12 Days of Christmas GMAT Competition - Day 7: Ross is biking at

Question

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

Ross is biking at a constant rate of x miles per hour along a straight route when he is overtaken by Rachel, who is also traveling along the same route at a constant speed 4 times faster than his. After driving for t minutes, Rachel takes a break and stops to wait for Ross to catch up. If she had to wait 45 minutes for Ross to catch up, what is the value of t?

A. 10
B. 12
C. 15
D. 20
E. 45

A. 10
B. 12
C. 15
D. 20
E. 45

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

Ross travels at x miles per hour, and Rachel travels at 4x miles per hour.

The relative speed between them is 4x - x = 3x miles per hour.

In t minutes (or t/60 hours), the distance between Ross and Rachel becomes (t/60) * 3x = tx/20 miles.

For Ross to catch up, he needs (tx/20)/x = t/20 hours. Since this time equals 3/4 hours (45 minutes), we have:

t/20 = 3/4, which simplifies to t = 15 minutes.

Alternatively, we can think of it this way: In 45 minutes, Ross covered the distance at his speed of x miles per hour. This distance originated when Rachel was driving at the relative speed of 3x miles per hour. Therefore, Rachel must have been driving for a third of the time it took Ross to catch up, or 15 minutes.

Answer: C.

Karanjotsingh · Answer

Solve this step by step:

Let Ross's speed be x mph Rachel's speed is 4x mph 
 In t minutes (t/60 hours):
 
 Rachel travels: 4x * (t/60) miles 
 Ross travels: x * (t/60) miles  
 
 During the 45-minute wait:
 
 Ross travels: x * (45/60) = 0.75x miles  
 
 For Rachel to meet Ross: Distance covered by Rachel = Distance covered by Ross 4x * (t/60) = x * (t/60) + 0.75x 
 Solve: 4xt/60 = xt/60 + 0.75x 3xt/60 = 0.75x t = 15

Answer: C. 15

Kinshook · Answer

Ross is biking at a constant rate of x miles per hour along a straight route when he is overtaken by Rachel, who is also traveling along the same route at a constant speed 4 times faster than his. After driving for t minutes, Rachel takes a break and stops to wait for Ross to catch up. If she had to wait 45 minutes for Ross to catch up, what is the value of t?

A. 10
B. 12
C. 15
D. 20
E. 45

If she had to wait 45 minutes for Ross to catch up, what is the value of t?

Ross's speed = x miles per hour
Rachel's speed = 4x miles per hour
After t minutes, distance between Rachel and Ross = 4xt/60 - xt/60 = xt/20 miles
It takes 45 minutes = 3/4 hours for Ross to cover xt/20 miles at x miles per hour
xt/20 = 3x/4
t = 3*20/4 = 15 minutes

IMO C

skry209 · Answer

distance covered by ross in (t+45) = distance covered by rachel in t
x*(t+45) = 4x*t
=> t = 45/3 = 15 
 Option C

A_Nishith · Answer

Step 1:  Define the speeds and distances
Ross's biking speed = x miles per hour.
Rachel's driving speed = 4x miles per hour (4 times Ross's speed).

The distance traveled by Rachel in t minutes is the same distance Ross needs to travel during the total time it takes for him to catch up. 
Since Rachel waits 45 minutes for Ross to catch up, Ross travels for a total time of:
t+45minutes.

Step 2:  Convert minutes to hours
Speeds are given in miles per hour, so we convert t and 45 minutes into hours:

t minutes= t/60 hours,45minutes = 45/60 = 0.75hours

Step 3 : Distance relationships
The distance Rachel travels in t minutes is:

Distance by Rachel = speed×time = 4x* t/60
The distance Ross travels in t+45 minutes is:

Distance by Ross= speed×time = x*( t/60+0.75)
Since Rachel waits for Ross to catch up, the two distances are equal:

4x*t/60 = x*(t/60 +0.75)

Step 4:  Simplify the equation
Cancel x (as x≠0):

4* (t/60) = t/60 + 0.75
Multiply through by 60 to eliminate the denominators: 4t=t+45
Simplify:
⟹ 3t=45⟹t=15.

Final Answer: 
The value of t is  C. 15

Krunaal · Answer

Avg. Speed of Ross: x mph
Avg. Speed of Rachel: 4x mph

Rachel post overtaking Ross, drives for t minutes (t/60 hrs) and stops.

Distance travelled by Rachel for t minutes= 4x * (t/60)
Distance travelled by Ross for t minutes= x * (t/60)

She waits for Ross for 45 minutes (3/4 hrs)

Ross travels   -  : 3x * (t/60) distance in 3/4 hrs at the speed of x mph

Therefore, x*3/4 = 3x*(t/60) =>  t = 15 minutes

Answer C.

nikiki · Answer

rate 
 time 
 distance 
 
  ross 
 x 
 t+45 
 d 
 
  rachel 
 4x 
 t 
 d

xt + 45x = 4xt
3xt = 45x
 t = 45/3 = 15minutes

Option C

nishantswaft · Answer

The stem tells us that;
 
  Ross is biking at a constant rate of x miles per hour   
  He is overtaken by Rachel   
  Rachel is also traveling along the same route at a constant speed 4 times faster   
  After driving for t minutes, Rachel takes a break and stops to wait for Ross to catch up   
  she had to wait 45 minutes for Ross to catch up

And we have to select;  
 
  the value of t

Speed of Ross=  x miles per hour 
 Speed of Rachel = 4x miles per hour

Since in this scenario the distance of each will be the same let's calculate their individual distances.

Distance of Rachel =4x*t/60

Distance of Ross = x*t/60+x*45/60

Since they are equal; 
 Let's equate these two.

4x*t/60 = x*t/60+x*45/60

4x*t/60 - x*t/60 = x*45/60

3xt/60 = 3x/4

t=15

Answer is C

Nikhil17bhatt · Answer

IMO C
 
 To solve this problem, let's break it down step by step using the given information and the relationship between distance, speed, and time. 
  Step 1: Define the speeds and times  
 
  Ross's speed:   x   miles per hour  
  Rachel's speed:   4x   miles per hour (since she is traveling 4 times faster than Ross)  
  Rachel travels for   t   minutes before stopping.  
  Rachel waits for 45 minutes for Ross to catch up.   
  Step 2: Convert time to hours  
 Since the speeds are given in miles per hour, we need to convert the time from minutes to hours. 
 
  T   minutes is   t/60  hours.  
  45 minutes is   45/60=3/4  hours.   
 Step 3: Calculate the distance Rachel travels before stopping 
The distance Rachel travels before stopping is:
 Distance=Speed×Time=4x×t/60=4xt/60
 Step 4: Calculate the distance Ross travels while Rachel waits 
During the 45 minutes that Rachel waits, Ross continues to travel at his constant speed x miles per hour. The distance Ross travels in 45 minutes is:
 Distance=Speed×Time=x×3/4=3x/4
 Step 5: Calculate the total distance Ross needs to travel to catch up 
When Rachel stops, she has traveled 2xt/30miles. Ross needs to travel this same distance to catch up to Rachel. Since Ross is already traveling while Rachel waits, the total distance Ross needs to cover is:2xt/30−3x/4
 Step 6: Set up the equation for Ross's travel time 
The time it takes Ross to catch up to Rachel is the same as the time Rachel waits, which is 45 minutes or 3/4hours. Therefore, we set up the equation:2xt/30=x×(t/60+3/4)
 Step 7: Solve for t 
T=15

campbellgarrison9 · Answer

The total distance for Ross and Rachel is the same so it can be set equal to eachother.

The total time traveled for Ross is equal to the initial t value plus the 45 minutes Rachel waited. Rachels travel time is only going to take into account the initial t because she was no longer traveling during the 45 minutes.

x(t+3/4) = 4x(t)

xt + 3/4 x = 4xt

3/4 x = 3 xt

3/12 = t

3/12 of one hour equals 15 minutes. Answer is C

OmerKor · Answer

Hi everyone:)

This is a distance/rate=time question
Ross Speed = x
Rachel Speed = 4x
Driving t minutes. and then 45 mins for Ross to catch Rachel.

Let's work backward:
t=12mins (1/5h)
x=5 mph 4x= 20mphs
Ross drove 1 Mile
Rachel drove 4 Miles
4-1=3 Miles is the distance between them.

Ross needs to catch this distance in 45 minutes. let's check that:
5mph*3/4 = 15/4 is not 3 Miles.
Eliminate.

Let's try 
t=15mins (1/4h)
x=4 mph 4x= 16 mphs
Ross drove 1 Mile
Rachel drove 4 Miles
4-1=3 Miles is the distance between them.

Ross needs to catch this distance in 45 minutes. let's check that:
4mph*3/4 = 3. This is match the distance we need to find.

t=15 minutes is our Answer - C

HelpfulChocolate · Answer

Speed = Distance/Time

Ross 
Speed (S1) = x mph
d = d
t1 = t + 45
Since Ross didn't stop cycling while Rachel passed and continued biking.

Distance = Speed * Time
d = x * (t+45) -> 1

Rachel 
Speed (S2) = 4x mph
d = d
t2 = t

Distance = Speed * Time
d = 4x * t -> 2

Equate 1 & 2 (since distance is the same)
x * (t + 45) = 4x * t
(t + 45) = 4 * t
t + 45 = 4t
45 = 4t - t
3t = 45
 t = 15 mins

is the correct answer.

pintukr · Answer

Ross is biking at a constant rate of x miles per hour along a straight route when he is overtaken by Rachel, who is also traveling along the same route at a constant speed 4 times faster than his. After driving for t minutes, Rachel takes a break and stops to wait for Ross to catch up. If she had to wait 45 minutes for Ross to catch up, what is the value of t?

A. 10
B. 12
C. 15
D. 20
E. 45

A. 10
B. 12
C. 15
D. 20
E. 45

Suppose, Ross' speed = x miles/hr
 Rachel's speed = 4x miles/ hr

In t minutes, Rachel is ahead by = (4x-x) * (t/60) miles 
In 45 minutes, Ross covers this distance for catching up with Rachel = x * (45/60)

thus, (4x-x) * (t/60) = x * (45/60)
 t/20 = 3/4
 t = 15

(C) is the CORRECT answer

Lizaza · Answer

Since Rachel passing Ross, the distance covered in t was at a speed that's the difference between Rachel's and Ross's:  v=4x-x=3x  
 Then, this distance  s=t*3x

After that, Ross covered this distance in 45 min, so his speed was  x=\frac{s}{45}=\frac{3tx}{45}  
 Which means that  45x=3tx , and  3t=45  
 Therefore,  t=15   and    the answer is C.

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