You have 48 minutes to take an exam of 21 questions. After you finish the first 10 questions, you realize you must answer the remaining questions twice as fast to finish the exam.
How many questions per minute were you answering?
Also, how many questions per minute must you now answer to finish the exam on time?
q = first questions/min
2 q = final que/min
t min for first ten
48 - t min for final 11
q t = 10 so t = 10/q
q t + 2q(48-t) = 21
10 + 96q - 20 = 21
96 q = 31
q = 31/96 questions/min
so twice that = 62/96 to finish
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check
t = 960/31 = 30.97 min
48-t = 17.03 min
30.97 * 31/96 = 10
17.03 * 62/96 = 11
sum = 21 sure enough
To find the number of questions you were answering per minute initially, you can divide the total number of questions (21) by the total time (48 minutes) given for the exam:
Questions per minute initially = Total questions / Total time = 21 / 48
To find the number of questions you must now answer per minute to finish the remaining questions twice as fast, you can calculate the remaining time available after you finished the first 10 questions.
The remaining time available is 48 minutes - the time taken to answer the first 10 questions. Let's call this time "rt" for remaining time.
rt = 48 minutes - time taken for first 10 questions
To finish the remaining 11 questions in this remaining time "rt", you need to divide the number of remaining questions (11) by the remaining time (rt):
Questions per minute now = Remaining questions / Remaining time
But since you want to finish the remaining questions twice as fast, you can multiply the initial questions per minute by 2:
Questions per minute now = 2 * (21 / 48)
Therefore, to finish the exam on time, you were initially answering approximately 0.4375 questions per minute (or 7/16 questions per minute), and you must now answer approximately 0.9166 questions per minute (or 11/12 questions per minute).