AMC 10 · 2021 · #2

Easy mode Grade 3

Problem

A card is a rectangle, $4$ inches on one side and $6$ inches on the other.

Make one of its sides $1$ inch shorter. The card that is left has area $18$ square inches.

Now go back to the original card and make the other side $1$ inch shorter instead. What is the area of that card, in square inches?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: An index card measures $4$ inches by $6$ inches. One of its two side lengths gets cut down by $1$ inch, and the card that remains has area $18$ square inches. The problem never says which side was cut. Find the area of the card you would get if the other side were the one cut down by $1$ inch instead.

Givens: The card starts as a $4 \times 6$ rectangle, so its area is $4 \times 6 = 24$ square inches; Shortening one side by $1$ inch leaves a card of area $18$ square inches; Answer choices: (A) $16$, (B) $17$, (C) $18$, (D) $19$, (E) $20$ square inches

Unknowns: The area, in square inches, of the card obtained by shortening the other side by $1$ inch

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #3 Eliminate Possibilities, #1 Draw a Diagram

The missing piece of information — which side got shortened — has only two candidates, so Tool #2 (Make a Systematic List) writes down both cards and both areas in a two-row table instead of guessing. Tool #3 (Eliminate Possibilities) then uses the one number the problem hands over, the area $18$, to cross off the row that does not fit; the surviving row identifies which side was actually cut, and the crossed-off row is exactly the card the question asks about. Tool #1 (Draw a Diagram) keeps the two cuts straight: sketching the rectangle shows that trimming a side removes a thin strip along one edge, which makes it obvious the two cuts are not interchangeable.

Execute — Answer: E

#1 Draw a Diagram 3.MD.C.7 Step 1

See the card and its two cuts

Sketch the card as a rectangle $4$ inches on one pair of sides and $6$ inches on the other pair.
Its area is side times side, $4 \times 6 = 24$ square inches.
Shortening a side by $1$ inch means shaving a $1$-inch strip off one edge, and there are only two edges to shave: the $4$-inch dimension can drop to $3$, or the $6$-inch dimension can drop to $5$.

$$4 \times 6 = 24 \text{ sq in};\qquad \text{cut A: } 4 \to 3,\qquad \text{cut B: } 6 \to 5$$

💡 A rectangle has just two different side lengths, so shortening "a side" can only mean one of two things.

#2 Make a Systematic List 3.MD.C.7 Step 2

List both resulting cards

Write out a two-row table.
Cut A shrinks the $4$ to $3$ and leaves the $6$ alone, giving a $3 \times 6$ card of area $3 \times 6 = 18$ square inches.
Cut B shrinks the $6$ to $5$ and leaves the $4$ alone, giving a $4 \times 5$ card of area $4 \times 5 = 20$ square inches.
Both numbers come from the same rule: area is the product of the two side lengths.

$$\text{Cut A: } 3 \times 6 = 18;\qquad \text{Cut B: } 4 \times 5 = 20$$

💡 With only two possibilities, computing both is faster and safer than trying to reason out which one is meant.

#3 Eliminate Possibilities 3.OA.D.8 Step 3

Let the $18$ pick the row

The problem says the cut actually performed produced area $18$.
Only Cut A gives $18$, so Cut A is what happened: the side that was shortened was the $4$-inch side.
That eliminates Cut B as the description of what she did — and Cut B is precisely the "other side" the question turns to.

$$18 = 3 \times 6 \;\Rightarrow\; \text{the shortened side was the } 4\text{-inch side}$$

💡 The given area is not decoration — it is the clue that tells you which of the two cuts the story is describing.

#2 Make a Systematic List 3.MD.C.7 Step 4

Read off the other row

Shortening the other side means leaving the $4$-inch side alone and cutting the $6$-inch side down to $5$.
That is the $4 \times 5$ card already in the table, with area $4 \times 5 = 20$ square inches.
As a check, the original card is $24$ square inches, and this cut removes a strip $1$ inch wide and $4$ inches long, so $24 - 4 = 20$ agrees.
The answer is (E).

$$4 \times 5 = 20 \text{ sq in} \;=\; 24 - (1 \times 4) \;\Rightarrow\; \textbf{(E)}$$

💡 Once you know which cut was described, the answer is the row of the table you did not use.

[1] #1 3.MD.C.7 Sketch the card as a rectangle $4$ inches on one pair of sides and $6$ inches on

[2] #2 3.MD.C.7 Write out a two-row table. Cut A shrinks the $4$ to $3$ and leaves the $6$ alone

[3] #3 3.OA.D.8 The problem says the cut actually performed produced area $18$. Only Cut A gives

[4] #2 3.MD.C.7 Shortening the other side means leaving the $4$-inch side alone and cutting the

Review

Reasonableness: Cutting $1$ inch off a side always removes a strip whose area equals the length of the other side, so the two cuts remove $6$ and $4$ square inches from the original $24$. The cut that removed $6$ gave $18$, so the cut that removed only $4$ must give something larger than $18$ but smaller than $24$ — and $20$ sits exactly there. Another guard: the new area has to be a product of whole side lengths taken from $\{3,4\}$ and $\{5,6\}$, which rules out (D) $19$ (prime) and (B) $17$ (prime) immediately. Choice (C) $18$ is the trap of repeating the area the problem already gave, and (A) $16$ would require shrinking both sides rather than one.

Alternative: Skip the table and work with the strip that gets removed. Trimming $1$ inch off a side deletes a rectangle $1$ inch wide whose length is the other side. Since the area fell from $24$ to $18$, the removed strip had area $6$, so its length was $6$ — meaning the trimmed side was the $4$-inch one. The other trim removes a strip of area $1 \times 4 = 4$, leaving $24 - 4 = 20$ square inches.

CCSS standards used (min grade 3)

3.MD.C.7 Relate area to multiplication and addition operations (Finding the area of each rectangular card by multiplying its two side lengths ($4 \times 6$, $3 \times 6$, $4 \times 5$) and seeing a trimmed edge as a removed strip of area.)
3.OA.D.8 Solve two-step word problems using four operations within 100 (Using the given area $18$ to decide which of the two cuts the story describes, then carrying that decision into a second multiplication for the other cut.)

⭐ When a problem hides which side changed, try both — the number it hands you tells you which case really happened, and the case you crossed off is usually what it asks for next.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

See the card and its two cuts

List both resulting cards

Let the $18$ pick the row

Read off the other row

Review

CCSS standards used (min grade 3)

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