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The moisture content of wet wood is the ratio of the weight of the water in the wood to the weight of the wood when it is dry, expressed as a percent. For example, a piece of wet wood weighing 60 pounds, 30 pounds of which is water, has a moisture content of 100 percent. What is the weight, in pounds, of the water in 12,000 pounds of wet wood with a moisture content of 20 percent?
Let's start by understanding what moisture content really means in everyday terms. Think of wet wood as having two parts: the actual wood material (dry wood) and the water absorbed into it.
The problem tells us that moisture content is calculated as: the weight of water divided by the weight of dry wood, then expressed as a percentage.
Let's verify this with the given example: 60 pounds of wet wood contains 30 pounds of water. This means the dry wood weighs \(60 - 30 = 30\) pounds. So moisture content = \(\frac{30\,\mathrm{pounds\,of\,water}}{30\,\mathrm{pounds\,of\,dry\,wood}} \times 100\% = 100\%\).
Now for our problem: We have 12,000 pounds of wet wood with 20% moisture content. We need to find how much of that 12,000 pounds is water.
Process Skill: TRANSLATE - Converting the problem's definition into a workable understanding
Let's think through this step by step using plain language:
If we call the weight of water "W" pounds, then:
Since moisture content is 20%, we know that:
The weight of water divided by the weight of dry wood equals 0.20
In mathematical terms: \(\mathrm{W} \div (12,000 - \mathrm{W}) = 0.20\)
Now let's solve this equation step by step:
Starting with: \(\mathrm{W} \div (12,000 - \mathrm{W}) = 0.20\)
Multiply both sides by (12,000 - W):
\(\mathrm{W} = 0.20 \times (12,000 - \mathrm{W})\)
Distribute the 0.20:
\(\mathrm{W} = 0.20 \times 12,000 - 0.20 \times \mathrm{W}\)
\(\mathrm{W} = 2,400 - 0.20\mathrm{W}\)
Add 0.20W to both sides:
\(\mathrm{W} + 0.20\mathrm{W} = 2,400\)
\(1.20\mathrm{W} = 2,400\)
Divide both sides by 1.20:
\(\mathrm{W} = 2,400 \div 1.20 = 2,000\)
Let's verify: If there are 2,000 pounds of water, then dry wood weighs \(12,000 - 2,000 = 10,000\) pounds. Moisture content = \(2,000 \div 10,000 = 0.20 = 20\%\) ✓
The weight of water in the wet wood is 2,000 pounds.
Looking at our answer choices, this corresponds to choice B: 2,000.
1. Misinterpreting the moisture content definition: Students often confuse the definition and think moisture content = (weight of water) ÷ (total weight of wet wood) instead of the correct definition: moisture content = (weight of water) ÷ (weight of dry wood). This leads them to set up the equation as \(\mathrm{W} \div 12,000 = 0.20\), giving an incorrect answer of 2,400.
2. Incorrectly identifying what the 12,000 pounds represents: Some students may think the 12,000 pounds refers to the dry wood weight rather than the total wet wood weight, leading to a completely wrong setup where they treat 12,000 as the denominator in the moisture content ratio.
1. Algebraic manipulation errors: When solving \(\mathrm{W} \div (12,000 - \mathrm{W}) = 0.20\), students commonly make mistakes when cross-multiplying or distributing. For example, they might incorrectly write \(\mathrm{W} = 0.20 + (12,000 - \mathrm{W})\) instead of \(\mathrm{W} = 0.20 \times (12,000 - \mathrm{W})\).
2. Arithmetic errors in the final calculation: Students may correctly set up \(1.20\mathrm{W} = 2,400\) but then make calculation errors when dividing \(2,400 \div 1.20\), potentially getting 2,000 wrong or not recognizing that \(1.20 = \frac{6}{5}\), leading to incorrect final answers.
1. Selecting the wrong computed value: If students made the initial setup error (using total weight instead of dry weight), they would get 2,400 and select choice C instead of the correct choice B (2,000). They might feel confident because 2,400 appears as an answer choice, not realizing their conceptual error from the beginning.