AMC 8 · 2000 · #4

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

graph-readingpercentagesystematic-enumeration systematic-enumerationcasework ↑ Prerequisites: graph-readingpercentage

📏 Short solution 💡 2 insights 📊 Diagram

Problem

In Carlin City, the share of working adults who worked at home was 5% in 1960, 8% in 1970, 15% in 1980, and 30% in 1990. Five line graphs are shown. Pick the one whose four plotted points match all four percentages.

Pick an answer.

(A)

Graph A: 10%, 12.5%, 20%, 30%

(B)

Graph B: 5%, 10%, 20%, 25%

(C)

Graph C: 5%, 8%, 25%, 30%

(D)

Graph D: 5%, 10%, 28%, 30%

(E)

Graph E: 5%, 8%, 15%, 30%

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Five graphs, only one correct: this is the textbook setup for Tool #3 (Eliminate Possibilities). Read each graph's 1960 point against the target 5% first — that single quick check kills most of the graphs immediately. Tool #1 (Draw a Diagram) supports the read by treating each gridline as 10% so the points can be located precisely. No arithmetic is needed beyond reading values off the axis.

1STEP 1

List the four targets the answer must hit: (1960, 5%), (1970, 8%), (1980, 15%), (1990, 30%).

(1960, 5%), (1970, 8%), (1980, 15%), (1990, 30%)

2STEP 2

Easiest filter first: check the 1960 point against 5%. Graph (A) starts at 10% — too high — so (A) is out.

Graph (A): 1960 → 10% ≠ 5% → out

3STEP 3

Now test the 1970 point of 8% (just under the first gridline). Graphs (B) and (D) sit on 10%, so (B) and (D) are out.

Graph (B): 1970 → 10% ≠ 8%; Graph (D): 1970 → 10% ≠ 8% → both out

4STEP 4

Two left: (C) and (E). At 1980 the value must be 15%. Graph (C) jumps to about 25%, so (C) is out and (E) survives.

Graph (C): 1980 → 25% ≠ 15%; Graph (E): 1980 → 15% = 15% ✓

5STEP 5

Confirm (E): its four points read 5%, 8%, 15%, 30% — every year matches, so the answer is (E).

Graph (E): (1960,5), (1970,8), (1980,15), (1990,30) → (E)

Answer

Graph E: 5%, 8%, 15%, 30%

The four percentages climb slowly at first (5 → 8, a gain of only 3 over a decade) and then steeply (15 → 30, doubling in the last decade). Graph (E)'s line is nearly flat between 1960 and 1970, rises more from 1970 to 1980, and shoots up sharpest from 1980 to 1990. That increasing-steepness shape matches the data, while the eliminated graphs either start too high or rise too evenly.

💡Key takeaway

When the question asks "which graph?", do not redraw everything — pick one data point, read it off each option, and eliminate. The first mismatched point kills the graph. Here, checking 1960 → 5% and 1970 → 8% already leaves only (E).

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