Distance/time/rate problem (solved but not elegant)?

Question

I'm usually a pro at these. But there is a detail at the end and I'm not sure how to write the equation around it. I already figured out the answer but I want a "prettier" equation.

Truck leaves town A at 2 p.m. heading for town B, traveling at 40 mi/h. At 3:15 car leaves town B heading to town A traveling at 55 mi/h. (Easy) But then it says - the truck and car pass when the car is 2 hours from town A. How far apart are the towns?

40t = 55(t - 1.25) gets me nowhere.

I know 2 hours represents 110 miles left in the car's journey. The answer is 192.5 miles because I just gridded the problem out. Any takers?

Hi Cattbarf - that's exactly how I solved it also. Great minds think alike! But I want to teach this problem tomorrow and need it to look like proper Alg 2 ! Thanks!

Answer 1

You learned an important lesson. Pretty is nice but there are other ways to skin cats.

Lets try this one, slightly more elegant.
So if TruckA starts at 2 PM, he will travel 110 miles when he is passed by Car B going the other way. So what time is it when they meet?
TA DA 110/40= 2-3/4 hours later or 4:45
Since car B starts at 3:15, he has been on the road for 1-1/2 hours when be passes Truck A.
So the distance he travels is 55*1.5=82.5 miles.
The sum of the two trips is the distance from the two towns, or 192.5 miles.

Answer 2

The car is two hours from town A, so at 55 mph it is 110 ten miles from A. The truck has thus traveled 110 miles since it left town A. At 40 mph the truck must have been traveling for 2.75 hours (two hours and 45 minutes) {110 miles divided by 40 mph).

Since the truck left 1 hour and 15 minutes before the car left its town, the car must have been traveling for 2.75 hours minus 1.25 hours, i.e. 1.5 hours. In 1.5 hours at 55 mph, the car has traveled 82.5 miles. So the distance between A and B is 110 miles plus 82.5 miles = 192.5 miles.

Check my math to be sure I got the details right. Best wishes and good luck.

Answer 3

Ok, I'll give it a go.
If the car has 2 hours left, then it has 110 miles left to town A when they pass. That means that the truck has traveled 110 miles from town A.
d=vt, so the time elapsed for the truck is 2.75 h.
The time elapsed for the car is 2.75-1.25 = 1.5 h.
1.5h x 55mi/h = 82.5 mi traveled by the car.
82.5 mi traveled + 110 mi left = 192.5 mi.

It's possible to put that all in one equation, but it's ugly. I like it better stepwise.

{[(2 h x 55mi/h)/(40 mi/h) - 1.25 h] x 55 mi/h} + (2 h x 55 mi/h)

Answer 4

this is a good question
you're almost there take that 110 miles thats how far the truck has gone. how long did that take? at 40mph thats 2.75 hours so that makes the time that they meet 4:45. So then you know the car has been driving for an hour and a half so its gone 82.5 miles. Finally 82.5 + 110 gives you you're 192.5 answer

hope this helped

Answer 5

Sorry, but you're wrong.

After two hours (4:00 pm), the truck is 80 miles from town A. The car left town B at 3:15 pm, so it will only travel 45 min. (3/4 hr.) until it meets with the truck. The car is traveling 55 mph, so in 3/4 hr. it travels 41.25 miles.

The towns then, are 121.25 (80 + 41.25) miles apart