Competition · AMC preparation · step 4 of 4
AMC 8 · 2020 · #25
Grade 8 geometry-2dalgebra
Pick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The figure is given but it pays to redraw it and label every segment with s₁, s₂, s₃ — that is Tool #1. Tool #7 (Identify Subproblems) splits the geometry into two independent equations: one from the horizontal direction (total width) and one from the vertical direction (height of the middle column). With two equations linking s₁, s₂, s₃ to the known numbers 3322 and 2020, Tool #13 (Convert to Algebra) lets us add or subtract the equations to eliminate s₁ and s₃ in a single move and read off s₂ — no need to solve for all three side lengths.
Redraw and label the squares
Redraw and label: the three squares' widths sit side by side across the whole top edge, S₁ then S₂ then S₃.
Labeling each segment turns a picture into a sentence: "these three widths together make the long side."
4.G.A.1Draw A DiagramWrite the width equation
The labeled top edge equals the given width 3322 — that is the first equation.
Writing the picture's width as a sum of unknowns is exactly using variables to describe a real quantity.
6.EE.B.6Identify SubproblemsWrite the height equation
Each side column is 2020 tall, so reading the middle gap gives s₁ - s₂ + s₃ = 2020 — the second equation.
Stacking two side columns of height 2020 and reading what is left for the middle square turns the vertical layout into one clean equation.
6.EE.B.7Identify SubproblemsSubtract to eliminate two sides
Subtract Eq. 2 from Eq. 1 — s₁ and s₃ cancel, s₂ doubles: 2 s₂ = 1302.
When two equations share most variables, subtracting them lines up the unknowns so they vanish — a classic simultaneous-equations move.
Subtracting the height equation s₁ - s₂ + s₃ = 2020 from the width equation s₁ + s₂ + s₃ = 3322 makes s₁ and s₃ cancel and leaves 2s₂ = 1302.
▸ Why?
Both equations are true totals, so taking away the whole of the second from the whole of the first keeps a valid equation, and on the number side that difference is 3322 - 2020 = 1302.
▸ Why?
On the symbol side the s₁ and s₃ terms are identical in both equations so they subtract away, while the s₂ term, added in one equation and taken away in the other, builds up to 2s₂.
▸ Why?
Combining the two s₂ terms means pulling out the shared s₂: s₂ - (-s₂) = s₂ (1 + 1) = 2s₂, which is just gathering two equal parts.
▸ Why?
For s₁ and s₃ the pulled-out count is 1 - 1 = 0, so each side length is taken zero times and adds nothing to the result.
Solve for the middle square
Divide by 2: s₂ = 651, matching choice (A).
Dividing a 4-digit number by 2 is straightforward whole-number division.
5.NBT.B.6Convert To AlgebraThis AMC 8 problem only needs Grade 8 simultaneous equations you already know — two pictures of the rectangle's edges become two equations, and subtracting them cancels the squares you don't care about!
- Redraw and label the squares
- Write the width equation
- Write the height equation
- Subtract to eliminate two sides
- Solve for the middle square
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