Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #8
Grade 6 geometry-2dPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The answer doesn't depend on the actual size of the rectangle, so Tool #9 (Solve an Easier Related Problem) says: replace L and W with friendly numbers — pick L = W = 10 — and compute both areas directly. Tool #1 (Draw a Diagram) keeps the picture honest: sketch the 10 × 10 rectangle, then the new 11 × 9 rectangle, and compare areas by counting. This skips the algebra route (Tool #13) entirely and shows why +10% and -10% don't cancel.
Pick easy side lengths
Pick friendly numbers: let L = 10 and W = 10, so the old area is 100.
Using 100 as the old area is a deliberate trick: any final answer in "percent" becomes the new area itself, no extra division needed.
4.MD.A.3Solve An Easier Related ProblemApply the percent changes
10% of 10 is 1, so the new length is 11 and the new width is 9.
Draw the new 11 × 9 rectangle next to the original 10 × 10 square — one side got a little longer, the other a little shorter.
6.RP.A.3Draw A DiagramCompute the new area
The new area is just 11 × 9 = 99 — one less than the old 100.
An 11 × 9 grid has 99 unit squares — one less than the original 100, even though both sides only changed by 1.
With the sides taken as 10 and 10, growing one side by 10% and shrinking the other by 10% makes the new 11 × 9 rectangle cover a little less than the original 10 × 10 square, so the two percentage changes do not undo each other.
▸ Why?
The new rectangle measures 11 by 9, and its area is the number of unit squares filling an 11 × 9 grid.
▸ Why?
An 11 × 9 rectangle holds 11 equal rows of 9 unit squares, so counting the squares is the product 11 × 9.
▸ Why?
That product 11 × 9 equals 10 × 10 minus 1, so it sits just below the original area of 10 × 10.
▸ Why?
Writing the sides as (10 + 1) and (10 - 1), the product opens into 10 × 10 - 10 × 1 + 1 × 10 - 1 × 1, where the two middle strips cancel and only -1 is left over.
▸ Why?
Multiplying a sum by a sum lets you multiply each part of one factor by each part of the other and add all the pieces.
▸ Why?
The strip gained on the longer side, 10 × 1, and the strip lost on the shorter side, 1 × 10, are the same size, so their difference is zero.
▸ Why?
10 × 1 and 1 × 10 count the very same block of squares just turned on its side, so they are equal.
Compare to the old area
Since the old area is exactly 100, the new area 99 is already the percent: 99%.
The whole reason we picked L = W = 10 was to make this last conversion free.
6.RP.A.3Solve An Easier Related ProblemThis AMC 8 problem only needs Grade 6 percent reasoning — and a clever Grade 4 trick: pick L = W = 10 so the old area is exactly 100 and the answer falls out!
- Pick easy side lengths
- Apply the percent changes
- Compute the new area
- Compare to the old area
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