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08 Aug 2026, 17:24
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Why aren't we taking -ve sign into consideration even when it is already mentioned that students enrolments have decreased from fall 1999 to spring 2004?
10 Aug 2026, 14:33
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Hi Amity007,
Great question, and it's the exact spot where almost everyone in this thread got stuck too. The key is what the variable S actually stands for.
S is the percent change, not the size of the decrease. The problem defines it with a signed formula: (X_New − X_Old) / X_Old × 100. Because enrollment fell, that formula already spits out a negative number for S. So the minus sign you're looking for isn't missing - it's baked inside S.
That's why the students-in-2004 factor is written as (100 + S)/100, not (100 − S)/100:
- "100" means the original 100% of students.
- Adding S adjusts for the change.
- Since S is already negative, adding it lowers the total - exactly the decrease you expect.
If you instead wrote (100 − S), you'd be subtracting a negative, which flips it back to an increase - the opposite of what happened.
See it with real numbers
Say enrollment dropped 20%. Then by the formula, S = -20.
- Correct form: 100 + S = 100 + (-20) = 80 → 80% of the original. ✓ (a 20% drop)
- Wrong form: 100 − S = 100 − (-20) = 120 → 120%. ✗ (a 20%rise)
So (100 + S) faithfully captures the decline; (100 − S) would undo it.
This is why the answer uses (100+S)/(100+F) × R rather than the option with (100−S). The algebra holds no matter the sign - you just plug in the signed value of S, and the decrease takes care of itself. Same logic applies to F: faculty grew, therefore F is positive, and 100 + F correctly increases that count.
Answer: Column 1 = 5,500/R; Column 2 = (100+S)/(100+F) × R
Great question, and it's the exact spot where almost everyone in this thread got stuck too. The key is what the variable S actually stands for.
S is the percent change, not the size of the decrease. The problem defines it with a signed formula: (X_New − X_Old) / X_Old × 100. Because enrollment fell, that formula already spits out a negative number for S. So the minus sign you're looking for isn't missing - it's baked inside S.
That's why the students-in-2004 factor is written as (100 + S)/100, not (100 − S)/100:
- "100" means the original 100% of students.
- Adding S adjusts for the change.
- Since S is already negative, adding it lowers the total - exactly the decrease you expect.
If you instead wrote (100 − S), you'd be subtracting a negative, which flips it back to an increase - the opposite of what happened.
See it with real numbers
Say enrollment dropped 20%. Then by the formula, S = -20.
- Correct form: 100 + S = 100 + (-20) = 80 → 80% of the original. ✓ (a 20% drop)
- Wrong form: 100 − S = 100 − (-20) = 120 → 120%. ✗ (a 20%rise)
So (100 + S) faithfully captures the decline; (100 − S) would undo it.
This is why the answer uses (100+S)/(100+F) × R rather than the option with (100−S). The algebra holds no matter the sign - you just plug in the signed value of S, and the decrease takes care of itself. Same logic applies to F: faculty grew, therefore F is positive, and 100 + F correctly increases that count.
Answer: Column 1 = 5,500/R; Column 2 = (100+S)/(100+F) × R
Amity007
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