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During a certain month, 40 percent of all the oil used by a certain factory was used by the oil heater. The cost per gallon of the oil used by the oil heater was half the cost per gallon of the rest of the oil used by the factory. For that month, the cost of the oil used by the oil heater was what fraction of the cost of all the oil used by the factory?
Let's break down what the problem is telling us in plain English. We have a factory that uses oil, and we need to figure out what fraction of the total oil cost goes to the oil heater.
Here's what we know:
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
Let's use simple numbers to make this concrete. Imagine the factory uses \(100\) gallons of oil total in a month.
This means:
Using variables: If total oil volume = \(\mathrm{V}\), then:
Now let's think about the price per gallon. The problem says heater oil costs half as much per gallon as the other oil.
Let's say the other oil costs \(\$2\) per gallon (picking an easy number).
Then heater oil costs \(\$1\) per gallon (half of \(\$2\)).
Using variables: If other oil costs \(\mathrm{P}\) dollars per gallon, then:
Now we multiply volume × price per gallon to get total costs.
Using our concrete example:
Using variables:
Finally, we find what fraction the heater oil cost represents of the total cost.
Using our concrete example:
Fraction = (Cost of heater oil) ÷ (Total cost) = \(\$40 ÷ \$160 = \frac{40}{160} = \frac{1}{4}\)
Using variables:
Fraction = \((0.2\mathrm{VP}) ÷ (0.8\mathrm{VP}) = \frac{0.2\mathrm{VP}}{0.8\mathrm{VP}} = \frac{0.2}{0.8} = \frac{1}{4}\)
Notice how the \(\mathrm{V}\) and \(\mathrm{P}\) cancel out - this means our answer doesn't depend on the actual volume or price, just on the relationships given in the problem!
The cost of oil used by the heater represents \(\frac{1}{4}\) of the total cost of all oil used by the factory.
This matches answer choice C: \(\frac{1}{4}\).
To verify: Even though the heater uses \(40\%\) of the oil volume, it only accounts for \(25\%\) of the cost because the oil it uses is much cheaper (half the price per gallon).
Students often confuse which oil is cheaper. The problem states "the cost per gallon of the oil used by the oil heater was half the cost per gallon of the rest of the oil." Some students might incorrectly think the heater oil costs twice as much, rather than half as much. This fundamental misunderstanding would lead them to set up the wrong price relationship from the start.
A common trap is thinking that since the heater uses \(40\%\) of the oil volume, it must also account for \(40\%\) of the cost. Students might jump to answer choice E (\(\frac{1}{2}\)) by incorrectly reasoning that \(40\%\) is close to \(50\%\). They fail to recognize that volume and cost are different measures that need to be calculated separately.
Some students might set up the fraction backwards, calculating (total cost) ÷ (heater cost) instead of (heater cost) ÷ (total cost). This would give them \(4\) instead of \(\frac{1}{4}\), leading them to look for an answer that doesn't exist among the choices.
Even with the correct setup, students often make calculation mistakes. For example, when computing \(0.4\mathrm{V} \times \frac{\mathrm{P}}{2}\), they might get \(0.4\mathrm{VP}\) instead of \(0.2\mathrm{VP}\), or when adding \(0.2\mathrm{VP} + 0.6\mathrm{VP}\), they might get \(0.7\mathrm{VP}\) instead of \(0.8\mathrm{VP}\). These small errors compound to give wrong final fractions.
When students arrive at the fraction \(\frac{0.2\mathrm{VP}}{0.8\mathrm{VP}}\), they might struggle to simplify it correctly. Some might forget to cancel out the VP terms, while others might incorrectly simplify \(\frac{0.2}{0.8}\) as \(\frac{1}{5}\) instead of \(\frac{1}{4}\), leading them to choose answer B.
No likely faltering points
This problem involves percentage relationships and cost ratios that can be effectively solved using carefully chosen concrete numbers.
Since we're dealing with \(40\%\) of oil going to the heater, let's choose a total oil volume that makes percentage calculations clean:
The problem states that heater oil costs half as much per gallon as other oil. Let's choose:
Now we can calculate the total cost for each type of oil:
The fraction of total cost represented by heater oil:
Fraction = Cost of heater oil / Total cost = \(\$40 / \$160 = \frac{1}{4}\)
Our smart numbers were chosen logically:
Answer: C. \(\frac{1}{4}\)