AMC 8 · 2020 · #25
Grade 8 geometry-2dalgebraProblem
Rectangles R1 and R2, and squares S1,S2, and S3, shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of S2 in units?
Pick an answer.
AMC 8 2020 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 big rectangle measuring $3322$ wide and $2020$ tall is tiled by three squares $S_1, S_2, S_3$ (side lengths $s_1, s_2, s_3$) and two rectangles $R_1, R_2$. From the diagram, $S_1$ sits in the upper-left, $R_1$ directly below it, $S_2$ is the middle square sharing the same vertical column as $R_1$'s right edge through $S_1$'s right edge, $S_3$ sits in the lower-right, and $R_2$ is above $S_3$. Find $s_2$.
Givens: Outer rectangle width $= 3322$; Outer rectangle height $= 2020$; $S_1, S_2, S_3$ are squares with side lengths $s_1, s_2, s_3$; $R_1$ is directly below $S_1$ and $R_2$ is directly above $S_3$; Answer choices: (A) $651$, (B) $655$, (C) $656$, (D) $662$, (E) $666$
Unknowns: The side length $s_2$ of the middle square $S_2$
Understand
Restated: A big rectangle measuring $3322$ wide and $2020$ tall is tiled by three squares $S_1, S_2, S_3$ (side lengths $s_1, s_2, s_3$) and two rectangles $R_1, R_2$. From the diagram, $S_1$ sits in the upper-left, $R_1$ directly below it, $S_2$ is the middle square sharing the same vertical column as $R_1$'s right edge through $S_1$'s right edge, $S_3$ sits in the lower-right, and $R_2$ is above $S_3$. Find $s_2$.
Givens: Outer rectangle width $= 3322$; Outer rectangle height $= 2020$; $S_1, S_2, S_3$ are squares with side lengths $s_1, s_2, s_3$; $R_1$ is directly below $S_1$ and $R_2$ is directly above $S_3$; Answer choices: (A) $651$, (B) $655$, (C) $656$, (D) $662$, (E) $666$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems, #13 Convert to Algebra
The figure is given but it pays to redraw it and label every segment with $s_1, s_2, s_3$ — 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 $s_1, s_2, s_3$ to the known numbers $3322$ and $2020$, Tool #13 (Convert to Algebra) lets us add or subtract the equations to eliminate $s_1$ and $s_3$ in a single move and read off $s_2$ — no need to solve for all three side lengths.
Execute — Answer: A
4.G.A.1 Step 1 - Redraw and label.
- Mark the side lengths $s_1, s_2, s_3$ on each square in the figure and identify which segments line up along the top edge of the big rectangle.
- The three squares' widths sit side-by-side across the entire top, with $S_1$ on the left (above $R_1$), $S_2$ in the middle, $S_3$ on the right (above which is $R_2$).
💡 Labeling each segment turns a picture into a sentence: "these three widths together make the long side."
6.EE.B.6 Step 2 - Read the width equation.
- The labeled top edge equals the given width $3322$, giving 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.7 Step 3 - Read the height.
- The left column ($S_1$ above $R_1$) has total height $2020$, so $R_1$'s height is $2020 - s_1$.
- The right column ($S_3$ below $R_2$) has total height $2020$, so $R_2$'s height is $2020 - s_3$.
- In the middle column, $S_2$ exactly fills the vertical gap between the top of $R_1$ (height $2020 - s_1$ from the bottom) and the bottom of $R_2$ (height $s_3$ from the bottom).
💡 Stacking two side columns of height $2020$ and reading what is left for the middle square turns the vertical layout into one clean equation.
8.EE.C.8 Step 4 - Eliminate $s_1$ and $s_3$ in one stroke.
- Subtract Eq.
- 2 from Eq.
- 1: the $s_1$ and $s_3$ terms cancel, and the $s_2$ terms reinforce.
💡 When two equations share most variables, subtracting them lines up the unknowns so they vanish — a classic simultaneous-equations move.
5.NBT.B.6 Step 5 Solve for $s_2$.
💡 Dividing a 4-digit number by $2$ is straightforward whole-number division.
4.G.A.1 Redraw and label. Mark the side lengths $s_1, s_2, s_3$ on each square in the fi 6.EE.B.6 Read the width equation. The labeled top edge equals the given width $3322$, giv 6.EE.B.7 Read the height. The left column ($S_1$ above $R_1$) has total height $2020$, so 8.EE.C.8 Eliminate $s_1$ and $s_3$ in one stroke. Subtract Eq. 2 from Eq. 1: the $s_1$ an 5.NBT.B.6 Solve for $s_2$. Review
Reasonableness: The answer must lie between $0$ and $\min(3322, 2020) = 2020$, and $651$ fits comfortably. As a sanity check, since $s_1 + s_2 + s_3 = 3322$ and $s_1 + s_3 - s_2 = 2020$, adding the two equations gives $2(s_1 + s_3) = 5342$, so $s_1 + s_3 = 2671$. That leaves $s_2 = 3322 - 2671 = 651$. Matches. The figure also implies $s_1, s_3 > s_2$ (the middle square is the smallest), and any split of $2671$ into two values each above $651$ is consistent (e.g. $s_1 \approx 1369, s_3 \approx 1302$).
Alternative: Tool #3 (Eliminate Possibilities) on the choices: from Eq. 1 + Eq. 2 we know $s_1 + s_3 = 2671$, and from Eq. 1 we have $s_2 = 3322 - (s_1 + s_3) = 3322 - 2671 = 651$. Substituting each of (A)-(E) into Eq. 1 and Eq. 2 simultaneously, only $s_2 = 651$ leaves a consistent $s_1 + s_3 = 2671$ that satisfies both, confirming (A) without re-deriving the algebra.
CCSS standards used (min grade 8)
4.G.A.1Draw points, lines, line segments, rays, angles, and identify in figures (Reading and labeling the line segments inside the composite rectangle so each side of every square is associated with a variable.)6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the unknown square side lengths $s_1, s_2, s_3$ and writing the top edge as the expression $s_1 + s_2 + s_3$.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Translating the vertical layout into the single equation $s_1 - s_2 + s_3 = 2020$ from the heights of the three columns.)8.EE.C.8Analyze and solve pairs of simultaneous linear equations (Subtracting the two linear equations in $s_1, s_2, s_3$ to eliminate $s_1$ and $s_3$ together and isolate $2 s_2 = 1302$.)5.NBT.B.6Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing $1302 \div 2 = 651$ to get the final side length.)
⭐ This 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!
⭐ This 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!
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