AMC 10 · 2016 · #10
Grade 6 geometry-2d
Pick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The inner length is the one number that controls everything, so tool #4 (Introduce a Variable) names it x and turns every area into an expression in x. The picture is three rectangles sitting inside one another, so tool #7 (Identify Subproblems) breaks the job into pieces: first the size of each nested rectangle, then each color's area as one rectangle minus the next smaller one. Once the three colored areas are written in x, the arithmetic-progression rule (equal gaps) becomes a single equation that pins down x.
Name the inner length
Let the inner rectangle's length be x and its width 1, so its area is x.
Give the one unknown a name and every area in the picture can be measured against it.
6.EE.B.6Use Matrix LogicSize the three rectangles
Each 1-foot border adds 2 to both dimensions, so the middle rectangle is 3x+6 and the whole rug is 5x+20.
A border of width 1 on all sides stretches each dimension by 2, so the rectangles grow in a steady pattern.
4.MD.A.3Identify SubproblemsFind each color's area
Subtract each rectangle from the one around it: the middle color is 2x+6 and the outer color is 2x+14.
Each ring of color is the leftover space after you remove everything it surrounds.
6.EE.A.2Identify SubproblemsUse the equal-gap rule and solve
Equal gaps mean the first difference equals the second, giving x+6=8, so x=2, choice (B).
Equal jumps means the two consecutive differences must match, which leaves one simple equation.
6.EE.B.7Use Matrix LogicName the inner length x, write each color's area in x, then the equal-jumps rule turns into x+6=8, so x=2.
- Name the inner length
- Size the three rectangles
- Find each color's area
- Use the equal-gap rule and solve
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