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13 Mar 2026, 13:20
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B
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Difficulty:
65%
(hard)
Question Stats:
59% (02:18) correct
41%
(02:37)
wrong
based on 106
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As a result of rezoning, the area and population of Brownville increased by 37 1/2% and 12 1/2% respectively. By what fraction of its original value did the population density (total population per total area) decrease?
A. 1/11
B. 2/11
C. 2/9
D. 1/4
E. 3/8
A. 1/11
B. 2/11
C. 2/9
D. 1/4
E. 3/8
16 Mar 2026, 01:54
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Let's pick simple numbers to make this easy.
Step 1: Convert the percentages to fractions.
- 37 1/2 % = 3/8
- 12 1/2 % = 1/8
Step 2: Assume original Area = 8 and original Population = 8 (picking 8 makes the fractions clean).
- New Area = 8 + (3/8 × 8) = 8 + 3 = 11
- New Population = 8 + (1/8 × 8) = 8 + 1 = 9
Step 3: Calculate the population densities.
- Original density = Population / Area = 8/8 = 1
- New density = 9/11
Step 4: Find the decrease as a fraction of the original.
Decrease = Original density − New density = 1 − 9/11 = 2/11
As a fraction of its original value: (2/11) / 1 = 2/11
Answer: B
Key principle: When both numerator and denominator of a ratio change by different percentages, don't try to subtract percentages directly. Instead, compute the new ratio and compare it to the old one. Here, the new density became 9/11 of the original, meaning it dropped by 2/11 of its original value.
Common mistake: Some students subtract the percentages (37.5% − 12.5% = 25%) and pick 1/4. That's wrong because density is a ratio (population divided by area), so the percentage changes don't simply subtract.
Step 1: Convert the percentages to fractions.
- 37 1/2 % = 3/8
- 12 1/2 % = 1/8
Step 2: Assume original Area = 8 and original Population = 8 (picking 8 makes the fractions clean).
- New Area = 8 + (3/8 × 8) = 8 + 3 = 11
- New Population = 8 + (1/8 × 8) = 8 + 1 = 9
Step 3: Calculate the population densities.
- Original density = Population / Area = 8/8 = 1
- New density = 9/11
Step 4: Find the decrease as a fraction of the original.
Decrease = Original density − New density = 1 − 9/11 = 2/11
As a fraction of its original value: (2/11) / 1 = 2/11
Answer: B
Key principle: When both numerator and denominator of a ratio change by different percentages, don't try to subtract percentages directly. Instead, compute the new ratio and compare it to the old one. Here, the new density became 9/11 of the original, meaning it dropped by 2/11 of its original value.
Common mistake: Some students subtract the percentages (37.5% − 12.5% = 25%) and pick 1/4. That's wrong because density is a ratio (population divided by area), so the percentage changes don't simply subtract.
