Percentages: There are 600 students in a school. The number of girls was at least 40% of the total…

Think of this as: we know how many boys and girls picked each sport, and everyone picked at least one. The number doing both is fixed once we fix the number of girls — so the problem reduces to finding the allowed range for the number of girls, then checking the endpoints.

Step 1 — Set up variables and translate the constraints.

Let $G$ = number of girls, $B$ = number of boys. We know $B + G = 600$, so $B = 600 - G$.

The problem gives two conditions on $G$:

1. Girls are at least 40% of total: $G \ge 0.40 \times 600 = 240$

2. Girls do not exceed boys: $G \le B = 600 - G \implies 2G \le 600 \implies G \le 300$

So $G$ can range over the interval $[240, 300]$.

Step 2 — Express the number of students in each sport.

• Swimming: $0.45B + 0.85G$
• Running: $0.75B + 0.55G$

Step 3 — Use the inclusion–exclusion principle to find "both."

Since every student chose at least one sport, the union (at least one sport) equals the entire school = 600.

$\text{Both} = \text{Swimming} + \text{Running} - \text{Union}$

$\text{Both} = (0.45B + 0.85G) + (0.75B + 0.55G) - 600$

$\text{Both} = 1.20B + 1.40G - 600$

Step 4 — Substitute $B = 600 - G$ to get "both" in terms of $G$ alone.

$\text{Both} = 1.20(600 - G) + 1.40G - 600$

$= 720 - 1.20G + 1.40G - 600$

$= 120 + 0.20G$

💡 Teacher tip: The expression $120 + 0.20G$ is linear in $G$ with a positive coefficient ($0.20 > 0$). That means "both" increases as $G$ increases. So the minimum and maximum must occur at the two endpoints of $G$'s range — no interior value can beat them.

Step 5 — Evaluate at the endpoints.

• Minimum number of students doing both $= 168$ (when $G = 240$, $B = 360$)
• Maximum number of students doing both $= 180$ (when $G = 300$, $B = 300$)

Step 6 — Match with the options.

The minimum and maximum are 168 and 180, respectively, which corresponds to Option A.

Answer: 168 and 180 (Option A)

💡 Why this works so cleanly: Once "both" turned out to be a linear function of $G$, the entire problem collapsed to checking two endpoints. Whenever a quantity is linear in a variable that ranges over an interval, the extremes always sit at the boundary — never in the middle.