Hi Abaline;

*1. Casey leaves home at 10 a.m. and drives at an average speed of 25 mph. Marshall leaves the same house 15 minutes later, and drives the same route, but twice as fast as Casey.*

*At what time will Marshall pass Casey, and how far will they be from home when he does?*

Casey...t hours...25 miles/hour

Marshall...(t-0.25)hours...50 miles/hour

t hours=time when they pass each other.

Note the fact that we cannot use time in minutes while speed is in miles/hour.

25t=50(t-0.25)

25t=50t-12.5

Subtract 50t from both sides...

25t-50t=50t-12.5-50t

-75t=-12.5

Divide both sides by -75...

(-75t)/-75=-12.5/-75

t=0.17

**They will pass each other 0.17 hours after Marshall leaves.**

(0.17 hours)[(60 minutes)/(1 hour)]=10.2 minutes

Note the fact that the unit of hours is in the numerator and denominator. It cancels.

distance=(rate)(time)

distance=(50 miles/hour)(0.17 hours)

distance=8.5 miles

**They will pass each other in 8.5 miles.**

*2. Pittsburgh is 470 miles from Chicago and 350 miles from Philadelphia. Trains leave Chicago and Philadelphia at the same time, but the Chicago train travels 40 mph faster then the Philadelphia one. Both trains reach Pittsburgh at the same time. Find each train's speed, rounding answers to the nearest tenth.*

distance=(rate)(time)

We have the distance and time. We need the rate.

Chicago-to-Pittsburgh...470 miles....(x+40) miles/hour

Philadelphia-to-Pittsburgh...350 miles...x miles/hour

We can use the formula...

distance/time=distance/time

or...

(470 miles)/(350 miles)=[(x+40) miles/hour]/(x miles/hour)

On the left side, the unit of miles is in the numerator and denominator. It cancels...

On the right side, the unit of miles/hour is in the numerator and denominator. It cancels...

(470 miles)/(350 miles)=[(x+40) miles/hour]/(x
miles/hour)

470/350=(x+40)/x

Cross-multiply...

470x=350(x+40)

470x=350x+14000

Subtract 350x from both sides...

470x-350x=350x-350x+14000

120x=14000

Divide both sides by 120...

(120x)/120=14000/120

x=116.7

x+40=156.7

The speed of the train from Philadelphia is 116.7 miles/hour.

The speed of the train from Chicago is 156.7 miles/hour.

*3. At 7:00 a train leaves a station, traveling at 55 mph. At 7:30 an express train leaves the same station, traveling the same route at 70 mph. How long will it take the express train to overtake the other train, and how far will they be from the station when it does?*

Train...55 miles/hour.....t

Express train...70 miles/hour...(t-0.5)

Please note that the unit of speed is in miles/hour. Therefore, the unit of time is in hours, not minutes.

(t hours)(55 miles/hour)=(70 miles/hour)[(t-0.50) hours]

The units are aligned. The unit of miles/hour is on both sides of the equation. The unit of hours is on both sides of the equation. All cancel...

(t hours)(55 miles/hour)=(70 miles/hour)[(t-0.50)
hours]

55t=70(t-0.50)

After alignment of units, calculations may be made.

55t=70t-35

Subtract 70t from both sides...

55t-70t=70t-35-70t

-15t=-35

Divide both sides by -15...

(-15t)/-15=-35/-15

t=2.3

**The trains will meet in 2.3 hours.**