Question

In the second year, students at a business school opt for exactly one of three electives: Systems, Operations, or HR.

The number of girls who opted for Operations plus the number of boys who opted for Systems is 37. Twenty-two students opted for the Operations elective. Twenty girls opted for the Systems and Operations electives put together. The number of students who opted for the Systems elective plus the number of boys who opted for the Operations elective is 37. Twenty-five students opted for the HR elective.

The number of students in the second year is

• A. 73
• B. 74
• C. 75
• D. 76

Hint:

Write four equations from the four given sums, using girls-in-Systems, girls-in-Operations, boys-in-Systems and boys-in-Operations as your four unknowns, then add Systems, Operations and HR.

Answer 1

The Correct Option is D

Step 1: Translate the clues, keeping the goal in mind.
Let p = girls in Systems, q = girls in Operations, r = boys in Systems, s = boys in Operations. Since Operations = 22 and HR = 25 are already given directly, the only thing missing for the total is the Systems count, which is $p + r$.

Step 2: Write Systems in two different ways.
From "girls in Operations plus boys in Systems is 37": $r = 37 - q$. From "twenty girls opted for Systems and Operations": $p = 20 - q$. Adding these:
\[ \text{Systems} = p + r = (20 - q) + (37 - q) = 57 - 2q \]
This is one expression for Systems, written purely in terms of $q$.

Step 3: Get a second expression for Systems from the fourth clue.
"Students in Systems plus boys in Operations is 37" means:
\[ \text{Systems} + s = 37 \implies \text{Systems} = 37 - s \]
From "22 students opted for Operations": $q + s = 22$, so $s = 22 - q$. Substituting:
\[ \text{Systems} = 37 - (22 - q) = 15 + q \]

Step 4: Set the two expressions equal and solve for q.
Both expressions describe the same quantity, Systems, so:
\[ 57 - 2q = 15 + q \implies 42 = 3q \implies q = 14 \]

Step 5: Find Systems and add up the total.
Using either expression, Systems $= 15 + 14 = 29$ (check with the other: $57 - 28 = 29$, matches). Adding all three electives:
\[ \text{Total} = 29 + 22 + 25 = 76 \]

Final Answer:
The number of students in the second year is 76.\[ \boxed{76} \]