There are 360 people in my school. 15 take calculus, physics, and chemistry, and 15 don't take any of them. 180 take calculus. Twice as many students take chemistry as take physics. 75 take both calculus and chemistry, and 75 take both physics and chemistry. Only 30 take both physics and calculus. How many students take physics?
There are 360 people and 345 people taking courses. 15 take all 3 and 60 only take Calculus because 75+30+15+60=180.
Next, to find the total number of people who are only taking physics or only taking chemistry, I subtracted 240 from 345 because 240 (69+75+75+30) is everyone taking courses except for the people only taking physics of chemistry. I found that 105 are either only taking phy. or only taking chem.
lets say z=only physics
105=y+z or y=105-z
Also, x=the total number of people taking physics (so 2x is the total number of people taking chemistry)
75+2z=y and from before: y=105-z
z=10 so 10 people only take physics
Finally, the number of people taking physics is 10 (only) + 75 (phy. and chem) +30 (and calc) + 15 (all three) = 130. 130 students are taking physics.