When Professor Sum was asked by Mr. Little how many students were in his classes, he answered, “All of them study either languages, physics, or not at all. One-half of them study languages only, one-fourth of them study French, one-seventh of them study physics only, and 20 do not study at all.” How many students does Professor Sum have?
1/2 + 1/4 + 1/7 = X - 20
Solve for X.
I hope this helps.
To solve this problem, we need to use a method called "set theory" or "Venn diagrams." Let's break down the given information step by step:
Let's assume the total number of students in Professor Sum's classes is 'x'.
1) "One-half of them study languages only":
From this, we can conclude that the number of students who study ONLY languages is (1/2) * x.
2) "One-fourth of them study French":
This implies that the number of students who study French is (1/4) * x.
3) "One-seventh of them study physics only":
This tells us that the number of students who study ONLY physics is (1/7) * x.
4) "20 do not study at all":
According to this statement, 20 students do not study any subject.
Now let's put all this together using a Venn diagram:
-------------------------------------
| Languages |
| |
| --------------- |
| | | |
| | | |
| French Common physics |
| | x/4 |
| | |
--------------- |
| | |
| | |
| | | |
| | | |
| | | |
| --------------- |
| | |
-------------------------------------
From the diagram, we can see that the number of students who study only languages is (1/2) * x, and the number of students who study only physics is (1/7) * x. The number of students who study French is (1/4) * x.
Also, it's mentioned that 20 students do not study at all.
Now, we can form an equation using the given information:
(1/2) * x + (1/4) * x + (1/7) * x + 20 = x
To solve this equation and find the value of x, we can multiply through by the least common denominator (LCD) which is 28:
14x + 7x + 4x + 560 = 28x
25x + 560 = 28x
560 = 3x
x = 560 / 3
x ≈ 186.67
Therefore, Professor Sum has approximately 187 students in his classes.