AMC 10 · 2008 · #23
Grade 7 geometry-2dA rectangular floor measures a by b feet, where a and b are positive integers and b>a. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair (a,b)?
Pick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A floor is a rectangle measuring a by b feet, with a and b positive integers and b > a. An artist paints a smaller rectangle inside, leaving an unpainted border exactly 1 foot wide all the way around. That border takes up exactly half of the floor's area. Count how many ordered pairs (a, b) make this possible.
Givens: The floor is a by b feet, where a and b are positive integers and b > a; The painted rectangle sits inside with its sides parallel to the floor's sides; The unpainted part is a border of uniform width 1 foot all the way around; The border's area equals half the total floor area
Unknowns: How many ordered pairs (a, b) satisfy all the conditions
Understand
Restated: A floor is a rectangle measuring a by b feet, with a and b positive integers and b > a. An artist paints a smaller rectangle inside, leaving an unpainted border exactly 1 foot wide all the way around. That border takes up exactly half of the floor's area. Count how many ordered pairs (a, b) make this possible.
Givens: The floor is a by b feet, where a and b are positive integers and b > a; The painted rectangle sits inside with its sides parallel to the floor's sides; The unpainted part is a border of uniform width 1 foot all the way around; The border's area equals half the total floor area
Plan
Primary tool: #13 Convert to Algebra
Secondary: #1 Draw a Diagram, #2 Make a Systematic List, #3 Eliminate Possibilities
The wording hides a clean equation. A quick sketch shows the 1-foot border shaves 2 feet off the width and 2 feet off the length, so the painted rectangle is (a-2) by (b-2). The 'half the area' condition then becomes an equation in a and b; rearranged and factored it turns into a product of two integers equal to 8. Counting the valid pairs is just listing the factor pairs of 8 and keeping the ones with b > a.
Execute — Answer: B
6.EE.A.2 Step 1 Sketch the border, size the inner rectangle
- The unpainted border has width 1 foot and runs all the way around the painted rectangle.
- On the left-right direction it removes 1 foot on each side, so 2 feet total off the width; the same happens top-to-bottom, removing 2 feet off the length.
- So if the floor is a by b, the painted rectangle inside is (a-2) by (b-2) feet.
💡 A border of width 1 on both opposite sides eats 2 units off that whole dimension.
7.EE.B.4 Step 2 Turn 'half the area' into an equation
- The border is half the floor, so the painted rectangle must be the other half.
- Its area (a-2)(b-2) therefore equals half of the whole floor's area a*b.
- Multiplying both sides by 2 clears the fraction and gives a tidy equation to work with.
💡 If the border is exactly half, the paint is the other half, so their two areas must be equal.
7.EE.A.1 Step 3 Expand, then factor with the +16 trick
- Expand the left side: 2(a-2)(b-2) = 2ab - 4a - 4b + 8.
- Set it equal to ab and move everything to one side, leaving ab - 4a - 4b + 8 = 0.
- The left side almost factors; adding 8 to both sides makes the constant 16, and then it factors cleanly as (a-4)(b-4).
- This is Simon's Favorite Factoring Trick: choose the constant that lets a two-variable expression become one product.
💡 Adding just the right constant lets a stubborn two-variable expression collapse into a single neat product.
4.OA.B.4 Step 4 List the factor pairs, keep b > a
- Now a-4 and b-4 are two integers whose product is 8, and since b > a we need a-4 < b-4.
- The positive factor pairs of 8 with the smaller factor first are 1 x 8 and 2 x 4.
- From 1 x 8: a-4 = 1 and b-4 = 8, so (a, b) = (5, 12).
- From 2 x 4: a-4 = 2 and b-4 = 4, so (a, b) = (6, 8).
- Negative factor pairs would force a to be 0 or less, which no real floor can have, so they are thrown out.
- Both surviving pairs satisfy b > a and a > 2, so there are 2 ordered pairs, choice (B).
💡 Every way to split 8 into two whole-number factors is one candidate floor, and the rule b > a discards the mirror-image pairs.
6.EE.A.2 The unpainted border has width 1 foot and runs all the way around the painted re 7.EE.B.4 The border is half the floor, so the painted rectangle must be the other half. I 7.EE.A.1 Expand the left side: 2(a-2)(b-2) = 2ab - 4a - 4b + 8. Set it equal to ab and mo 4.OA.B.4 Now a-4 and b-4 are two integers whose product is 8, and since b > a we need a-4 Review
Reasonableness: Test both floors against the story. For (5, 12): total area is 60, the painted rectangle is 3 by 10 = 30, exactly half, so the border is also 30 - correct. For (6, 8): total area is 48, the painted rectangle is 4 by 6 = 24, exactly half - correct. Could there be more? The number 8 has only the positive factor pairs {1, 8} and {2, 4}; their reverses give b < a and are excluded, and negative pairs give impossible floor sizes. So exactly 2 pairs survive, matching choice (B) and ruling out 1, 3, 4, and 5.
Alternative: Skip the factoring and guess-and-check a in order. From (a-2)(b-2) = ab/2 you can solve for b at each a. a = 3 and a = 4 give no valid b (b comes out negative or the equation is impossible). a = 5 gives 3(b-2) = 2.5b, so b = 12. a = 6 gives 4(b-2) = 3b, so b = 8. a = 7 gives a non-integer b, and from a = 8 on the solution has b less than or equal to a, which violates b > a. Scanning stops, and again only (5, 12) and (6, 8) work.
CCSS standards used (min grade 7)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Writing the painted rectangle's dimensions and area as expressions (a-2)(b-2) in the unknown side lengths)7.EE.B.4Use variables to represent quantities and construct simple equations to solve problems (Translating the 'painted area is half the floor' condition into the equation 2(a-2)(b-2) = ab)7.EE.A.1Apply properties of operations to add, subtract, factor, and expand expressions (Expanding the product and using the +16 completing-the-constant step to factor into (a-4)(b-4) = 8)4.OA.B.4Find all factor pairs for a whole number in the range 1-100 (Listing every factor pair of 8 to generate the candidate (a, b) floors)
⭐ A one-foot border shrinks each side by two; write the half-area rule as an equation, nudge it into a neat product, and count the factor pairs.
⭐ A one-foot border shrinks each side by two; write the half-area rule as an equation, nudge it into a neat product, and count the factor pairs.
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