The value of a minted coin is directly proportional to its thickness and inversely proportional to its area. What is the

[B BTGmoderatorLU]

The value of a minted coin is directly proportional to its thickness and inversely proportional to its area. What is the value of a coin with radius \(10\, \text{mm}\) and thickness \(1\,\text{mm}?\)

1. Value of a coin with radius \(5\, \text{mm}\) and thickness \(2\, \text{mm}\) is \(\$1\)

2. Value of a coin with radius \(2.5\,\text{mm}\) and thickness \(1\,\text{mm}\) is \(\$2\)

The OA is D

[BTG Expert Team]

What we see: a Data Sufficiency value question. We need the value of a coin with radius 10 mm and thickness 1 mm.

Step 1. Turn the words into a formula.
What we see: value is directly proportional to thickness and inversely proportional to area. A coin's face is a circle, so its area is \(\pi r^2\).
What it means: \(V = k \cdot \dfrac{t}{\pi r^2}\), where \(k\) is one fixed number we do not know yet.
What we do and why: our coin is worth \(V = \dfrac{k \cdot 1}{\pi \cdot 10^2} = \dfrac{k}{100\pi}\). The only unknown is \(k\). Any one coin with a known radius, thickness and value gives us \(k\).

Step 2. Statement 1 alone.
What we see: radius 5, thickness 2, value $1.
What we do: \(1 = \dfrac{k \cdot 2}{25\pi}\), so \(k = \dfrac{25\pi}{2}\). Then our coin is worth \(\dfrac{25\pi/2}{100\pi} = \dfrac{1}{8}\), or $0.125.
What it means: one value. Sufficient.

Step 3. Statement 2 alone.
What we see: radius 2.5, thickness 1, value $2.
What we do: \(2 = \dfrac{k}{6.25\pi}\), so \(k = 12.5\pi\). Our coin is worth \(\dfrac{12.5\pi}{100\pi} = \dfrac{1}{8}\) again.
What it means: one value. Sufficient.

The trap: C. Two numbers seem to be missing, but proportional means just one hidden constant, and either coin fixes it.
Faster way: compare with the target coin. In statement 1 the radius doubles, so the area is 4 times as big (value divided by 4), and the thickness halves (value divided by 2). $1 divided by 8. No need to finish the math in Data Sufficiency once you see one constant is all you need.
Takeaway: "Directly" or "inversely proportional" means one unknown constant. One complete example pins it down.

Answer: D