SOLUTION: A car stopped at an railway crossing for 16 minutes. To get to the location 80 km away on time, it had to speed up an additional 10 km/h. Calculate the planned speed.

Question 971499: A car stopped at an railway crossing for 16 minutes. To get to the location 80 km away on time, it had to speed up an additional 10 km/h. Calculate the planned speed.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the planned speed is r.
the planned time is t hours.
rate * time = distance, so r*t = d
d is equal to 80, so r*t = 80
that's the planned rate and the planned time.

since he was stopped at the railway for 16 minutes, then he had 16 minutes less to reach the desintation at the planned time.

16 minutes is equal to 16/60 hours which is equal to 4/15 hours.

his new time is t - 4/15 hours.
his new rate is r + 10 kilometers per hour.

he still has to travel the same distance, so the new formula is:

(r+10) * (t-4/15) = 80

you have two equations that need to be solved simultaneously.

they are:

r*t = 80 and (r+10) * (t-16) = 80

solve for r in r*t = 80 to get r = 80/t

replace r with 80/t in (r+10) * (t-4/15) = 80 to get:

(80/t + 10) * (t-4/15) = 80.

multiply both sides of the equation by 15*t to get:

15*t * (80/t + 10) * (t-4/15) = 80 * 15*t

re-associate the terms to get:

t*(80/t+10) * 15*(t-4/15) = 80*15*t

simplify to get:

(80+10t) * (15t - 4) = 80*15*t

simplify to get:

1200t - 320 + 150t^2 - 40t = 1200t

the 1200t on the left and right side of the equation cancel out and you are left with -320 + 150t^2 - 40t = 0
commute the terms to get:
150t^2 - 40t - 320 = 0

divide both sides of the equation by 10 and you get:

15t^2 - 4t - 320 = 0

factor the quadratic equation using the quadratic formula and you get:

t = 1.6

multiply this by 15/15 and you get the equivalent fraction of t = 24/15.

solve for r in r*t = 80 which becomes r*24/15 = 80 which becomes r = 15*80/24 which gets you r = 50.

t-4/15 is therefore equal to 20/15 and r+10 is equal to 60.

you have r * t = 80 becomes 50 * 24/15 = 80 which becomes 80 = 80 which is true, and you have (r+10) * (t-4/15) = 80 becomes 60 * 20/15 = 80 which becomes 80 = 80 which is also true.

you can do the math to confirm that both these equations are true.

the planned speed is 50 kilometers per hour.

the delay at the railroad crossing caused the speed to be increased to 60 kilometers per hour because there were 16 less minutes to reach the objective at the same time.

i used fractions to avoid using repeating decimals.

if you use decimals rather than fractions, then store the results temporarily n the calculator so you can work with the unrounded numbers until you get to the end where it is then ok to round.

a key factor was to convert 16 minutes to equivalent hours so all the measurements were consistent with each other.

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