At a particular university with 150 students 70% receive loans and 40%

Question

At a particular university with 150 students 70% receive loans and 40% receive scholarships. What percent of the students receive neither scholarships nor loans?

(1) At most 20% of the students received scholarships as well as loans.
(2) No fewer than 30 students that receive loans also receive scholarships.

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ManifestDreamMBA · Answer

T = 150 or 100%
L = 70%
S = 40%
T = L+S - B+N
100 = 70+40-B+N
N=B-10 - (1)

We need B to solve this

Statement 1
B<= 20
Using eqn 1 we know a relation b/w B and N
But we don't know how much B can vary b/w 10 and 20, since N cannot be negative. N can be b/w 0 and 10
Insufficient

Statement 2
Bmin = 30
Bmin = 30*100/150 =20%
This tells me N is at-least 10, but we don't know the N max
Insufficient

Combining we know N = 10
Sufficient

Gmat750aspirant · Answer

We need to determine what percentage of the 150 students receive neither scholarships nor loans.

Given: 
- 70% receive loans → 0.7 × 150 = 105 students 
- 40% receive scholarships → 0.4 × 150 = 60 students 
- Let x be the number of students who receive both scholarships and loans. 
- Students receiving either a scholarship or a loan = (105 + 60 - x). 
- Students receiving neither = 150 - (105 + 60 - x) = 150 - 165 + x = x - 15. 
- We need to determine x to find the percentage of students who receive neither.

Evaluating the Statements

(1) At most 20% of the students received both scholarships and loans. 
- 20% of 150 = 30, so x ≤ 30. 
- If x is at most 30, then students receiving neither = x - 15 ≤ 30 - 15 = 15. 
- However, we don’t know the exact value of x, so we cannot determine a specific percentage. 
- Not sufficient.

(2) No fewer than 30 students that receive loans also receive scholarships. 
- This means x ≥ 30. 
- If x is at least 30, then students receiving neither = x - 15 ≥ 30 - 15 = 15. 
- Again, we don’t have an exact value for x, so we cannot determine a specific percentage. 
- Not sufficient.

Evaluating Both Statements Together 
- Statement (1) says x ≤ 30, and Statement (2) says x ≥ 30. 
- This means x = 30. 
- Students receiving neither = 30 - 15 = 15. 
- Percentage of students receiving neither = (15 / 150) × 100 = 10%. 
- Since we can now determine a specific percentage, the statements together are sufficient.

Conclusion 
Answer: C (Statements 1 and 2 together are sufficient, but neither alone is sufficient).

NextstopISB · Answer

Total students = 150 
Students receiving loans = 70% of 150 = 105 
Students receiving scholarships = 40% of 150 = 60 
We need to find the number of students who receive neither scholarships nor loans.

Using the formula for the union of two sets: 
Total students = (Students with loans) + (Students with scholarships) - (Students with both) + (Students with neither) 
=> 150 = 105 + 60 - (Students with both) + (Students with neither) 
=> Students with neither = 150 - (165 - Students with both) 
=> Students with neither = 150 - 165 + Students with both 
=> Students with neither = (Students with both) - 15

Statement (1): 
- At most 20% of students received both scholarships and loans. 
- 20% of 150 = 30, so at most 30 students received both. 
- Using the equation: Students with neither = 30 - 15 = 15 (maximum). 
- But we only know the upper bound, not the exact value. 
- INSUFFICIENT.

Statement (2): 
- At least 30 students received both scholarships and loans. 
- This means Students with both is at least 30. 
- Using the equation: Students with neither = (at least 30) - 15 = at least 15. 
- We only know the lower bound, not the exact value. 
- INSUFFICIENT.

Combining (1) and (2): 
- From (1), Students with both is at most 30. 
- From (2), Students with both is at least 30. 
- So, Students with both = exactly 30. 
- Substituting in the equation: 
 Students with neither = 30 - 15 = 15. 
- SUFFICIENT.

Final Answer: (C) Both statements together are sufficient.

At a particular university with 150 students 70% receive loans and 40%

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GMAT Data Sufficiency (DS) Questions

At a particular university with 150 students 70% receive loans and 40%

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