Ben and Sam set out together on a bicycle traveling at 15 and 12 miles

_________
Around 650.

1. Let the total distance traveled be x.
2. Total time taken by sam = x/12 = 2/3 +1+ t where t is the time Ben needs to catch up after restarting.
3. Time gained by Ben is due to the difference in speed i.e., x/12 - x/15
4. Time lost by Ben is the repair time = 1
5. (3) and (4) are equal when Ben catches up. i.,e., x/12 - x/15 =1. x=60
6.Substituting x in (2) we have t= 3.33 hrs.

Ben and Sam set out together on a bicycle traveling at 15 and 12 miles per hour, respectively. After 40 minutes, Ben stops to fix a flat tire. If it takes Ben one hour to fix the flat tire and Sam continues to ride during this time, how many hours will it take Ben to catch up Sam and assuming he resumes his ride at 15 miles per hour?

We need to find how long Ben and Sam travel until Ben needs to change his tire. Then, we need to get their respective rates to see how long it will take Ben to catch up to Sam.

Rate = 15/60
Rate = 1/4 miles/minute

Rate = 12/60
Rate = 1/5 miles/minute

Distance (Ben) = 1/4*40
Distance (Ben) = 10 miles

Distance (Sam) = 1/5*40
Distance (Sam) = 8 miles

Ben stops for one hour. Sam Travels for 1/5*60 = 12 miles. Because Ben was two miles ahead of Sam when he stopped, Sam traveled 10 miles ahead of Ben.
Ben's rate of gain on Sam is 3 miles/hour (15-12). Therefore, the time it will take for Ben to catch up to Sam: Time = distance/rate ==> Time = 10/3 ==> Time = 3.33 hours.

Answer: B. 3.33

Diff style of thinking...

T = Time sam cycles after fixing tyre to catch up with ben

Total time Sam Cycled at 15mph = 40mins+t = t + 40/60 hrs
Total Distance travelled = 15 * (t+ 40/60) ------- (1)

Total time Ben cycled at 12mph = 40mins + 60mins (When sam was fixing tyre) + t = t + 100/60 hrs
Total Distance traveled = 12 * (t+100/60) --------- (2)

(1) = (2) and solving t = 3.33

Sb = 15 * 40/60 = 10
Ss = 12 * 40/60 = 8 . 2 mile behind.
then 1 hour more, Ss = 12 * 1 = 12
after 1 hour Sam will go x more mile in t hours. so, x = 12t ..................(2)
and same has to go, 10+x = 15*t ..............(1)

from (1) and (2) , t = 10/3 = 3.333333 = B

Ben after 40 min = 15*2/3=10 miles
Sam after 40 min=12*2/3=8 miles

Sam goes 8+12=20 miles, so Ben should cover 20-10=10 miles with speed difference in 3m/h

10/3=3.33

B

Great Bunuel, if I may ask, do we subtract Sam's time in the eqution above because he keeps moving?

Yes.

This is the concept of Relative Speed.

When two objects move in same direction, their speeds get subtracted.

When two objects (speeds V1 and V2) move in opposite directions (towards each other or away from each other), they cover the distance between them (or create distance between them) at the rate of (V1 + V2).

Distance taken by Ben and Same is equal to each other. Since distance is equal to speed and time,
15 *(40min/60min + t running time after the fix)+ 0*(1 hr of pause time)= 12 * (40/60+1hr + t)
hence, 10+15t=8+12+12t
3t=10
t=3.33 hr

My solution:

Before the tire break: Ben did 10 miles & Sam did 8 miles
During the repair: Ben did 0 mile & Sam did 12 miles

When Ben resumed, he had 10 miles to catch-up at a relative speed of 3 miles per hour.

Answer is 10/3

Ben and Sam both travel for 40 minutes or 2/3 of an hour. During that time, Ben rides 15 x 2/3 = 10 miles and Sam rides 12 x 2/3 = 8 miles. During the one hour that Ben is fixing his flat tire, Sam rides another 12 miles. So, after that hour, Sam has traveled 8 + 12 = 20 miles and Ben has traveled 10 miles.

We can use the following formula to determine how long it will take Ben to catch Sam:

difference in distance/difference in rate

(20 - 10)/(15 - 12) = 10/3 = 3.33

Answer: B

Quick solution problem once you break the question

-> make a table to simplify life before you jump into the equation

In the first 40 mins that both travel the distance each of them has travelled is

B= 15*(40/60)=10
S= 12*(40/60)=8

=> we now know that B is currently 2m ahead of S

Next
B stops
S continues and does 12m
But B had a 2m lead on S so now the difference between the two is 10M

so now a simple formula

The distance of S + miles he has gained = the distance of B

Distance = speed * time
-> S distance is 10 + 12*T
-> B distance is 15*T
-> 10 + 12T = 15T
-> 10 = 15T - 12T
-> 10 = 3T
-> 10/3 = T
-> 3.333 = T

ANS 3.333333

in 40 min Ben has traveled 10 km and Sam has traveled 8km. Time sam will take to reach ben would be 2/3 (2=distance; 3=relative speed)
now for one hour, Ben has been traveling solo hence he has covered 12 km.
time for Ben to reach sam would be 12/3= 4
we will subtract the initial 2/3 from this as that's a gain time for Ben when he was ahead
4-2/3 = 3/10

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