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Algebra 1 Questions

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?
 
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.
 
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?
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4 Answers

1) and 2) are done so I'll do 3).
 
3) d=r*t
    d=55*t
    d=70*(t-0.5)
55t=70(t-0.5)
55t=70t-35
15t=35
t=35/15=2 1/3 hours at 55mph
d=55*(2 1/3)=128 1/3 miles from the station
t-(1/2)=2 1/3- 1/2=1 5/6 hours to overtake the other train
d=70*(1 5/6)=128 1/3 miles as a check
 
These all fall into the category of rate x time = distance problems. If you set up a chart and fill in what you know (the givens) about each driver, airplane, train etc. it will help organize the information. The key in each problem is to determine which variable (distance, time, rate) is the same for both items. That determines how you write your equation to solve for the unknowns.
 
Problem 1
 
                           Rate         x       Time     =   Distance
Casey                 25 m/h                t           =         d
Marshall              50 m/h                t - 0.25          =         d
 
NOTE: the time is in hours to be consistent with the rate which is in hours
 
You are solving for the time t, and you know the distances are the same. Therefor: distance = distance
25(t) = 50(t-0.25)
Solve using distributive property and properties of equality
25t= 50t - 12.5
-25t = -12.5
t = 0.5 hours
 
Plug t back into either of the original equations and d will equal 12.5 miles
___________________________________________________________________
 
Question 2
Let c be the Chicago train speed rate 
 
                        rate    x     time    =    distance to Pittsburgh
Chicago Train       c      x     t          =      470 mi
Phila Train        (c-40)   x    t         =       350 mi
 
This time you are solving for c and you know that time is the same so:  Chicago train t  =  Phila train t
 470/c  =  350/(c-40)  Solve as a proportion ( easiest way is to use cross-products)
 
350c = 470c - 18800
 
c= 156.7 miles per hour (Chicago train speed rounded to nearest tenth)
 
Plug 156.7 back into either of the original equations to get 
 
Philly train speed = 116.7 miles per hour (rounded to the nearest tenth)
 
Out of time - If I get a chance will update later with the third problem.
 
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.
 
 
 
 
 
 
 
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?

Dc = 25*(Tc-10)
Dm = 2*25*(Tm-10.25)

When Tm = Tc := T, then Dc = Dm.

25*(T-10) = 2*25*(T-10.25)

T-10 = 2T-20.5

Add -T+20.5 to both sides:

10.5 = T = 10:30 am

Dc = 25(10.5-10) = 12.5 miles

Dm = 50(10.5-10.25) = 12.5 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.

Cd = Cr * Ct; Pd = Pr * Pt

Cr = Pr + 40 mph

Ct = Pt := T

Cd = 470 miles, Pd = 350 miles

Ct = Cd/Cr = 470/(Pr+40) = Pt = Pd/Pr = 350/Pr

470/(Pr+40) = 350/Pr

Multiply by Pr(Pr+40)

470 Pr = 350 (Pr + 40)

47 Pr = 35 Pr + 35 * 40

12 Pr = 350 * 4

Pr = 350/3 mph

Cr = Pr + 40 = 470/3 mph

check:

Ct = Cd/Cr = 470/(470/3) = 3 hrs

Pt = Pd/Pr = 350/(350/3) = 3 hrs

Ct = Pt √


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?

Nd = 55 * (Nt-7), Ed = 70 * (Et-7.5)

When Nd = Ed, Nt = Et := T.

55 * (T-7) = 70 * (T-7.5)

55 T - 55*7 = 70 T - 70*7.5

Add -55T + 70*7.5 to both sides:

70*7.5 - 55*7 = 15 T

Divide by 5 both sides:

70*3/2 - 11*7 = 3 T

3 T = 35*3 - 77 = 105 - 77 = 28

T = 28/3 = 9:20

Nd = 55(28/3-21/3) = 55*7/3 = 5*7*11/3

Ed = 70(28/3-15/2) = 70(56/6-45/6) = 70*11/6 = 35*11/3 = 5*7*11/3

Nd = Ed √

385/3 = 128 + 1/3 miles from the station.