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A quick diagram makes this easier to visualize!

|Home| --- (10 mi) --- |Cyclist's current position| --- (x miles) --- |farthest the cyclist can go|

The cyclist needs to get from his current position, to the farthest point, and then back, and he has 48 minutes to do so.

His rate is 3 minutes per mile, which is 1/3 of a mile per minute. 10 miles / (1/3 mile/min) = 30 minutes. So from his current position to return home, he needs 30 minutes. Let's subtract that from the 48 minutes he has, leaving us with 18 minutes.

In these 18 minutes, he needs to travel forward, but then he needs to move backwards too. So we can only utilize half of the 9 minutes to go forward.

Almost there! So 9 minutes cycling at 1/3 miles per minute = 3 miles. He can go forward 3 more miles before he needs to turn around, which is our answer!
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30 minutes to travel 10 miles, left with 18 minutes to go which he can travel 3 miles farther and back.
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Total distance can be travelled in 48 minutes at a constant speed of 3 min/mile is: 16 miles
Assume, Farther continued distance from 10 miles before turn around: X miles
Then, 10 + 2X = 16
Then, X= 3 miles (Answer)
Bunuel
At 10:00 p.m., a cyclist is 10 miles from home on a straight road. From that time on, the cyclist continues riding farther from home at a constant speed of 3 minutes per mile, then turns around and rides back toward home at the same speed along the same road. If the cyclist must arrive back home at 10:48 p.m., how many additional miles can the cyclist travel away from home after 10:00 p.m. before turning around?

A. 2
B. 3
C. 6
D. 9
E. 12
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Bunuel
At 10:00 p.m., a cyclist is 10 miles from home on a straight road. From that time on, the cyclist continues riding farther from home at a constant speed of 3 minutes per mile, then turns around and rides back toward home at the same speed along the same road. If the cyclist must arrive back home at 10:48 p.m., how many additional miles can the cyclist travel away from home after 10:00 p.m. before turning around?

A. 2
B. 3
C. 6
D. 9
E. 12
Let x be the distance travelled away from home after 10:00 pm

Cyclist reached home at 10:48 PM, which means he travelled 10+2x distance in 48 minutes.

rate is 3 minutes per mile, so for 10 miles he spends 30 minutes.

To cover 2x distance he spends 18 minutes

To cover x distance he spends 9 minutes

t/d = rate in minutes per mile

9/x = 3

x = 3

Correct Answer B

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I believe is the option E
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Bunuel
At 10:00 p.m., a cyclist is 10 miles from home on a straight road. From that time on, the cyclist continues riding farther from home at a constant speed of 3 minutes per mile, then turns around and rides back toward home at the same speed along the same road. If the cyclist must arrive back home at 10:48 p.m., how many additional miles can the cyclist travel away from home after 10:00 p.m. before turning around?

A. 2
B. 3
C. 6
D. 9
E. 12
Lets say x is the extra miles the cyclist can go. This clearly means from 10pm to 10:48pm he has to drive 2x + 10. His rate is 1mile / 3minutes and he has to travel for 48min. D=r*t

2x + 10 = 1/3 * 48
2x + 10 = 16
2x = 6
x= 3
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in 48 minutes (from 10pm to 10:48pm), a cyclist rides 48/3= 16(miles)
16 miles also included 10 miles from home which is the starting point.
so the additional miles the cyclist travel away from home after 10pm before turning around is (16-10)/2=3 miles
Bunuel
At 10:00 p.m., a cyclist is 10 miles from home on a straight road. From that time on, the cyclist continues riding farther from home at a constant speed of 3 minutes per mile, then turns around and rides back toward home at the same speed along the same road. If the cyclist must arrive back home at 10:48 p.m., how many additional miles can the cyclist travel away from home after 10:00 p.m. before turning around?

A. 2
B. 3
C. 6
D. 9
E. 12
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Answer C
10 miles is 30 minutes, so the cyclist has 18 mins to bike away and back to their starting position. This means that it's 9 minutes each way, so 3 miles
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B
1mile=3min
10miles=xmin

x=(10*3)/1=30min

48min-30min=18min
18/2=9min

1mile=3min
x miles=9min

x=9/3=3miles
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d = additional miles
3 minutes/mile

3d(from home) + 3d(toward home) + 3*10 = 48 minutes
6d = 48 - 30 = 18 minutes
d = 3 additional miles

IMO B
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Cyclist is travelling at 3 minutes per mile, i.e. Speed = 1/3 miles/min
Let additional distance = D
He travels 2D+10 miles in 48 mins
(2D+10)/(1/3) = 48
D = 3 miles
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10:48 - 10:00 = 48 minutes

He is 10 miles from home so he will take 10*3 = 30 minutes to ride toward home.
But he will ride another x miles so we have to add twice that distance (go and return): 2x*3 minutes.

30 + 2x*3 = 48
6x = 18
x = 3

Answer B
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Let x miles be the distance between home and rider:
If he travels in stright line he will travel:
Distance = Time/ speed
=> (48 min )/ (3 min/miles)
=> 12 miles
So he will turn after 2 miles

Option: A
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Bunuel
At 10:00 p.m., a cyclist is 10 miles from home on a straight road. From that time on, the cyclist continues riding farther from home at a constant speed of 3 minutes per mile, then turns around and rides back toward home at the same speed along the same road. If the cyclist must arrive back home at 10:48 p.m., how many additional miles can the cyclist travel away from home after 10:00 p.m. before turning around?

A. 2
B. 3
C. 6
D. 9
E. 12
Total time to return to home = 48 mins
Distance covered = 10+x
time for the total distance covered after 10 pm= 48mins = 3x+3(10+x)= 3x+30+3x=48 mins
x=3
additional mile after 10:00pm is 3 miles
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total distance = additional miles (go) + additional miles (return) + 10
total minutes = 3 x total distance = 3 x 2 x additional miles + 3 x 10
48 = 6 x additional miles + 30
additional miles = 18/6 = 3

The answer is B
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Since the Cyclist rides at the same speed both ways , every extra mile away from home costs 2 miles (1 Mile going away + 1 Mile coming back ).
At 10 pm cyclist is already 10 miles away , Time = 10x 3 = 30 minutes.
Extra time beyond return = 48-30 = 18 minutes .
3 minute = 1 mile
18 min = 6 miles of riding time .
Every extra miles away needs 2 miles of riding (out +back)
6/2 = 3
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If he rides zero additional miles he takes 3*10=30 minutes.
In 48-30=18 minutes he can ride 18/3=6 miles.
As any additional mile he rides from home, he must also ride it back toward home, 6/2=3 additional miles.

The correct answer is B
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