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
\(\frac{240}{(240 - 4x)} = \frac{(x + 5)}{x}\)
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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[r(r - 5)] = 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
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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
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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

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)
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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:

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

\(240/16 = 15\)