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The mass of \(1 \text{ cubic meter}\) of a substance is \(800 \text{ kilograms}\) under certain conditions. What is the volume, in cubic centimeters, of \(1 \text{ gram}\) of this substance under these conditions? (\(1 \text{ kilogram} = 1{,}000 \text{ grams}\) and \(1 \text{ cubic meter} = 1{,}000{,}000 \text{ cubic centimeters}\))
Let's break down what we're given and what we need to find using everyday language.
We know that 1 cubic meter of this substance weighs 800 kilograms. Think of this like knowing that a standard-sized box of cereal weighs a certain amount - we have the relationship between the size of the container and how much stuff fits in it.
What we need to find is: if we take just 1 gram of this substance (a very tiny amount), how much space will it take up when measured in cubic centimeters?
This is essentially flipping the relationship around - instead of "how much does this volume weigh?" we're asking "how much space does this weight take up?"
Process Skill: TRANSLATE - Converting the problem from density language to a volume-finding question
Let's establish the basic relationship in simple terms first.
We know: 1 cubic meter contains 800 kilograms of the substance
This means the substance is quite dense - it packs a lot of weight into each unit of space.
In mathematical terms, we can write this fundamental relationship as:
\(\mathrm{Density} = 800 \text{ kg per } 1 \text{ m}^3\)
This tells us how much mass fits in a given volume. To find how much volume a given mass takes up, we'll need to work with the inverse of this relationship.
Now we need to convert our density relationship to match the units in our question - grams and cubic centimeters instead of kilograms and cubic meters.
Start with what we know: 800 kg per 1 cubic meter
Convert kilograms to grams:
\(800 \text{ kg} = 800 \times 1{,}000 \text{ g} = 800{,}000 \text{ g}\)
Convert cubic meters to cubic centimeters:
\(1 \text{ m}^3 = 1{,}000{,}000 \text{ cm}^3\)
So our density relationship becomes:
\(800{,}000 \text{ g per } 1{,}000{,}000 \text{ cm}^3\)
We can simplify this by dividing both numbers by 1,000:
\(800 \text{ g per } 1{,}000 \text{ cm}^3\)
This means every 1,000 cubic centimeters contains 800 grams of the substance.
Now we use our converted density to find the volume of 1 gram.
We know: 800 grams occupy 1,000 cubic centimeters
To find how much space 1 gram occupies, we can set up a simple proportion:
If 800 grams take up 1,000 cubic centimeters
Then 1 gram takes up ? cubic centimeters
Using basic proportional reasoning:
\(\mathrm{Volume\;for\;1\;gram} = \frac{1{,}000 \text{ cm}^3}{800 \text{ g}} \times 1 \text{ g}\)
\(\mathrm{Volume\;for\;1\;gram} = \frac{1{,}000}{800} \text{ cm}^3\)
\(\mathrm{Volume\;for\;1\;gram} = 1.25 \text{ cm}^3\)
Process Skill: SIMPLIFY - Using proportional reasoning instead of complex formulas
The volume of 1 gram of this substance is 1.25 cubic centimeters.
Looking at our answer choices, this matches choice (B) 1.25.
We can verify this makes sense: since the substance is quite dense (800 kg per cubic meter), we expect a small amount like 1 gram to take up a relatively small volume, which 1.25 cubic centimeters certainly is.
1. Misinterpreting what needs to be found: Students often confuse density problems and try to find "how much does 1 cubic centimeter weigh?" instead of "what volume does 1 gram occupy?" This leads them down the wrong path entirely, calculating mass per unit volume instead of volume per unit mass.
2. Forgetting about unit conversions: Students may recognize this as a density problem but fail to plan for the extensive unit conversions required. They might attempt to work directly with the given units (kilograms and cubic meters) and the target units (grams and cubic centimeters) without realizing they need to convert systematically.
3. Not recognizing the inverse relationship: Students may understand they need to find volume per gram but fail to realize this requires using the inverse of the given density relationship. They might try to use the given ratio directly instead of flipping it.
1. Unit conversion mistakes: Students frequently make errors when converting between metric units, such as incorrectly converting 1 cubic meter to 100,000 cubic centimeters instead of 1,000,000 cubic centimeters, or converting 1 kilogram to 100 grams instead of 1,000 grams.
2. Arithmetic errors in division: When calculating 1,000 ÷ 800, students may make computational mistakes, getting 0.8 or 8.0 instead of 1.25. This is especially common when students don't simplify the fraction properly or misplace decimal points.
3. Setting up proportions incorrectly: Students may set up the proportion backwards or with incorrect corresponding units, such as writing "800 grams : 1 gram = ? cubic centimeters : 1,000 cubic centimeters" instead of the correct "800 grams : 1,000 cubic centimeters = 1 gram : ? cubic centimeters".
1. Selecting the reciprocal of the correct answer: Students who correctly calculate 1.25 might second-guess themselves and select 0.80 (choice A) because they think they may have inverted the relationship. This happens when students lose confidence in their approach at the final step.
2. Choosing answers that result from unit conversion errors: If students made errors in their unit conversions, they might arrive at 12.50 (choice D) or 80.00 (choice E), which are the result of systematic conversion mistakes but still seem plausible given the magnitude of the numbers in the problem.