AMC 8 · 2000 · #4
Grade 6 arithmeticProblem
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.
AMC 8 2000 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: 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
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)
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
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 $30$.
💡 Grade 5 coordinate-plane reading: each data point is a $(x, y)$ pair where $x$ is the year and $y$ is the percent.
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.
💡 One wrong point is enough. Reading the leftmost point on the line is the quickest check.
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.
💡 $8\%$ should sit a little below the first gridline; a point sitting on the gridline ($10\%$) is visibly wrong.
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.
💡 $15\%$ is the midpoint between gridlines $10$ and $20$. (C) overshoots; (E) lands right.
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.
💡 After elimination, the survivor must still be verified end-to-end — never skip the confirmation.
5.G.A.2 List the four target points the right graph must hit. The vertical axis is in pe 6.SP.B.4 Use the easiest filter first: check each graph's $1960$ value against the target 6.SP.B.4 Check the remaining graphs against the $1970$ value of $8\%$ (just below the fir 6.SP.B.4 Two graphs remain: (C) and (E). Check the $1980$ value, which must be $15\%$ (ha 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.2Represent 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.4Display 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).
⭐ 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).
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