2021 AMC Fall 10A Problem 2

P1-10~1 minPrint

Linear Equations & Word Problems Quadrilaterals & Polygon Areas Test small/specific values or special cases to find or verify the answer

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?

Show solution without answering

Key insight

Try both sides. Only cutting the 4-inch side gives 3 by 6 = 18, so the other side is the 6-inch side, and 4 by 5 = 20.

Solution

The card is $4$ by $6$ , so there are only two sides she could shorten. Try both and see which one gives $18$ .

Shorten the $4$ -inch side: the card becomes $3$ by $6$ , with area $3 \times 6 = 18$ .
Shorten the $6$ -inch side: the card becomes $4$ by $5$ , with area $4 \times 5 = 20$ .
The problem says the area would be $18$ , so the first change was to the $4$ -inch side.
The question asks about the other side, which is the $6$ -inch side. From step 2, that gives area $20$ .

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

Why this works

The area $18$ is not extra information. It tells you which side was shortened first. When a problem allows only two possibilities, work out both, and let the fact the problem gives you choose between them.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(E)}\ 20}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Geometry, Chapter 8: Quadrilaterals
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Introduction to Geometry, Chapter 9: Polygons (§9.3)