Recycling Claim: Percentage Points vs. Percent Change
A town council candidate claims that municipal recycling has 'doubled' since 2010. The public works department reports that in 2010 the town generated 4,800 tons of municipal solid waste, of which 28% was recycled. In 2020 the town generated 5,400 tons, of which 40% was recycled. During a mathematics class, a student writes: 'The recycling rate increased from 28% to 40%, which is a 12% increase, so the town recycled 12% more material in 2020 than in 2010.' A second student argues that the candidate is roughly correct because 40% is close to twice 28%. The teacher wants students to distinguish percentage points from percent change and to evaluate the claim using quantities rather than rates alone. The class has calculators and the two waste-total figures; no other data are available.
1. Identify the two separate errors in the first student's statement. Then compute the actual number of tons recycled in each year and the percent change in recycled tons from 2010 to 2020.
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Error 1: 28% to 40% is an increase of 12 percentage points, not a 12% increase. Error 2: a 12% increase in the recycling rate would not mean a 12% increase in recycled tons, because total waste also changed. Tons recycled in 2010: 0.28(4,800) = 1,344 tons. In 2020: 0.40(5,400) = 2,160 tons. Change: 2,160 - 1,344 = 816 tons. Percent change: 816 / 1,344 is about 0.607, or about 60.7%. A 12% increase from 1,344 would be only 1,505.28 tons, so the student's conclusion is far off.
2. The candidate's 'doubled' claim: calculate the recycling rate that would have been needed in 2020 for the recycled tonnage to be double the 2010 figure. Is the claim supported? Explain.
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Double the 2010 recycled tonnage is 2(1,344) = 2,688 tons. With 5,400 tons generated in 2020, the required rate is 2,688 / 5,400, which is about 0.4978, or about 49.8%. The actual 2020 rate is 40%, so recycled tonnage is 2,160 tons, not 2,688 tons. The increase is about 60.7%, not 100%. If total waste had stayed at 4,800 tons, a 40% rate would give 1,920 tons, which is 42.9% above 1,344 tons, still not double. The claim is not supported by these data.
3. Describe two graph-design choices that could make the 28%-to-40% change look like a doubling, and explain how each distorts the comparison. What design would you require?
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A truncated vertical axis, say starting at 25% instead of 0%, makes the 12-point rise occupy a large fraction of the plotted height; on an axis from 25% to 40%, the bar for 40% uses the full scale while 28% uses only 3/15 of it, so the visual ratio is 5:1 even though the values are 40:28. A second choice is using area or 3D bars, so increasing the height also increases width or depth and squares the visual impression. Both exaggerate a 12-percentage-point change. A fair graph should start the vertical axis at 0%, use equal intervals, show percentage recycled as bars or points, and label the change as 12 percentage points, not 12%, with the tonnage totals reported separately.