, will Calvin catch up with Hobbes before he reaches the mile marker? (Assume a straight track and average speed.)
Convert minutes into fractions of an hour first.
6 minutes =6/60 or 1/10 hour to run 1 mile.
8 minutes =8/60 or 2/15 hour to run 1 mile.
Convert the time to rum 1 mile into miles per hour by multiplying it by a quantity which turns the time into 1 hour. This quantity is then equivalent to the miles run in 1 hour, or miles per hour.
1/10 hour x 10 = 1 hour 10 x 1 mile = 10 mph
2/15 hour x 15/2 = 1 hour 15/2 x 1 mile = 7 1/2 mph
Thus Hobbes’s rate was 7 ½ mph and Calvin’s rate was 10 mph. Let x = time in hours for Calvin to catch Hobbes.
|Hobbes||x + 1/60||15/2||15/2(x + 1/60)|
Hobbes takes 1 minute more than Calvin. One minute is 1/60 of an hour. Distances are equal and unknown in this problem, and we can use this equality for our equation.
10x = 15/2 (x + 1/60)
10x = 15x/2 + 1/8
80x = 60x + 1
x = 1/20 hour
Therefore it takes Calvin 1/20 hour
to catch up. Thus, Calvin will catch up with Hobbes before he reaches the mile
10x = 15/2 (x + 1/60)
10(1/20) = 15/2 (1/20+ 1/60)
1/2 = 15/2 (3/60 + 1/60)
1/2 = 15/2 (4/60)
1/2 = 1/2
21. Rhoda Davidson rides her bike to the bus station where she barely makes it in time to catch the bus to work. She spends half an hour on her bike and two-thirds of an hour on the bus. If the bus travels 39 mph faster than she travels on her bike, and the total distance from home to work is 40 miles, find the rate of the bike and the rate of the bus.
Let x = rate of bicycle
x + 39 = rate of bus in mph.
The two distances are equal.
|Bus||x + 39||x + 39||2/3 (x + 39)|
Write 1/2 x as x/2 in your equation.
Distance on the bike plus the distance on the bus equals the total distance.
x/2 + 2(x + 39)/3 = 40
x/2 + (2x + 78)/3 = 40
Multiply by the LCD of 6, to clear fractions.
6 (x/2) + 6[(2x + 78)/3]= 6(40)
3x + 4x + 156 = 240
7x = 84
x = 12 mph
x + 39 = 51mph
12/2 + [2(12 + 39)]/3 = 40
6 + 102/3 = 40
6 + 34 = 40
40 = 40
22. Harry and Kerry started
from the same point at the same time. They traveled in opposite directions on
their bicycles. Harry traveled at the rate of 9 km/h, and Kerry traveled at 11
km/h. After how many hours were they
60 km apart?
23. Two trains leave Trackville
at the same time. One travels north at 90 km/h. The other travels south at 110
km/h. After how many hours will the trains be 900 km apart?
24. Two steamships sailing in
opposite directions pass each other. One ship is sailing at 32 knots (nautical
miles per hour). The other ship is sailing at 28 knots After how many hours will
the ships be 1 50 nautical miles apart?
25. Two jets are traveling
toward each other and are 3400 km apart. One jet is flying at 875 km/h. The
other jet is flying at 825 km/h. In how many hours will the jets pass each
26. A train left Podunk and
traveled west at 70 km/h. Two hours later, another train left Podunk and
traveled east at 90 km/h. How many hours had the first train traveled when they
were 1420 km apart?
27. A train left Podunk and
traveled north at 75 km/h. Two hours later, another train left Podunk and
traveled in the same direction at 100 km/h. How many hours had the
first train traveled when the second train overtook it?
28. Joe Spout left a campsite
on a trip down the river in a canoe, traveling at 6 km/h. Four hours later,
Joe’s lather set out after him in a motorboat. The motorboat traveled at 30
km/h. How long after Joe’s father started did he overtake the canoe?
In the question above, how far
had Joe traveled down the river when his father overtook him?
29. Two trucks left Bucks
Trucks traveling in opposite directions. One truck traveled at a rate of 70
km/h. the other at 80 km/h. After how many hours were the trucks 900 km apart?
30. A truck left Hucks Trucks
and traveled north at 80 km ft One hour later, another truck left Hucks Trucks
and traveled south at 60 km/h. How many hours had the first truck traveled when
they were 150 km apart?
31. Steve McSpoke left home on
his bicycle at 8:00 A.M., traveling at 18 km h. At 10:00 A.M., Steves brother
set out after him on a motorcycle, following the same route. The motorcycle
traveled at 54 km/h. How long had Steve traveled when his brother overtook him?
In Exercise 31, how far had
Steve traveled when his brother overtook him?
32. Dr. Pepper left Oakville at
9:00 A.M. and drove to Central City at 60 km/h. H. Salt left Oakville at 11:00
A.M. and traveled the same route to Central City. If both men arrived in Central
City at 4:00 P.M., at what rate did H. Salt travel?
33. Two jets are traveling
toward each other and are 4000 km apart. The rate of one jet is 1 00 km h faster
than the rate of the other. If the jets pass each other after 2.5 hours, what is
the rate of the faster jet?
34. Ms. Driva Reck drove from
her home to a service station at 48 km h. She returned home by bicycle at 16 km
h. The entire trip took 4 hours. How far was the service station from Ms. Recks
35. A plane on a search mission
flew east from an airport, turned, and flew west back to the airport. The plane
cruised at 300 km/h when flying east, and 400 km/h when flying west. The plane
was in the air for 7 hours. How far from the airport did the plane travel?
36. A motorboat can travel
upstream on a river at 18 km/h and downstream at 30 km h. How far upstream can
the boat travel if it leaves at 8:00 A.M. and must return by noon?
37. A boat travels 60 km
upstream (against the current) in 5 hours. The boat travels the same distance
downstream in 3 hours. What is the rate of the boat in still water? What is the
rate of the current?
38. When a plane flies into the
wind, it can travel 3000 km in 6 hours. When it flies with the wind, it can
travel the same distance in 5 hours. Find the rate of the plane in still air and
the rate of the wind.
39. When Lucy swims with the
current, she swims 18km in 2 hours. Against the current, she can swim only 14 km
in the same time. How fast can Lucy swim in still water? What is the rate of the
40. With the wind, a jet can
fly 2500 km in 2 h 30 mm. Against the wind, it can fly only 2000 km in the same
time. Find the rate of the jet in still air and the rate of the wind.
41. On an upstream trip, a
canoe travels 40 km in 5 hours. Downstream, it travels the same distance in half
the time. What is the rate of the canoe in still water and the rate of the
42. A duck can fly 2400 m in 10
mm with the wind. Against the wind, t can fly only two thirds of this distance
in 10 mm. How fast could the duck fly in still air? What is the rate of the
43. With the wind, a plane flew
1400 km in 4 hours. On the return trip, the pilot was forced to land after 1
h 30 mm, having traveled only 450 km. Find the rate of the plane in still
air and the rate of the wind.
44. A salmon swims 100 m in 8
min downstream. Upstream, it would take the fish 20 min to swim the same
distance. What is the rate of the salmon in still water? What is the rate of the