Kate bakes a -inch by -inch pan of cornbread. The cornbread is cut into pieces that measure inches by inches. How many pieces of cornbread does the pan contain?
- A)
- B)
- C)
- D)
- E)
Answer
A
Key insight
Count how many 2-inch pieces fit along each side of the pan, 10 along one side and 9 along the other, then multiply.
Solution
Picture the pan cut into a grid of little squares, then count rows and columns.
- Along the -inch side, each piece takes up inches, so pieces fit in a row.
- Along the -inch side, rows fit.
- Both divisions come out even, so the grid fills the pan with nothing left over.
- A grid with pieces in each of rows has pieces.
The answer is .
Why this works
To count things laid out in a grid, multiply the number in one row by the number of rows. First check that the piece fits each side evenly. When it does, dividing the areas gives the same answer.
Alternative approach
Divide areas. The pan is square inches. Each piece is square inches. So there are pieces.
The trap
Dividing the area 360 by 2, the side of a piece, instead of by 4, the area of a piece. That gives 180.
Concise solution
Along the -inch side the pan is cut into pieces; along the -inch side into pieces. Since both dimensions divide evenly, the pieces form a grid with nothing left over:
The answer is .
Review in the AoPS books
- Introduction to Counting & Probability, Chapter 1: Counting Is Arithmetic
- Introduction to Geometry, Chapter 8: Quadrilaterals
- Introduction to Counting & Probability, Chapter 2: Basic Counting Techniques
- Introduction to Geometry, Chapter 9: Polygons (§9.3)
Techniques
Set up the equation/formula and compute; no special trick needed