AMC 8 · 2000 · #4

Grade 6 arithmetic

Archetype Small-Domain Constrained Search

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

📏 Short solution 💡 2 insights 📊 Diagram

Problem

In 1960 only 5% of the working adults in Carlin City worked
at home. By 1970 the "at-home" work force had increased to
8%. In 1980 there were approximately 15% working at home,
and in 1990 there were 30%. The graph that best illustrates
this is:

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%

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Data points (year, percent): $(1960, 5\%)$, $(1970, 8\%)$, $(1980, 15\%)$, $(1990, 30\%)$; Each option (A)-(E) is a line graph with the same axes (year on horizontal, percent on vertical) and gridlines at $10\%$, $20\%$, $30\%$; Answer choices: five graphs labeled (A), (B), (C), (D), (E)

Unknowns: Which graph plots all four of the given data points correctly

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #1 Draw a Diagram

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.

Execute — Answer: E

#1 Draw a Diagram 5.G.A.2 Step 1

List the four target points the right graph must hit.
The vertical axis is in percent, with gridlines at $10$, $20$, and $30$.

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

💡 Grade 5 coordinate-plane reading: each data point is a $(x, y)$ pair where $x$ is the year and $y$ is the percent.

#3 Eliminate Possibilities 6.SP.B.4 Step 2

Use the easiest filter first: check each graph's $1960$ value against the target $5\%$ (half a gridline up).
Graph (A) starts at $10\%$ — too high — eliminated.

$$\text{Graph (A): } 1960 \to 10\% \ne 5\% \;\Rightarrow\; \text{out}$$

💡 One wrong point is enough. Reading the leftmost point on the line is the quickest check.

#3 Eliminate Possibilities 6.SP.B.4 Step 3

Check the remaining graphs against the $1970$ value of $8\%$ (just below the first gridline).
Graph (B) shows $1970 \to 10\%$ — eliminated.
Graph (D) shows $1970 \to 10\%$ — eliminated.

$$\text{Graph (B): } 1970 \to 10\% \ne 8\%; \quad \text{Graph (D): } 1970 \to 10\% \ne 8\% \;\Rightarrow\; \text{both out}$$

💡 $8\%$ should sit a little below the first gridline; a point sitting on the gridline ($10\%$) is visibly wrong.

#3 Eliminate Possibilities 6.SP.B.4 Step 4

Two graphs remain: (C) and (E).
Check the $1980$ value, which must be $15\%$ (halfway between the first and second gridlines).
Graph (C) jumps to about $25\%$ in $1980$ — eliminated.
Graph (E) sits at $15\%$ — passes.

$$\text{Graph (C): } 1980 \to 25\% \ne 15\%; \quad \text{Graph (E): } 1980 \to 15\% = 15\% \;\checkmark$$

💡 $15\%$ is the midpoint between gridlines $10$ and $20$. (C) overshoots; (E) lands right.

#3 Eliminate Possibilities 5.G.A.2 Step 5

Confirm (E) by reading all four of its points.
(E) plots $5\%$, $8\%$, $15\%$, $30\%$ across the four years — every point matches.

$$\text{Graph (E): } (1960,5),\,(1970,8),\,(1980,15),\,(1990,30) \;\Rightarrow\; \textbf{(E)}$$

💡 After elimination, the survivor must still be verified end-to-end — never skip the confirmation.

[1] #1 5.G.A.2 List the four target points the right graph must hit. The vertical axis is in pe

[2] #3 6.SP.B.4 Use the easiest filter first: check each graph's $1960$ value against the target

[3] #3 6.SP.B.4 Check the remaining graphs against the $1970$ value of $8\%$ (just below the fir

[4] #3 6.SP.B.4 Two graphs remain: (C) and (E). Check the $1980$ value, which must be $15\%$ (ha

[5] #3 5.G.A.2 Confirm (E) by reading all four of its points. (E) plots $5\%$, $8\%$, $15\%$, $

Review

Reasonableness: The four percentages climb slowly at first ($5 \to 8$, a gain of only $3$ over a decade) and then steeply ($15 \to 30$, 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.

Alternative: Tool #1 (Draw a Diagram): on scratch paper, mark the four target points $(1960,5), (1970,8), (1980,15), (1990,30)$ on the same axes, connect them with line segments, and then scan the five options for the one that matches the picture. The hand-drawn line is nearly flat early and steep late — exactly the shape of (E).

CCSS standards used (min grade 6)

5.G.A.2 Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane (Reading each $(\text{year}, \text{percent})$ data point as a point on the coordinate plane so the target graph can be matched to the four given values.)
6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots (Comparing each option's plotted percentages against the target percentages year by year to eliminate graphs whose plotted values do not match.)

⭐ 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 $1960 \to 5\%$ and $1970 \to 8\%$ already leaves only (E).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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