A visual representation of the MATHCOUNTS Sprint Round competition. Display a stopwatch set to 40 minutes next to a sheet of paper with 30 problem numbers written down, the first 20 of these are marked as completed. An empty sheet with 10 problems is there to indicate the remaining ones. Also depict a pencil and calculator placed nearby. The scene takes place in a well-lit setting with a wooden desk. Make sure the image does not contain any text.

The MATHCOUNTS Sprint Round competition consists of 30 problems with a time limit of 40 minutes. Suppose you complete 20 Sprint Round problems in 25 minutes. On average, how many times as long will you be able to spend on each remaining problem than you did on each of the first 20? Express your answer as a common fraction.

6/5

1 1/5

Answer: 6/5

25 minutes for 20 questions which means 25/20 for each question which is 5/4 or 1 minute and 15 seconds. You have 15 minutes left and 10 questions so you will spend 15/10 minutes or 1 and a half minutes. Divide (15/10)/(5/4) = 6/5

6/5

Well, if you completed 20 problems in 25 minutes, that means you spent 1.25 minutes on each problem. So let's see how many problems you have remaining: 30 problems - 20 problems = 10 problems.

Now, if you have 10 problems remaining and a time limit of 40 minutes, that means you can spend 40 minutes / 10 problems = 4 minutes on each remaining problem.

So, how many times longer is 4 minutes than 1.25 minutes? Let's find out: 4 minutes / 1.25 minutes = 16/5.

Therefore, you will be able to spend 16/5 times as long on each remaining problem than you did on each of the first 20.

Or in other words, the answer is 16/5.

To find the average amount of time you can spend on each of the remaining 10 problems, we need to determine how much time you have left after completing the first 20 problems.

Since you have 40 minutes to complete the entire competition, and you have already spent 25 minutes tackling the first 20 problems, the time remaining is 40 minutes - 25 minutes = 15 minutes.

To find the average time you can spend on each of the remaining 10 problems, we divide the total time available (15 minutes) by the number of problems remaining (10 problems):

Average time per problem = Total time available / Number of problems remaining

Average time per problem = 15 minutes / 10 problems = 1.5 minutes per problem

Therefore, you will be able to spend an average of 1.5 minutes on each of the remaining problems.

To answer the question, we need to compare this average time to the time spent on the first 20 problems. You spent 25 minutes on the first 20 problems, so the ratio of the time spent on each remaining problem to the time spent on the first 20 problems is 1.5 minutes / 25 minutes.

Simplifying this fraction gives us 1.5 / 25 = 3/50.

Therefore, you will be able to spend 3/50 times as long on each of the remaining 10 problems compared to each of the first 20 problems.

6/5