Question 1111087
let A = number of sections of ancient history
let B = number of sections of medieval history
let C = number of sections of modern history


you get 3 equations that need to be solved simultaneously.


they are:


A + B + C = 44 (first equation)
100A + 50B + 200C = 4900 (second equation)
A + B + 2C = 58 (third equation)


first equation is number of sections
second equation is number of students
third equation is number of teachers


subtract the first equation from the third equation to get:


A + B + 2C = 58 minus (A + B + C = 44) results in C = 14


since you know the value of C, rewrite your 3 original equations with the value of C being evaluated in each equation.


your equations become:


A + B + 14 = 44 (first equation)
100A + 50B + 2800 = 4900 (second equation)
A + B + 28 = 58 (third equation)


subtract the constants on the left side of the equal sign from both sides of the equations to get:


A + B = 30 (first equation)
100A + 50B = 2100 (second equation)
A + B = 30 (third equation)


subtract the second equation from 100 times the third equation to get:


100A + 100B = 3000 minus (100A + 50B = 2100) results in 50B = 900


solve for B to get B = 18


you now have:


C = 14
B = 18


replace B and C in the first equation to get:


A + B + C = 44 becomes A + 18 + 14 = 44


solve for A to get A = 12


you now have:


A = 12
B = 18
C = 14


that's your solution.


number of sections is A + B + C which becomes 12 + 18 + 14 which is equal to 44


number of students in each course is 100A + 50B + 200C = 1200 + 900 + 2800 which is equal to 4900


number of teachers in each course is A + B + 2C which becomes 12 + 18 + 28 which is equal to 58.


12 sections in ancient history have 1200 students and 12 teachers.
that's 1 teacher per course and 100 students per section.


18 sections in medieval history have 900 students and 18 teachers.
that's 1 teacher per course and 50 students per section.


14 sections in modern history have 2800 students and 28 teachers.
that's 2 teachers per course and 200 students per section.


answer to your questions is:


12 sections of ancient history and 18 sections of medieval history and 14 sections of modern history.