# How to estimate the time needed to solve a basic mathematics problem

I want a way to know the time needed to solve addition, subtraction, multiplication and division problems.

Examples

1. $15 + 10$

2. $500 - 132$

3. $10 \cdot 10$

4. $20 \, / \, 10$

Is it possible to create a formula that take the operation type and the operands values and calculate the time needed?

Thanks

• I don't think there is a general formula, but I've heard 4 to 5 times your own speed is usual. – Chris C Jan 24 '15 at 15:45
• Are these really questions for university students? – Dag Oskar Madsen Jan 24 '15 at 15:49
• I don't know about arithmetic. But the rule-of-thumb we used for freshman calculus exams was: do time yourself doing the exam, then multiply by 10. – Gerald Edgar Jan 24 '15 at 17:10
• @ChrisC: That sounds pretty fast. I suppose you meant taking 4 to 5 times as long as you do (or 4 to 5 times slower than your own speed). – J W Jan 24 '15 at 17:27
• @DagOskarMadsen: Perhaps it should be retagged (primary-education)? – J W Jan 24 '15 at 17:29

I will formulate my comment into a more fully fledged answer.

I do not believe that there can be a formulaic answer to how long does it the average student to do any particular problem or exam. It depends on far too many factors that are individualistic for each student such as 'how does this student study?', 'what are their motivations to study and do well?', or 'did they get enough sleep?'. It even depends on the type and difficulty of the problem itself and how to quantify that.

The best way to gauge how long it takes students to do particular problems is to just watch them doing it. Assign some quizzes with such problems and time how long it takes half to three-fourths or even all to finish. After doing this several times and comparing how long they take on average to how quickly you can do the same problems, you can get an estimate for how much longer you should give them compared to yourself.

Some estimates that I frequently hear are:

• University Students: They usually are given 4 to 5 times how long the instructor takes in the introductory courses. It might be 2 to 3 in higher level courses.

• High School Level or Equivalent: This answer by benblumsmith gives an estimate of 6 to 8 times.

• I've not heard any estimates given for younger students nor do I have any experience to venture a guess. Jasper's answer is a good reference for this.

Addendum: I just realized that some online assignment systems (i.e., WebAssign, Connect, etc) can provide instructors averages for how long students take on assignments (I believe from first access to last access on the assignment). This could be used to gauge how long students take for homework assignments, a harder thing to measure for instructors.

• I can confirm "x4" for university. – Jasper Jan 25 '15 at 21:30

As discussed in a question about teaching mental arithmetic, my fourth grade math class included a daily 3-minute exercise with 50 - 100 problems. Each day's problems were either addition (of one- or two-digit whole numbers), subtraction (of one- or two-digit whole numbers), multiplication (of integers between 0 and 12), or division (of integers between 0 and 144 by integers between 0 and 12, yielding integers between 0 and 12).

If the students have been through such a program to encourage them to memorize basic math facts and perform arithmetic quickly, it is therefore reasonable to expect students age 11 or higher to correctly answer items 1, 3, and 4 in 2 - 10 seconds per question, with an error rate on the order of one percent. Question 2 might require more steps, such as writing out the problem, and two different carries. Thus, question 2 might take 5 - 60 seconds.

I would not be surprised if the time to manually perform addition or subtraction is proportional to the total number of digits involved. Similarly, I would not be surprised if the time to manually perform multiplication is proportional to the product of the number of digits involved, multiplied by a different proportionality constant. I expect long division to take even longer than the corresponding multiplication problem.

• Would you share some rough statistics about the results? Knowing about what percent did 10 problems correctly, or 20, or 30, etc. and how that improved over the school year would be useful to some of us here. Gerhard "Tutoring A Fourth Grader Currently" Paseman, 2015.01.26 – Gerhard Paseman Jan 26 '15 at 22:37
• @Gerhard Paseman -- It has been a while since I was in fourth grade. After filling out as much of the sheet as we could in the three minutes, we would swap sheets with our neighbors to mark. I do not remember, but it is likely that the teacher passed out a second sheet with the answers. Typically, each student would get 0 - 3 answers wrong. At the beginning of the year, it was normal for students to only finish about 20% of the worksheet. By the Spring, it was not surprising that some students finished the worksheet. It was an honor to be able to finish the worksheet; … – Jasper Jan 26 '15 at 23:51
• … it was an honor to get however far you could without making any mistakes; and it was a feat to finish the worksheet without making any mistakes. I do not remember if or how the exercise affected our grades. My impression is that trying hard was the point, and improvement was encouraged. – Jasper Jan 26 '15 at 23:53
• My grade schools used tracking for Math and English. (This was especially important, because many students did not speak English at home.) There were typically 26 - 30 students in each class. I was in the high track for Math. My fourth grade teacher helped supervise recess. Specifically, about half of the fourth-grade boys would play touch football, with him as the "all-time quarterback". Most of the plays were Hail Maries, and a good time was had by all. (We continued doing this after fourth grade, but each team had to choose a quarterback.) – Jasper Jan 27 '15 at 0:00
• This group of boys formed the core of the high school football team. Our high school had the toughest academic standards for athletic eligibility in the county. The high school football coach emphasized academics. I don't think it was a coincidence that the football team went undefeated my senior year, and won the division championship. – Jasper Jan 27 '15 at 0:04