Dave rode 30 miles at 10 miles per hour and 10 miles at 30 miles per hour. What was his average speed, in miles per hour, for the entire ride?
Dave rode a total of $40$ miles.  The $30$ mile segment took $\frac{30}{10}=3$ hours, while the $10$ mile segment took $\frac{10}{30}=\frac{1}{3}$ hours, so the full ride took $3+\frac{1}{3}=\frac{10}{3}$ hours.

So the average speed was  $\frac{40}{\frac{10}{3}}=\boxed{12}$ miles per hour.
12
Level 4
Algebra
Problem: Dave rode 30 miles at 10 miles per hour and 10 miles at 30 miles per hour. What was his average speed, in miles per hour, for the entire ride?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the average speed, we need to calculate the total distance and the total time taken for the ride. 

First, determine the total distance. Dave rode 30 miles and then 10 miles, which sums to a total distance of 40 miles.

Next, calculate the time taken for each segment of the ride. For the first segment, he rode 30 miles at a speed of 10 miles per hour. The time taken for this segment is calculated as follows: 

Time = Distance / Speed = 30 miles / 10 miles per hour = 3 hours.

For the second segment, he rode 10 miles at a speed of 30 miles per hour. The time taken for this segment is:

Time = Distance / Speed = 10 miles / 30 miles per hour = 1/3 hours.

Now, add the time taken for both segments to find the total time:

Total time = 3 hours + 1/3 hours = 3 + 0.333... = 3.333... hours, which can also be expressed as 10/3 hours.

Now, we can calculate the average speed using the formula:

Average Speed = Total Distance / Total Time = 40 miles / (10/3 hours).

To simplify this, multiply by the reciprocal of the total time:

Average Speed = 40 miles * (3/10) = 120/10 = 12 miles per hour.

Thus, the average speed for the entire ride is 12 miles per hour.<|END_OF_LATENT|> The final answer is $12$.