MHB Solve Number Theory Problem to Find Time for Express Bus

AI Thread Summary
The discussion centers around a number theory problem involving two types of buses, Regular and Express, traveling between Station A and Station J. The Regular bus stops at every station and takes a total of 120 minutes, while the Express bus, which departs 40 minutes later, takes 80 minutes to reach the same destination. The calculations involve equating the travel times of both buses, factoring in their speeds and stop durations. The derived equations show that the distance and speed relationship leads to the conclusion that the Express bus takes 80 minutes. The solution emphasizes the importance of correctly accounting for the time differences and bus speeds in the problem.
Marcelo Arevalo
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On a particular bus line, between Station A and Station J, there are 8 other
stations. Two types of buses, Express and Regular, are used. The speed of an
Express bus is 1.2 times that of a Regular bus. Regular buses stop at every
station, while Express buses stop only once. A bus stops for 3 minutes. On a
particular day, a Regular bus departed from Station A. 40 minutes later an
Express bus departed from the same station. The two buses arrived at
Station J at the same time. How long did the Express bus take from Station A
to Station J?

- - - Updated - - -

As refer to answer key
the answer was 80
no idea how did they got it.
 
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The time $t$ (in minutes) traveled by the regular bus (including stops) is:

$$t_R=\frac{d}{v}+24$$

And for the express bus (including the 40 minute delay) is:

$$t_E=\frac{10d}{12v}+40$$

Since they arrived at the last stop at the same time, you can equate the two times ($t_R=t_E$), and then solve for $d$, and then substitute for $d$ into either equation above to find $t$. What do you find?
 
Is this the continuation of the given data above??
d/v + 24 = 10d/12v + 40
12d + 288v = 10d + 480v
2d = 480v -288v
d = 96vfrom 1 : tR = 96v/v + 24 = 120
from 2 : tE = 960v/12v + 40 = 120
 
My solution:

please comment, thank you.

Regular Speed = Y
Express Speed = 1.2Y

Time taken if Regular = x/y + 8(3)/60 (min/hr)
if Express = x/1.2y + 3/60 (min/hr)

Combining the two equation:
we have; x/y + 24/60 = x/1.2y + 3/60 + 40/60
x/y + 24/60 = x/1.2y + 43/60
x/y - x/1.2y = 19/60 multiply both sides by 6y
6x - 5x = 114/60 y
x = 1.9y

Substituting on Express = x/1.2y + 3/60 mins
= 1.9y/1.2y + 3/60
= 98 mins or 1hr 38mins
 
Marcelo Arevalo said:
Is this the continuation of the given data above??
d/v + 24 = 10d/12v + 40
12d + 288v = 10d + 480v
2d = 480v -288v
d = 96vfrom 1 : tR = 96v/v + 24 = 120
from 2 : tE = 960v/12v + 40 = 120

You have found the time it takes the regular bus...since the express bus left 40 minutes after the regular bus, you need to subtract 40 minutes to find the time it took the express bus.
 
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