A student answered 86 problems on a test correctly and received a grade $98\%$. How many problems were on the test, if all the problems were worth the same number of points?

Hint: Let us assume that the total number of problems on the test are N. It is given that a student correctly answered 86 problems and received a grade of $98\%$. As all the problems are worth the same number of points so the percentage of grade is same as the percentage of number of questions answered correctly out of the total questions. Now, we can equate $98\%$ of N to 86. And solving this equation will give us the value of N which is the total number of problems on the test.

Complete step-by-step solution:
Let us assume the total number of questions in the test are N.
And the student answered 86 problems and got $98\%$ grade. As all the questions worth the same number of points so the grade percentage is the same percentage as that of the percentage of questions answered correctly.
Due to the above information, we can write $98\%$ of N as 86. So, equating $98\%$ of N to 86 we get,
$\Rightarrow \dfrac{98}{100}\times N=86$
Cross multiplying in the above equation we get,
$\Rightarrow 98N=8600$
Dividing 98 on both the sides we get,
$\begin{align}
  & \Rightarrow N=\dfrac{8600}{98} \\
 & \Rightarrow N=87.75 \\
\end{align}$
Rounding off the number which we are getting for N we get,
$\Rightarrow 87$
Hence, we got the total number of questions in the test as 87.

Note: The mistake that could be possible in the above problem is that you might forget to round off the number of questions (which is the value of N). Generally, the examiner knows such weaknesses of the students and gives options in which you can find an option which is not rounded off so make sure you have rounded off the total number of questions because the number of questions are always positive integers.