SOLUTION: A motorist travels a distance of 84km. He finds that; if on the return journey he increases his average speed by 4km/h, he will take half an hour less. What was his average speed

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Question 1144577: A motorist travels a distance of 84km. He finds that; if on the return journey he increases his average speed by 4km/h, he will take half an hour less.
What was his average speed for the first part of the journey?
How long did he take for the double journey?
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
              SPEED         TIME        DISTANCE

firstpart     r             84/r         84

secondpart    r+4          84/(r+4)      84

difference                 1/2

84%2Fr-84%2F%28r%2B4%29=1%2F2
.
.
r%5E2%2B4r-672=0
.
.
highlight%28r=24%29

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
basic formuls ia r * t = d
r is the rate
t is the time
d is the distance

if d = 84 kilometers, then the formula becomes r * t = 84
that would be the formula going.
on the return, the speed is increased by 4 kilometers per hour and the time is half an hour less.
the formula for the return journey becomes:
(r + 4) * (t - .5) = 84
you have 2 equations that need to be solved simultaneously.
they are:
r * t = 84
(r + 4) * (t - .5) = 84
in the first equation, solve for r to get:
r = 84/t
in the second equation, replace r with that to get:
(84/t + 4) * (t - .5) = 84
multiply both sides of that equation by t to get:
(84 + 4 * t) * (t - .5) = 84 * t
simplify by completing the indicated multiplication to get:
84 * (t - .5) + 4 * t * (t - .5) = 84 * t
simplify further to get:
84 * t - 42 + 4 * t^2 - 2 * t = 84 * t
subtract 84 * t from both sides of the equation to get:
84 * t - 84 * t - 42 + 4 * t^2 - 2 * t = 0
combine like terms and order in descending order of degree to get:
4 * t^2 - 2 * t - 42 = 0
factor this quadratic equation to get:
t = 3.5 or t = -3
t has to be positive so t = 3.5.

in the first equation, solve for r to get r = 84 / 3.5 = 24.
first equation of r*t = 84 becomes 3.5 * 24 = 84 which becomes 84 = 84 which is true.

in the second equation, replace r with 24 and t with 3.5 to get:
(r + 4) * (t - .5) = 84 becomes 28 * 3 = 84 which becomes 84 = 84 which is true.

r = 24 and t = 3.5 is confirmed to be good.
the average speed for the first part of the journey is 24 kilometers per hour.
the journey there took 3.5 hours and the journey back took 3 hours for a total of 6.5 hours.