M31-14

Question

In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?

A. 144
B. 119
C. 113
D. 88
E. 31

A. 144
B. 119
C. 113
D. 88
E. 31

Bunuel · Accepted Answer

Official Solution:

In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology?

A. 144
B. 119
C. 113
D. 88
E. 31

A. 144
B. 119
C. 113
D. 88
E. 31

First of all, notice that since 144 took geometry and 119 took biology, the number of students who took both geometry and biology cannot be greater than 119.
 
Therefore, the  greatest possible number of students that could have taken both geometry and biology, is 119 .
 
Next:
 
{Total} = {geometry} + {biology} - {both} + {neither};
 
232 = 144 + 119 - {both} + {neither};
 
{both} = 31 + {neither}.
 
{both} is minimized when {neither} is 0. In this case {both} = 31.
 
Therefore, the  smallest possible number of students that could have taken both geometry and biology, is 31 .
 
Thus, the answer is 119 - 31 = 88.

Answer: D

anuharidas · Answer

HI Bunel,

I approached the problem the same way but eliminated the correct answer in the following grounds,

if the greatest number 0f students learning biology and geometry is 119 , how can the total be 232 ??

the total be just 144 where 119 students study both and 25 students study biology .. if that is the case the total wont be 232 ........please clarify this ambiguity .. Please make me understand where am I going wrong .

Bunuel · Answer

You missed the number of students who took neither geometry nor biology.

{Total} = {geometry} + {biology} - {both} + {neither}

For he greatest possible number of students that could have taken both geometry and biology (119):
232 = 144 + 119 - 31 + 0

For he smallest possible number of students that could have taken both geometry and biology (31):
232 = 144 + 119 - 119 + 88

ScottTargetTestPrep · Answer

Solution:

The greatest possible number of students that could have taken both courses is 119, which occurs if all the students who took biology also took geometry. The smallest possible number of students that could have taken both courses is 144 + 119 - 232 = 31, which occurs if we maximize the number of students taking only one course. Therefore, the difference is 119 - 31 = 88.

Answer: D

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

GobaChatro · Answer

Thank you. This explanation has completely cleared up my confusion.

Rahilgaur · Answer

Maximum number of students that could have taken both subjects = 119
Let minimum no. of such students = x
232=119-x+144
x= 31

Answer = 119-31= 88

Posted from my mobile device

AbhishekP220108 · Answer

Can be solved easily by just making matrix
M1
 G -G
 B 119 00 | 119
 
-B 25 88 | 113
_________
 144 88 T=232
M2
 G -G
 B 31 88 | 119
 
-B 113 00 | 113
_________
 144 88 T=232

119-31=88

phmahi1997 · Answer

I like the solution - it’s helpful.

pyreddy · Answer

M31-14

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