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: Name the four unknown groups.
Since every student picks exactly one of Systems, Operations or HR, split Systems and Operations by gender. Let p = girls in Systems, q = girls in Operations, r = boys in Systems, s = boys in Operations.

Step 2: Turn each sentence into an equation.
"Girls in Operations plus boys in Systems is 37" gives \(q + r = 37\). "Twenty-two students opted for Operations" gives \(q + s = 22\). "Twenty girls opted for Systems and Operations put together" gives \(p + q = 20\). "Students in Systems plus boys in Operations is 37" gives \((p + r) + s = 37\).

Step 3: Reduce to one unknown.
From the third equation, \(p = 20 - q\). From the second equation, \(s = 22 - q\). Substitute both into the fourth equation:
\[ (20 - q + r) + (22 - q) = 37 \] \[ 42 - 2q + r = 37 \] \[ r = 2q - 5 \]

Step 4: Solve for q, then the rest.
Put \(r = 2q - 5\) into the first equation, \(q + r = 37\):
\[ q + (2q - 5) = 37 \implies 3q = 42 \implies q = 14 \]
So \(r = 2(14) - 5 = 23\), \(p = 20 - 14 = 6\), and \(s = 22 - 14 = 8\).

Step 5: Add up the second year.
Systems total \(= p + r = 6 + 23 = 29\). Operations total is already given as 22, and HR total is given as 25. So the second year has:
\[ 29 + 22 + 25 = 76 \]
The nearby wrong options (73, 74, 75, 77) all come from a slip somewhere in this chain, such as mixing up which sum is 20 and which is 37, or forgetting that the Systems total is \(p + r\) and not just one of the two parts. Working through the four equations in the order above avoids that.

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