. In a group of 150 students, more students are taking a math class than are taking a science class this semester. If 80 are taking a science class and 25 are not taking either math or science this semester, what is the minimum number of students who could be taking both math and science this semester?

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In a group of 150 students, more students are taking a math class than are taking a science class this semester. If 80 are taking a science class and 25 are not taking either math or science this semester, what is the minimum number of students who could be taking both math and science this semester?
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There are four categories of students here:
1. Those taking math but not science.
2. Those taking science but not math.
3. Those taking both math and science.
4. Those taking neither math nor science.
1. Let x = the number of students taking math but not science.
2. Let y = the number of students taking science but not math.
3. Let z = the number of students taking both math and science.
4. Let w = the number of students taking neither math nor science.
We are asked:
>>.....what is the minimum number of students who could be
taking both math and science this semester?.....<<
So we want to know the smallest possible value of z.
>>......In a group of 150 students,......<<
Therefore, x + y + z + w = 150
>>......more students are taking a math class than are taking
a science class this semester......<<
Therefore x + z > y + z. Subtracting z from both sides,
x > y
>>......If 80 are taking a science class......<<
Therefore y + z = 80
>>......and 25 are not taking either math or science this
semester,......<<
Therefore w = 25
So we have these four facts:
x + y + z + w = 150
x > y
y + z = 80
w = 25
Substituting w = 25 in the first equation,
x + y + z + 25 = 150
x + y + z = 125
Substitute 80 for y + z in that:
x + 80 = 125
x = 45
Substitute x = 45 in x > y
45 > y
Solve y + z = 80 for y
y = 80 - z
Substitute 80 - z for y in 45 > y
45 > 80 - z
z > 80 - 45
z > 35
So the minimum value of z is 36, for
that is the smallest possible whole
number greater than 35.
Edwin</pre>

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