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| Country | Amount Recycled (tons) | Amount Disposed of (tons) |
|---|---|---|
| A | 16,700 | 142,800 |
| B | 8,800 | 48,000 |
| C | 13,000 | 51,400 |
| D | 3,900 | 20,300 |
| E | 3,300 | 16,200 |
The table above shows the amount of waste material, in tons, recycled by each of five counties in a single year and the amount of waste material, also in tons, that was disposed of in landfills by the five counties in that year. Which county had the lowest ratio of waste material disposed of to waste material recycled in the year reported in the table?
Let's start by understanding what we're looking for in everyday terms. We want to find which county is most efficient at recycling - meaning they send the least amount of waste to landfills compared to what they recycle.
Think of it this way: if a county recycles \(10\) tons and disposes of \(50\) tons, that's a ratio of \(50 ÷ 10 = 5\). This means for every \(1\) ton they recycle, they dispose of \(5\) tons. If another county recycles \(20\) tons and disposes of \(40\) tons, that's \(40 ÷ 20 = 2\), meaning they only dispose of \(2\) tons for every \(1\) ton recycled. The second county is more efficient because their ratio is lower.
So we need to calculate: \(\mathrm{Amount\,Disposed} ÷ \mathrm{Amount\,Recycled}\) for each county, and find the smallest result.
Process Skill: TRANSLATE - Converting the problem language "lowest ratio of waste material disposed of to waste material recycled" into the mathematical operation disposed ÷ recycled
Now let's calculate the ratio for each county by dividing disposed amount by recycled amount:
Country A:
Disposed: \(142,800\) tons, Recycled: \(16,700\) tons
Ratio = \(142,800 ÷ 16,700 ≈ 8.55\)
Country B:
Disposed: \(48,000\) tons, Recycled: \(8,800\) tons
Ratio = \(48,000 ÷ 8,800 ≈ 5.45\)
Country C:
Disposed: \(51,400\) tons, Recycled: \(13,000\) tons
Ratio = \(51,400 ÷ 13,000 ≈ 3.95\)
Country D:
Disposed: \(20,300\) tons, Recycled: \(3,900\) tons
Ratio = \(20,300 ÷ 3,900 ≈ 5.21\)
Country E:
Disposed: \(16,200\) tons, Recycled: \(3,300\) tons
Ratio = \(16,200 ÷ 3,300 ≈ 4.91\)
To make the calculations easier, I rounded to two decimal places, but the exact relationships will give us the same ranking.
Looking at our calculated ratios:
- Country A: \(≈ 8.55\) (highest - least efficient)
- Country B: \(≈ 5.45\)
- Country C: \(≈ 3.95\) (lowest - most efficient)
- Country D: \(≈ 5.21\)
- Country E: \(≈ 4.91\)
Country C has the lowest ratio at approximately \(3.95\). This means that for every \(1\) ton of material that Country C recycles, they only dispose of about \(3.95\) tons in landfills. All other countries dispose of more waste relative to what they recycle.
Country C had the lowest ratio of waste material disposed of to waste material recycled.
The answer is C.
Students often confuse which number should be the numerator and which should be the denominator. The question asks for "ratio of waste material disposed of to waste material recycled," which means \(\mathrm{disposed} ÷ \mathrm{recycled}\). However, students might calculate \(\mathrm{recycled} ÷ \mathrm{disposed}\) instead, leading them to look for the highest value rather than the lowest.
Students might mistakenly think that the county with the smallest absolute amount of disposed waste or recycled waste is the answer, rather than understanding that we need to compare the relative ratios between disposed and recycled amounts for each county.
When dividing large numbers like \(142,800 ÷ 16,700\), students may make computational mistakes, especially if doing mental math or rushing through calculations. These errors can lead to incorrect ratio values and wrong comparisons.
Students might round too aggressively during intermediate steps or inconsistently across different counties, potentially changing the relative order of the ratios and leading to an incorrect identification of the minimum value.
After correctly calculating all ratios, students might accidentally select the county with the highest ratio (worst recycling efficiency) instead of the lowest ratio (best recycling efficiency), especially if they lose track of what "lowest ratio" means in the context of the problem.