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?
Professor Sum has 56 students.
28 of then study languages - one half
(14 of which study French)
8 study physics - one-seventh
20 do not study at all.
To find the number of students Professor Sum has, we need to break down the information provided and solve step by step.
Let's assign variables to each category:
Let L be the number of students studying languages only.
Let F be the number of students studying French.
Let P be the number of students studying physics only.
Let N be the number of students not studying at all.
Given information:
1. "One-half of them study languages only": This means L = 1/2 * (L + F + P).
2. "One-fourth of them study French": This means F = 1/4 * (L + F + P).
3. "One-seventh of them study physics only": This means P = 1/7 * (L + F + P).
4. "20 do not study at all": This means N = 20.
Now, let's solve the equations:
From statement 1:
L = 1/2 * (L + F + P)
Multiplying both sides by 2:
2L = L + F + P
From statement 2:
F = 1/4 * (L + F + P)
Multiplying both sides by 4:
4F = L + F + P
From statement 3:
P = 1/7 * (L + F + P)
Multiplying both sides by 7:
7P = L + F + P
Now, substitute the values of L, F, and P in terms of L, F, and P from equations 1, 2, and 3, respectively:
2L = F + P
4F = L + F + P
7P = L + F + P
Combining these equations:
2L = 4F - F + 7P - P
2L = 3F + 6P
Since we have two equations and two variables (L and F), we can solve them simultaneously:
2L = 3F + 6P ...(Equation A)
4F = L + F + P ...(Equation B)
Rearranging equation B:
3F = L + P
Substituting this into equation A:
2L = 3(L + P) + 6P
2L = 3L + 3P + 6P
2L = 3L + 9P
2L - 3L = 9P
-L = 9P
L = -9P
Since the number of students cannot be negative, we can conclude that L = 0. This means that there are no students studying languages only.
Now, substituting L = 0 into equation B:
3F = P
From statement 4:
N = 20 (students not studying at all)
Now, to find the total number of students, we need to find the sum of all the categories:
Total number of students = L + F + P + N
= 0 + F + 3F + 20
= 4F + 20
= 4(3F) + 20
= 12F + 20
Since we know that 3F = P, we can rewrite the equation:
Total number of students = 12F + 20
= 12(3F) + 20
= 36F + 20
Therefore, the total number of students Professor Sum has is given by the equation 36F + 20, where F is the number of students studying French.