I finished grading tests for two classes when I noticed something interesting about the worst scoring student for each.

In the first class a student who got 46 problems wrong got 46 percent right (to the nearest whole percent.)

How many problems were there on this test?

In the second class there were 35 problems but again the lowest scoring student got the same number wrong as percent right (again rounded.)

How many questions did this student get wrong?

What is the smallest number of questions for which this kind of thing can happen?

(Aside from the trivial one question test.)

What if the percent must be exact instead of rounded?

[Note: This actually happened.]

For *the ***first part**, let the approximate number of problems be x.

Then, we have:

x/46 = 54/100, giving:

x = 4600/54 = 85.1851 ~ 85

Thus, the required total number of problems is 85.

For the **second part**, let y denote the approximate total number of problems answered incorrectly.

Then, by the problem:

y/35 = (100-y)/100

or, y/35 + y/100 = 1

or, y = 700/27 = 25.926~ 26

Consequently, the required total number of problems answered

incorrectly is 26.