Question: Bill and Ted in a Race

Comment on Bill and Ted in a Race

Let T time ted complete the race.

Rate Time Distance
bob (240/T) - 5 T+4 240
ted 240/T T 240

Rate * Time = Distance
((240/T) - 5) ( T+4) = 240
240-5T+(960/T)-20 = 240
T^2+4T-192 = 0
T can be -16 or 12 therefore T=12

now (240/12) - 5 = 12.
greenlight-admin's picture

I have a good feeling (based on your other solutions thus far) that your approach is valid :-)

However, in this solution, I'm not sure what you mean with some of the expressions. For example, I don't know what "(240/T) - 5 T+4 240" and "240/T T 240" are referring to.

Thank you so much, Brent!, tried to put in table, somehow those extra spaces i have used to form table didn't work,my bad.

Let T be the time Ted take to complete the race

Ted's rate => D/T = 240/T since 240 is the total distance.

Bob would be taking T+4 time to complete the race

Bob's rate will be 5 m/hr slower than Ted's rate, that is => (240/T) - 5

Bob's rate * Bob's time = distance
((240/T)-5) (T+4) ) = 240
240-5T+(960/T)-20 = 240
T^2+4T-192 = 0
T can be -16 or 12 therefore T=12

Ted's race completion time is 12

Bob's rate will be (240/T) - 5 => ( (240/12) ) - 5 = 15 m/hr
greenlight-admin's picture


What is the quicker strategy to answer this question? I know the GRE doesn't test our ability to do long calculations, and this is a pretty long calculation. So I've been trying to figure out what the fastest way to answer this is without doing long calculations. Has anyone figured it out?
greenlight-admin's picture

Hi Allison,

I agree with the fact that the GRE doesn't test our ability to perform long calculations. That said, this is a pretty high-level question, and the calculations (once you've created the equation) aren't that crazy.

You said at the beginning of the video that checking the answer choices is the easiest way to solve this question can you mention how?
greenlight-admin's picture

You bet.

Let's test answer choice A) 7.5
This means Bill's average speed was 7.5 mph
Bill's travel time = distance/speed = 240/7.5 = 32 hours

We're told that Bill's speed is 5 mph slower that Ted's speed. So, Ted's average speed was 12.5 mph
Ted's travel time = distance/speed = 240/12.5 = 19.2 hours

The question tells us that Ted's travel time was 4 hours less than Bill's time.
HOWEVER, when we plug in answer choice A, Ted's travel time is 12.8 hours less than Bill's time

So, answer choice A is incorrect.

Now let's test answer choice E) 15
This means Bill's average speed was 15 mph
Bill's travel time = distance/speed = 240/15 = 16 hours

We're told that Bill's speed is 5 mph slower that Ted's speed. So, Ted's average speed was 20 mph
Ted's travel time = distance/speed = 240/20 = 12 hours

The question tells us that Ted's travel time was 4 hours less than Bill's time.
That's EXACTLY what we get when we plug in answer choice E!

So, answer choice E is correct.

I solved it the same way as you did except that I made Bill's rate B-5 and Ted's rate is B. When I substuited in the time equation distance/rate , I got almost the same algebraic equation except one thing the results were -15 and 20 !!! I repeated the calculations several times but I don't know where is my mistake !! I just made the equation as follow 240/B-5 = 240/B +4 . I ended up by B2 -5B -300 ! INSTEAD OF B2 +5B -300 !! Could you please let me know where is the problem ? Thank you so much!
greenlight-admin's picture

Keep in mind that you used B and B-5 to denote the speeds, whereas I used B and B+5

Given this difference, we cannot expect our equations to be identical. Let's keep going with your equation:

You have: B² - 5B - 300 = 0
Factor: (B - 20)(B + 15) = 0
So, EITHER B = 20 or B = -15
Since the speed cannot be negative, the correct solution is B = 20

IMPORTANT: In your solution, you let B = TED's rate, and you let B-5 = BILL's rate.
Since, B = 20, we know that TED's rate is 20 miles per hour.
This means BILL's rate = 20 - 5 = 15 mph (which is the correct answer)

ASIDE: Be careful when assigning variables.
If you had let T = Ted's rate and T-5 = Bill's rate, you would have spotted the problem.


Thank you so much Brent. Yes,I have to be more cautious with naming the variables. I repeated it again from the scratch using the proper names for each variable and I got it right. Thanks!

Plz let me know if my approach is correct. Whenever the option's are values I try to fed the numbers to ques.
I start from option C: i.e.
Bill speed = 12
Therefore Bill's Time = 240/12 = 20
So Ted's time = 20 -4 = 16 and speed = 240/16 = 15, whereas Bill speed is 12 ( 12 + 5 =17) this is not possible

Similarly if I take Bill's speed as =15

therefore Bill's time = 16

And ted's speed = 20 , This is the answer because Bill's speed + 5 = 20 = Ted's speed.

I did left the decimal but I use to start from option C and deciding on the value either move to E or A.

greenlight-admin's picture

That approach is perfect. Nice work!


I approached this question the following way:
Bill: Avg speed = x - 5 , time = t
distance = speed x time
time = 240/*(x-5)

Ted: Avg speed = x, time = t-4
distance = speed * time
time = 240/x + 4

Now my confusion starts. In the previous videos, I saw that we need to make the times equal to solve the equation. So let's write what I learnt and recall

240/(x-5) = 240/x + 4 + (4)-> This additional 4 is for the difference to equalize the time. But this gives the incorrect answer

If i solve the question like : 240/(x-5) = 240/x + 4 then I get the correct answer. Can you please explain what am I doing wrong?
greenlight-admin's picture

You did a lot of things correctly in your solution. However, there are two errors.

Early in your solution, you say that Ted's time = 240/x + 4
This is not true.
Travel time = distance/speed
Ted's distance is 240 miles, and his speed is x mph
So, his travel time = 240/x

IMPORTANT: At this time, we aren't yet trying to create an equation. So, we don't yet need to add 4.

To avoid this from happening, you should START with a word equation.
You have decided to equate their travel times.
So, we can write: (Bill's travel time) = (Ted's travel time) + 4
NOTE: NOW it's our goal to create an EQUATION. To create equality, we must add 4 hours to Ted's travel time.

Now that we have our EQUATION in place, it's just a matter of filling in the blanks.

time = distance/rate
Distance = 240 miles (for each person)
Bill's speed = x - 5
Ted's speed = x
So, we can write: 240/(x-5) = (240/x) + 4

Does that help?


Hello Brent!
If there is any way to find the values of Xs
In this case 30 and 20
Instead of applying the formula ?
Thanks ! :)

greenlight-admin's picture

If by "formula" you mean the distance/rate/time formula, then the answer is "no, we need to use that formula"

Does that help?


The absolute hardest thing about these kinds of rate questions is interpreting what "5 miles per hour slower" and "4 hours sooner" mean. You would think that 5 miles per hour slower means "x - 5" (slow means less). I need to train myself to think the OPPOSITE. Slow means add, sooner (although it indicates less) means ADD as well. Very, VERY confusing!
greenlight-admin's picture

If you're not certain whether you need to ADD or SUBTRACT, plug in some values that satisfy the given information and then determine what you need to do to make those quantities equal.

For example: Joe is 3 years older than Sue.
Let J = Joe's present age, and S = Sue's present age
So, it could be the case that J = 13 and S = 10

At this point, in order to create an equation, we must EITHER add 3 years to Sue's age OR subtract 3 years from Joe's age.

Hello Brent,
Just to see another approach could we use average speed in this problem.
The way i set it up was 240/[(240/x-5)+240/x)] = (x-5) and then use this so solve for x.
Although this is a long approach and using the Answer choices is by far the best method, i just want to see if I have the right equation setup. Thanks.
greenlight-admin's picture

Hi Ravin,

Unfortunately this approach won't work.
Your denominator, [240/x-5) + 240/x] represents the COMBINED travel times of Bill and Ted.

If you want to use average speed, then the numerator would have to be 480 miles to represent the COMBINED distance traveled by Bill and Ted.

So, 480/[(240/x-5)+240/x)] represent Bill and Ted's average speed.

Unfortunately, don't have a problem since we don't know their average speed.
In your solution, you are saying that their average speed is x-5, but this represents Bob's speed only.

I set up my equation as

RT = 240 for Ted
(r-5)(t+4) = 240 for Bob and set them equal
rt= (r-5)(t+4)
rt = rt + 4r - 5t -20
20 + 5t = 4r
I don't know how to keep solving or if I am making incorrect assumptions?
greenlight-admin's picture

The problem with your solution is that you're not actually using the fact that both men traveled 240 miles.
Notice that, if we made one small change to the question and said the men traveled 240,000,000, your resulting equation would still be the same: 20 + 5t = 4r

When you have rt = 240 ( where r and t = Ted's rate and time), you can convert this to t = 240/r, which means t = 240/r
Now we're using the fact that both men traveled 240 miles.

Likewise, if we know that (r-5)(t+4) = 240 (where r-5 and t+4 = Bob's rate and travel time), we can also write: t+4 = 240/(r-5)
Or we can write: t = 240/(r-5) - 4

At this point we have two equations that are both set equal to t, which means we can write: 240/r = 240/(r-5) - 4

Does that help?

yes! So I must use some form of substitution if I do it that way.

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