Bill and Ted each competed in a 240-mile bike race.

Question

Bill and Ted each competed in a 240-mile bike race. Bill’s average speed was 5 miles per hour slower than Ted’s average speed. If Ted completed the race 4 hours sooner than Bill did, what was Bill’s average speed in miles per hour?

A) 7.5
B) 10
C) 12
D) 12.5
E) 15

*Kudos for all correct solutions.

A) 7.5
B) 10
C) 12
D) 12.5 
E) 15

ShashankDave · Accepted Answer

Another way to build up the quadratic...

Let Bill's speed = x
and Ted's speech = x+5

When Ted covers the full distance, Bill has yet to travel for 4 more hours to complete the distance
 \frac{240}{(240 - 4x)} = \frac{(x + 5)}{x}

ScottTargetTestPrep · Answer

We are given that Bill and Ted each competed in a 240-mile bike race. We are also given that Bill’s average speed was 5 miles per hour slower than Ted’s average speed.

Both Bill and Ted had distance of 240 miles. If we let Ted’s speed = r, we can let Bill’s speed = r - 5. We can use the above information to determine the time of Bill and Ted in terms of variable r.

Since time = distance/rate, Ted’s time = 240/r and Bill’s time = 240/(r - 5).

Since Ted completed the race 4 hours sooner than Bill did, we can create the following equation:

240/r + 4 = 240/(r - 5)

To eliminate the denominators of the fractions we can multiply the entire equation by r(r-5) and we have:

240(r - 5) + 4  = 240r

240r - 1,200 + 4r^2 - 20r = 240r

4r^2 - 20r - 1,200 = 0

r^2 - 5r - 300 = 0

(r - 20)(r + 15) = 0

r = 20 or r = -15

Since r must be positive, r = 20.

Thus, Bill’s rate = 20 - 5 = 15 mph.

Answer: E

rudra316 · Answer

Vb and Vt be the average speeds of Bill and Ted. 
Vb = Vt - 5

Tb and Tt be the time take,
Tt = Tb - 4

We have Distance = Speed * Time 
  Vb* Tb = Vt * Tt = 240 
 (Vt - 5) * ( Tt + 4) = Vt * Tt 
4Vt - 5 Tt = 20
Replacing value of Tt as 240/Vt

4 Vt - 5*(\frac{240}{Vt}) = 20 
 Vt^{2} - 5 Vt = 300 
 (Vt -20)* (Vt +15) = 0 
Vt = 20

Average speed of Bill Vb = Vt-5 = 20- 5 = 15.

BrentGMATPrepNow · Answer

Another approach....

Bill’s average speed was 5 miles per hour slower than Ted’s average speed.  
Let B = Bill's travel speed
So, B + 5 = Ted's average speed

Ted completed the race 4 hours sooner than Bill did  
Let's start with a word equation: 
 (Bill's travel time)  =  (Ted's travel time)  + 4 
time = distance/speed
So, we get:  240/B  =  240/(B + 5)  + 4 
Rewrite 4 as 4(B + 5)/(B + 5) to get:  240/B  =  240/(B + 5) + 4(B + 5)/(B + 5)  
Simplify:  240/B  =  240/(B + 5) + (4B + 20)/(B + 5)  
Combine terms: 240/B = (4B + 260)/(B + 5)
Cross multiply: 240(B + 5) = (B)(4B + 260)
Expand and simplify: 240B + 1200 = 4B² + 260B
Set equal to zero: 4B² + 20B - 1200 = 0
Divide both sides by 4 to get: B² + 5B - 300 = 0
Factor: (B + 20)(B - 15) = 0
So, EITHER B = -20 OR B = 15
Since B (Bill's speed) cannot be a negative value, we can conclude that B = 15.

Answer:  E

RELATED VIDEO
 https://www.youtube.com/watch?v=XwVu_TEXQqs

shashankism · Answer

Let Ted's average speed be x miles/hr.
Bill's average speed be x-5 miles/hr.

Distance = 240 miles

240/(x-5) - 240/x = 4
240*5/(x(x-5)) = 4 
x(x-5) = 300 
x = 20
x-5 = 15

Answer E

gps5441 · Answer

Easier method
Bills time be= t and speed=x
Ted's time=t-4 and speed=x+5
distance =240
Bill's time(t) = distance/speed=240/x
Teds time(t-4)=240/(x+5)

Plugin answer choices. you should start from c or answers which are whole numbers

Let bills speed be =15
we now have
Bill(t)=240/15=16
Now Plugin for ted
t-4=240/(15+5)=240/20=12 or t=12+4=16.
with above approach we can avoid quadratic equation.

gracie · Answer

let r=Bill's average speed
Bill: d=rt
Ted: d=(r+5)(t-4)
combining,
rt=(r+5)(t-4)➡
4r=5t-20
substituting,
4r=5(240/r)-20➡
r^2+5r-300=0
(r+20)(r-15)=0
r=15 mph
E

BrentGMATPrepNow · Answer

Another option is to start with the following word equation: 
 (Bill's average speed) + 5 = (Ted's average speed)

Let t = Bill's travel TIME (in hours)
So, t - 4 = Ted's travel TIME (in hours)

speed = distance/time

So, our word equation becomes: 240/t + 5 = 240/(t - 4)
Multiply both sides by t to get: 240 + 5t = 240t/(t - 4)
Multiply both sides by (t - 4) to get: 240(t - 4) + 5t(t - 4) = 240t
Expand: 240t - 960 + 5t² - 20t = 240t
Subtract 240 t from both sides: -960 + 5t² - 20t = 0
Rearrange: 5t² - 20t - 960 = 0
Divide both sides by 5 to get: t² - 4t - 192 = 0
Factor to get: (t - 16)(t + 12) = 0

So, EITHER t = 16 OR t = -12

Since the time cannot be NEGATIVE, we can conclude that t = 16
In other words,  Bill's travel time was 16 hours

What was Bill’s average speed in miles per hour?  
speed = distance/time
So, Bill's speed = 240/16 = 15 miles per hour

Answer: 15

Answer: E

RELATED VIDEO FROM OUR COURSE
 https://www.youtube.com/watch?v=XwVu_TEXQqs

KarishmaB · Answer

Usually, the way I do these questions is this - Go to the options directly.

If Bill's speed is 12, time taken is 240/12 = 20 hrs
Then Ted's speed is 17 which is not divisible by 240 so is not 16 hrs.

If Bill's speed is 15, time taken is 240/15 = 16 hrs
Then Ted's speed is 20 and time taken is 240/20 = 12 hrs
Works!

Answer (E)

BrentGMATPrepNow · Answer

As I mention in the video below, these kinds of questions can be solved in a variety of ways.

For our 3rd approach, let's start with a word equation involving distances traveled. 
Since Bill and Ted both traveled 240 miles, we can write: 
  Bill's travel distance  =  Ted's travel distance

Let  x  = Bill's speed
So,  x + 5  = Ted's speed

Let  t  = Ted's travel time (in hours)
So,  t + 4  = = Bill's travel time (in hours)

Distance = (speed)(time) 
Plug values into the word equation to get: ( x )( t + 4 ) = ( x + 5 )( t )
Expand to get: xt + 4x = xt + 5t
Subtract xt from both sides to get: 4x = 5t
Divide both sides by 5 to get:  4x/5 = t

Where do we go from here?

Well, we know that Bill traveled 240 miles
So, (Bill's speed)(Bill's travel time) = 240
In other words: ( x )( t + 4 ) = 240
Replace  t  with  4x/5  to get: (x)(4x/5 + 4) = 240
Expand: 4x²/5 + 4x = 240
Multiply both sides by 5 to get: 4x² + 20x = 1200
Divide both sides by 4 to get: x² + 5x = 300
Rewrite as: x² + 5x - 300 = 0
Factor: (x - 15)(x + 20) = 0
So, EITHER x = 15 OR x = -20
Since the speed cannot be negative, it must be the case that x = 15

Answer: 15

RELATED VIDEO FROM OUR COURSE
 https://www.youtube.com/watch?v=XwVu_TEXQqs

NinetyFour · Answer

My big issue with 700 level math questions isn't a matter of getting an answer, but more of a timing issue. Does anyone have any tips on how I can improve here? It takes me 2-3minutes to solve a 700 level question. My process is 1) read the question, 2) setup equations, 3) solve. Thats as simplified as I can make the process, but it still takes (more often than not) closer to 3 minutes to solve any question. On practice tests, most of my questions are in the 700 range, and as a result I never finish the test on time and always end up missing the last 5 or 6 questions.... Quant scores usually hover around 48-50.

Here's my thought process if it helps anyone:

distance traveled for both racers are the same hence we can set the two rates and times equal to each other:

Let  t  represent Bill's time, and  s-5  represent Bill's speed.
Let  t-4  represent Ted's time, and  s  represent Ted's speed.

t*(s-5) = (t-4)*s(

s=\frac{5t}{4}

Let's plug this back into the equation above and solve for the time bill spent on the race:

\frac{5t}{4}*t - 5t = 240 
 t=16 
 Average speed = distance/time

240/16 = 15

BrentGMATPrepNow · Answer

Your solution is perfect. So, we need not discuss that. 
I want to mention that guessing on (or, even worse, NOT answering) 5 or 6 questions in a row will kill your score. Your goal should be to spread out your guesses.

Please review my time-management strategy below: 
 https://www.youtube.com/watch?v=bWrUiBT83PQ

If you consistently don't have enough time to answer all of the quant questions, it's better to allow yourself to guess on a few questions. IMO, this is better than hurrying and answering all of the questions haphazardly (which can result in a lot of careless mistakes).

Cheers,
Brent

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Bill and Ted each competed in a 240-mile bike race.

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