On Monday a group of students took a test, and the average (arithmetic mean) score was exactly 80.4. A student who was absent on Monday took the same test on Tuesday and scored 90. The average test was then exactly 81. How many students took the test on Monday?

Let's denote the number of students who took the test on Monday as X.

Step 1: We know that the total score of all students who took the test on Monday is X * 80.4 (since 80.4 is the average score).

Step 2: When the student who was absent on Monday took the test on Tuesday and scored 90, the total score of all students became X * 80.4 + 90.

Step 3: We also know that the new average score is 81, so the total score of all students is now (X + 1) * 81 (since one more student, the one who was absent on Monday, has now taken the test).

Step 4: We can set up the equation X * 80.4 + 90 = (X + 1) * 81 and solve for X.

Step 5: Simplifying the equation gives X * 80.4 + 90 = X * 81 + 81.

Step 6: Rearranging the terms gives X * 81 - X * 80.4 = 81 - 90.

Step 7: Factoring out X gives X * (81 - 80.4) = 81 - 90.

Step 8: Simplifying the right side gives X * 0.6 = -9.

Step 9: Finally, solving for X gives X = -9 / 0.6 = -15.

However, the number of students cannot be negative. This means that there's a mistake in our calculation. Let's go back and check our steps.

In step 6, we should have 90 - 81, not 81 - 90. So, the correct equation should be X * 0.6 = 90 - 81 = 9.

Solving this corrected equation gives X = 9 / 0.6 = 15.

So, there were 15 students who took the test on Monday.