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2021 AMC Fall 10A Problem 2

2021 AMC Fall 10A Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?

A)
$16$
B)
$17$
C)
$18$
D)
$19$
E)
$20$

Answer

Key insight

Only shortening the 4-inch side gives 3 * 6 = 18, so the other side is the 6-inch one, and 4 * 5 = 20.

Solution

Test which side was shortened. Shortening the $4$ -inch side gives a $3 \times 6$ card of area $18$ ; shortening the $6$ -inch side gives a $4 \times 5$ card of area $20$ . Since the problem says the result was $18$ , the first change was to the $4$ -inch side.

"The other side" is therefore the $6$ -inch side, and shortening it by $1$ inch produces a $4 \times 5$ card with area $4 \cdot 5 = 20$ .

The answer is $\boxed{\textbf{(E)}\ 20}$ .

Why this works

The given area is a clue, not a distraction: it tells you which of the two possible changes actually happened. When a problem has only two scenarios, compute both and let the stated result pick one.

The trap

Assuming the first change shortened the longer side, then shortening the 4-inch side for the answer and getting 3 * 6 = 18, choice (C).

Common mistakes

Assuming the first change shortened the longer side, then shortening the 4-inch side for the answer and getting 3 * 6 = 18, choice (C).
Shortening both sides ( $3 \times 5 = 15$ ) or misreading "the other side" as the same side again.

Techniques

Test small/specific values or special cases to find or verify the answer