AMC 10 · 2016 · #11

Grade 6 geometry-2d

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

perimeterarea-rectanglessystems-of-equations convert-to-algebra ↑ Prerequisites: perimeter

📏 Medium solution 💡 2 insights

Problem

Carl fences a rectangular garden with 20 posts, one at each corner and the rest spaced evenly 4 yards apart along the edges. The longer side (counting its two corner posts) holds twice as many posts as the shorter side (counting its two corner posts). Find the area of the garden in square yards.

Pick an answer.

(A)

256

(B)

336

(C)

384

(D)

448

(E)

512

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The post counts on the two sides are linked by a 'twice as many' relationship and a fixed total, so naming the shorter-side count with a variable turns the words into one equation. A quick sketch keeps the shared corner posts from being counted twice, and once a side's post count is known the length is a small multiplication subproblem.

1STEP 1

Name the posts per side

Let n be the posts on a shorter side (corners included); a longer side then has twice as many, 2n posts.

n = posts on a shorter side, 2n = posts on a longer side

2STEP 2

Count posts around the rectangle

Summing all four sides gives 2n + 2(2n), but the 4 shared corners are double-counted, so subtract 4; the total is 20.

2n + 2(2n) - 4 = 20

3STEP 3

Solve for n

Combining, 6n - 4 = 20, so 6n = 24 and n = 4; a shorter side has 4 posts, a longer side 8.

6n - 4 = 20 → 6n = 24 → n = 4

4STEP 4

Turn posts into side lengths

Length comes from gaps, one fewer than posts: 4 posts → 3 gaps → 12 yards; 8 posts → 7 gaps → 28 yards.

3 × 4 = 12 yd, 7 × 4 = 28 yd

5STEP 5

Compute the area

The garden is 12 by 28 yards, so its area is 12 × 28 = 336 square yards — choice (B).

12 × 28 = 336 square yards

Answer

336

Check the post count directly: 4 + 8 + 4 + 8 = 24 side-tallies, minus 4 doubly counted corners = 20 posts, exactly what Carl bought. The 12 by 28 rectangle gives 336 square yards, which is answer (B) and sits sensibly between the listed choices.

💡Key takeaway

Count the gaps, not the posts: a row of posts always has one fewer space than posts, and corners get shared by two sides.

Name the posts per side
Count posts around the rectangle
Solve for n
Turn posts into side lengths
Compute the area

Name the posts per side

Count posts around the rectangle

Solve for n

Turn posts into side lengths

Compute the area

More like this