a motorist drives to a city in 4 hrs. and returns by another route which is 26 mi. longer. on the return trip he traveled 5 miles per hour faster and took 4 hours and 12 min. . find the length of the shorter route.

let d be the shorter distance.

d+26=(r+5)4.2 (4.2 hrs is 4hrs 12min)
d=r*4

two equations, two unknowns, solve for d.

length of shorter trip ---- x miles

speed for shorter trip ---- 4/x mph , since speed = distance/time


length of longer trip --- x+26
speed for longer trip --- 21/(5(x+26)) .....explanation : ( 21/5 hrs = 4hr. 12 min )

21/(5(x+26)) - 4/x = 5 , ... the difference is their speeds is 5 mph

multiply by 5x(x+26)

21x - 20(x+26) = 25x(x+26)
21x - 20x - 520 = 25x^2 + 650x
25x^2 + 649x + 520=0
x = (-649 ± 607.619)/50

this gave me two negative answers, but x has to be > 0
Unless I made some silly arithmetic error, this problem has no solution.

Never mind my solution, go with bobpursley

Found my "silly error" , just noticed I had my fractions upside - down. e.g. speed for shorter trip = x/4 , etc.
Can I blame it on the heat affecting my brain ?

could bob pursley finish the problem ? i still don't have the answer or could you, reiny, do it again without the error. i need to see the problem worked, thanks.

Yes, I could finish it, however, you don't want anyone thinking you are an answer grazer. You do the work, and I will critique it.

mr. pursley, Have you ever worked geo. & trig. etc with the heat on wheather you blew up the town or the target or the building collapased or was built in the right place. i'm not an answer graser. you don't know how long i worked on this problem. i'm trying to learn some different types of math now for fun. if you choose to not assist, please don't.

To find the length of the shorter route, let's first set up some equations based on the given information:

Let's call the length of the shorter route "x" miles.

We know that on the outbound trip, the motorist traveled at a certain speed for 4 hours. Since speed = distance / time, we can write the equation as:

x = (speed on outbound trip) * 4

On the return trip, the motorist traveled 26 miles longer, which means the distance becomes (x + 26) miles. The motorist also traveled 5 miles per hour faster on the return trip, which we'll call "speed on return trip + 5". Using the formula for speed, we can write the equation:

(x + 26) = (speed on return trip + 5) * (4 + 12/60) (convert 12 minutes to hours)

Simplifying the equation, we get:

(x + 26) = (speed on return trip + 5) * 4.2 (simplify 4 + 12/60 to 4.2)

Now, we have a system of two equations:

x = (speed on outbound trip) * 4
(x + 26) = (speed on return trip + 5) * 4.2

We need to solve this system to find the value of x.

To solve the system, we can make an assumption that the speed on the outbound trip is equal to the speed on the return trip.

Let's denote the speed as "s". So, we have:

x = s * 4
(x + 26) = (s + 5) * 4.2

We can substitute the first equation into the second equation:

s * 4 + 26 = (s + 5) * 4.2

Expand the equation:

4s + 26 = 4.2s + 21

Simplify:

4s - 4.2s = 21 - 26
-0.2s = -5

Divide both sides by -0.2:

s = -5 / (-0.2) = 25

Now, substitute the value of s back into the first equation:

x = 25 * 4
x = 100

Therefore, the length of the shorter route is 100 miles.