Linear Equations

Question

On a particular day. Peter has to leave home at a certain time and travel to town P for a meeting at 10 AM. If his average speed is 54 km/hr, he will be 12 minutes late. If his average speed is 72 km/hr he will be 18 minutes early. Find his average speed if he reaches town P exactly at 10 AM.

Answer 1

Sorry, my original answer confused minutes and hours, yielding meaningless answers. Andy C. calculated it correctly on another copy of this same problem. Here is my improved version. First identify the quantities involved. Since these are constant motion problems, they involve the equation Distance = Rate * Time D = distance from home to town T = travel time to arrive exactly on time Rate1 = 54 Time1 = t + 12/60 = t+1/5 Rate 2 = 72 Time 2 = t - 18/60 = t-3/10 Rate3 = r Time 3 = t So the equations are D = 54(t+1/5) D = 72(t-3/10) D = r * t Our goal is to discover r, but in order to do that, we need to discover D and t first. Solve for r, dividing by t D/t = r Now, once we solve for t and d, we just put them into R = d/t I assume you know how to solve D = 54(t+1/5) D = 72(t-3/10) But I’ll go ahead and finish. For one thing, since the distances are equal, dispose of them for a moment 54(t+1/5) = 72(t-3/10) Multiply everything out 54t + 54/5 = 72t - 216/10 Collect terms 54/5 + 216/10 = 18t (108+216)/10 = 18t 324/10 = 18t Divide by 18 18/10 = t Plug this t back in one of the d= equations D = 72(t - 3/10) D = 72(18/10-3/10) D= 72(15/10) = 108 R = 108/(18/10) = 1080/18 = 60

Answer 2

 

distance/rate = time

Let R be the ideal rate of speed so that he arrives on time

12 minutes = 12/60 hr = 1/5 hr
18 minutes = 18/60 hr = 3/10 hr

d/54 = T + 1/5 <--- late
So T = d/54 - 1/5

D/72 = T - 3/10 <--- early
So T = d/72 + 1/5

d/54 - 1/5 = d/72 + 3/10 >--- same times, sets them equal

d/54 - d/72 = 3/10 + 1/5

4320 is a common denominator; multiplies and distributes
4320 across entire equation

80d - 60d = 1296 + 864

20 d = 2160

d = 108 km is the distance

At 54 mph, he gets there in 2 hours, which is 12 minutes late.
So it should only take him 1 hour and 48 minutes, which as
a fraction is 1 and 48/60 hours = 1 and 4/5 hours = 9/5 hours

108 = (9/5) x R

R = 108 x 5 / 9 = 60.

He needs to go 60 km/hr

CHECK:

108 / 54 = 2 hours

108/60 = 54/30 = 27/15 hours = 1 hour 48 minutes <-- 12 mins difference

108/72 = 54/36 = 27/18 = 3/2 = 1 hour 30 minutes <-- 18 minutes difference