Rational ExpressionsRational ExpressionsChapter 10Chapter 10

Rational ExpressionsRational ExpressionsChapter 10Chapter 10

Distance – Rate - TimeDistance – Rate - Time

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Motion Problems Motion Problems

Motion problems involve distance, time and rate. The equation that links these concepts is called

The Distance Formula:

d = r td = distance

t = time

r = rate

• milesmiles• kilometers kilometers • metersmeters• feetfeet• inchesinches

• hourshours• minutesminutes• secondsseconds• daysdays• yearsyears

• miles/hourmiles/hour• km/min. km/min. • m/sm/s• ft./sec.ft./sec.• inches/sec.inches/sec.

Motion Problems Motion Problems

Motion problems involve distance, time and rate. The equation that links these concepts is called

The Distance Formula:

d = r td = distance

t = timer = rate

You’ll need to be comfortablewith using and manipulating

the distance formula.

• miles/hourmiles/hour• km/min. km/min. • m/sm/s• ft./sec.ft./sec.• inches/sec.inches/sec.

d

tr

d

tr

Rational ExpressionsRational ExpressionsChapter 10Chapter 10

Rational ExpressionsRational ExpressionsChapter 10Chapter 10

Distance – Rate - TimeDistance – Rate - Time

dt

r

dt

r

Here is a strategy to solve motion problems Here is a strategy to solve motion problems

1. Read the problem (three times) picking out key information1. Read the problem (three times) picking out key information

2. Draw a diagram of what is happening.2. Draw a diagram of what is happening.

3. Make a table that relates 3. Make a table that relates distance, rate and time.distance, rate and time.

5. Answer the question – include the units!5. Answer the question – include the units! Distance = rate x time

4. Use the table to write an equation.4. Use the table to write an equation.

Distance = rate x timeDistance = rate x time

BikerBiker

WalkerWalker

54 miles54 miles

18 miles18 miles

x x = speed of walker= speed of walker x + 8 x + 8 = speed of cyclist= speed of cyclist

5454

1818

Example 1Example 1Example 1Example 1

A bicyclist travels 8 miles per hour faster than a walker. The cyclist travels 54 miles in the same time it takes the walker to walk 18 miles. Find their speeds.

A bicyclist travels 8 miles per hour faster than a walker. The cyclist travels 54 miles in the same time it takes the walker to walk 18 miles. Find their speeds.

18x54

x 8x 8

x

Example 1Example 1Example 1Example 1

TimeWalker

TimeBiker

18x

54x 8

Distance = rate x timeDistance = rate x time

BikerBiker

WalkerWalker

5454

181818x54

x 8x 8

x

A bicyclist travels 8 miles per hour faster than a walker. The cyclist travels 54 miles in the same time it takes the walker to walk 18 miles. Find their speeds.

A bicyclist travels 8 miles per hour faster than a walker. The cyclist travels 54 miles in the same time it takes the walker to walk 18 miles. Find their speeds.

Example 1Example 1Example 1Example 1

18 x 8 54x

18x 144 54x 144 36x

4 x

Walker 4 m.p.h.

Cyclist 12 m.p.h.

Walker 4 m.p.h.

Cyclist 12 m.p.h.

18x

54x 8

A bicyclist travels 8 miles per hour faster than a walker. The cyclist travels 54 miles in the same time it takes the walker to walk 18 miles. Find their speeds.

A bicyclist travels 8 miles per hour faster than a walker. The cyclist travels 54 miles in the same time it takes the walker to walk 18 miles. Find their speeds.

# 2# 2# 2# 2 One car travels 20 mph faster than the other One car travels 20 mph faster than the other car. While the faster car goes 240 miles the car. While the faster car goes 240 miles the other car travels 180 miles. Find their speeds.other car travels 180 miles. Find their speeds.

One car travels 20 mph faster than the other One car travels 20 mph faster than the other car. While the faster car goes 240 miles the car. While the faster car goes 240 miles the other car travels 180 miles. Find their speeds.other car travels 180 miles. Find their speeds.

Distance = rate x time

240

x + 20

x180

240

x = speed of red (slower) car

180

180x

240x 20

Slow carSlow car

Fast carFast car

# 2# 2# 2# 2 One car travels 20 mph faster than the other One car travels 20 mph faster than the other car. While the faster car goes 240 miles the car. While the faster car goes 240 miles the other car travels 180 miles. Find their speeds.other car travels 180 miles. Find their speeds.

One car travels 20 mph faster than the other One car travels 20 mph faster than the other car. While the faster car goes 240 miles the car. While the faster car goes 240 miles the other car travels 180 miles. Find their speeds.other car travels 180 miles. Find their speeds.

Distance = rate x time

x + 20

x180

240

x = speed of red (slower) car

180x

240x 20

TimeSlowCar

TimeFastCar

180x

240x 20

Slow carSlow car

Fast carFast car

180x

240x 20

# 2# 2# 2# 2 One car travels 20 mph faster than the other One car travels 20 mph faster than the other car. While the faster car goes 240 miles the car. While the faster car goes 240 miles the other car travels 180 miles. Find their speeds.other car travels 180 miles. Find their speeds.

One car travels 20 mph faster than the other One car travels 20 mph faster than the other car. While the faster car goes 240 miles the car. While the faster car goes 240 miles the other car travels 180 miles. Find their speeds.other car travels 180 miles. Find their speeds.

Slow car 60 m.p.h.

Fast car 80 m.p.h.

Slow car 60 m.p.h.

Fast car 80 m.p.h.

180 x 20 240x

180x 3600 240x

3600 60x

60 x

An airplane can fly at a speed of 240 m.p.h. in calm air.An airplane can fly at a speed of 240 m.p.h. in calm air.It takes the same time to fly 1080 miles It takes the same time to fly 1080 miles withwith the wind the windas it does 840 miles as it does 840 miles againstagainst the wind. Find the speed the wind. Find the speed of the wind.of the wind.

An airplane can fly at a speed of 240 m.p.h. in calm air.An airplane can fly at a speed of 240 m.p.h. in calm air.It takes the same time to fly 1080 miles It takes the same time to fly 1080 miles withwith the wind the windas it does 840 miles as it does 840 miles againstagainst the wind. Find the speed the wind. Find the speed of the wind.of the wind.

Example 3Example 3Example 3Example 3

Wind

speed in “calm air” = 240speed in “calm air” = 240

w = ?w = ?

d = 1080 miles

d = 840 miles

An airplane can fly at a speed of 240 m.p.h. in calm air.An airplane can fly at a speed of 240 m.p.h. in calm air.It takes the same time to fly 1080 miles It takes the same time to fly 1080 miles withwith the wind the windas it does 840 miles as it does 840 miles againstagainst the wind. Find the speed the wind. Find the speed of the wind.of the wind.

An airplane can fly at a speed of 240 m.p.h. in calm air.An airplane can fly at a speed of 240 m.p.h. in calm air.It takes the same time to fly 1080 miles It takes the same time to fly 1080 miles withwith the wind the windas it does 840 miles as it does 840 miles againstagainst the wind. Find the speed the wind. Find the speed of the wind.of the wind.

Example 3Example 3Example 3Example 3

Distance = rate x timeDistance = rate x time

With

Against

1080

840

240 + w

240 - w

Let w = speed of windLet w = speed of wind

1080 840240 w 240 w

Timewithwithwind

Timeagainstagainstwind

1080240 w

840240 w

w 30w 30

The speed of a freight train is 14 km/h slower than the speed of a The speed of a freight train is 14 km/h slower than the speed of a passenger train. The freight train travels 330 km in the same time passenger train. The freight train travels 330 km in the same time that it takes the passenger train to travel 400 km. Find the speed of that it takes the passenger train to travel 400 km. Find the speed of each train.each train.

The speed of a freight train is 14 km/h slower than the speed of a The speed of a freight train is 14 km/h slower than the speed of a passenger train. The freight train travels 330 km in the same time passenger train. The freight train travels 330 km in the same time that it takes the passenger train to travel 400 km. Find the speed of that it takes the passenger train to travel 400 km. Find the speed of each train.each train.

One car travels 40 km/h faster than another. While one travels 150 One car travels 40 km/h faster than another. While one travels 150 km, the other goes 350 km. Find their speeds.km, the other goes 350 km. Find their speeds.

One car travels 40 km/h faster than another. While one travels 150 One car travels 40 km/h faster than another. While one travels 150 km, the other goes 350 km. Find their speeds.km, the other goes 350 km. Find their speeds.

A lab tested two high-speed trains. One travels 40 km/h faster than A lab tested two high-speed trains. One travels 40 km/h faster than the other train. While one train travels 70 km, the other travels 60 the other train. While one train travels 70 km, the other travels 60 km. Find their speeds. km. Find their speeds.

A lab tested two high-speed trains. One travels 40 km/h faster than A lab tested two high-speed trains. One travels 40 km/h faster than the other train. While one train travels 70 km, the other travels 60 the other train. While one train travels 70 km, the other travels 60 km. Find their speeds. km. Find their speeds.

A person traveled 120 miles in one direction. The return trip was A person traveled 120 miles in one direction. The return trip was accomplished at double the speed and took 3 hours less time. Find accomplished at double the speed and took 3 hours less time. Find the speed goingthe speed going

A person traveled 120 miles in one direction. The return trip was A person traveled 120 miles in one direction. The return trip was accomplished at double the speed and took 3 hours less time. Find accomplished at double the speed and took 3 hours less time. Find the speed goingthe speed going

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