Math word problem

Question

Some students stood evenly spaced in a circular formation. They counted off at 1 and continued by consecutive integers once around the circle. How many students were in the circle formation of student 19 was opposite student 99?

Answer 1

Think of looking down at the table.

Student 19 is at the top and student 99 is at the bottom, like so:   19

 

 

                                                                                                                     99

At the right of the table you have students 20,21,22,..,96,97,98.

How many students are to the right ?

98-20=78 but you include both 20 and 98 so there are 79 students to the right.

Look at it this way: 1-98 equals 98 students but you are missing numbers 1-19,so 98-19=79 again!

We could stop now because the students are equally spaced which means there are 79 students to the left of the table.

79 to the left plus 79 to the right plus the two at the top and the bottom make 79+79+2=160 students.

Which students are to the left of the table if there are 79 students ?

Students 1,2,3,...,16,17,18 for a total of 18 students so far.

79-18=61 more students are to the left of the table.

Going clockwise from 99, we have students 100,101,102,...158,159,160.

100-160 gives you 61 students; 101 to 160 gives 60 students plus number 100 gives you 61 total to be added to the 18 we got before for a total of 61+18=79 students to the left of the table which we determined before but now you know what numbered students are to the left of the table as well as to the right of the table.

 

 

Answer 2

If the difference between 99 and 19 is halfway around the circle, then how many students are in the full circle?