In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number I and the smallest possible number of students that could have taken both geometry and biology? Choose the correct choice
A) 144  B) 119  C) 113  D) 88  E) 31
OFFICIAL SOLUTION:   First of all, notice that since 144 took geometry and 119 took biology, then the number of students who took both geometry and biology cannot be greater than 119.   {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.   The greatest possible number I of students that could have taken both geometry and biology, is 119.   Thus, the answer is 119 - 31= 
<gadget id="calculator">119 - 31</gadget>
<output>88</output>
88.   Answer: 88.
<result>D</result>
In a graduating class of 232 students, 144 took geometry and 119 took biology. What is the difference between the greatest possible number I and the smallest possible number of students that could have taken both geometry and biology?