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2016 AMC 10B Problem 2

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2016 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Functions · difficulty 1 of 5 · about 1 min · position P1-10

If $n\heartsuit m=n^3m^2$ , what is $\frac{2\heartsuit 4}{4\heartsuit 2}$ ?

A)
$\frac{1}{4}$
B)
$\frac{1}{2}$
C)
$1$
D)
$2$
E)
$4$

Answer

Key insight

Apply the definition literally: 2 heart 4 = 2^3 * 4^2 = 128 and 4 heart 2 = 4^3 * 2^2 = 256, so the ratio is 1/2.

Solution

The first input is cubed and the second is squared.

$2\heartsuit 4 = 2^3 \cdot 4^2 = 8 \cdot 16 = 128, \qquad 4\heartsuit 2 = 4^3 \cdot 2^2 = 64 \cdot 4 = 256.$

Therefore
$\frac{2\heartsuit 4}{4\heartsuit 2} = \frac{128}{256} = \frac12.$

The answer is $\boxed{\textbf{(B)}\ \frac{1}{2}}$ .

Why this works

A made-up operation is just a two-variable function; the only skill is matching each input to its slot in the formula. Writing the two values out fully before dividing prevents slot mix-ups, and powers of $2$ make the arithmetic quick.

Alternative approach

Without computing each side: $\frac{n^3m^2}{m^3n^2} = \frac{n}{m}$ , so the ratio is simply $\frac{2}{4} = \frac12$ .

The trap

Swapping the roles of the two inputs (cubing the second and squaring the first), which flips the ratio to 2.

Common mistakes

Swapping the roles of the two inputs (cubing the second and squaring the first), which flips the ratio to $2$ .
Assuming the operation is commutative and answering $1$ without computing.

Techniques

Set up the equation/formula and compute; no special trick needed