Final answer:
The total time to complete the math homework with a student and friend working together at rates of 5 minutes per question and 10 minutes per question, respectively, will be 90 minutes.
Step-by-step explanation:
The student is seeking help with a math homework problem that involves calculating the total time it will take two people working at different rates to complete a set of questions. The student starts the homework and completes questions at a rate of 5 minutes per question. The friend starts half an hour later and completes questions at a rate of 10 minutes per question. We need to determine the total time, x minutes, taken from when the student begins working to complete all 24 questions.
To solve this, we'll first find out how many questions the student can complete before the friend joins. Since the student begins right away and works for 30 minutes before the friend joins, the student can finish 30 / 5 = 6 questions in that time. Now, there are 24 - 6 = 18 questions remaining. The student and the friend work together to complete these 18 questions.
We have to calculate the time it takes for both working together, which involves setting up an equation based on their combined work rates. If S is the student's work rate and F is the friend's, we get S+F=1/(S's time per question) + 1/(F's time per question) = 1/5 + 1/10 = 3/10 questions per minute. To complete 18 questions, they will take 18 / (3/10) = 60 minutes. Therefore, the total time, x, is 30 minutes alone + 60 minutes together = 90 minutes.