During a certain month, 40 percent of all the oil used by a certain factory was used by the oil...

1. Translate the problem requirements: We need to find what fraction the heater oil cost represents of the total oil cost. Given: \(40\%\) of oil volume goes to heater, heater oil costs half the price per gallon of other oil.
2. Set up volume relationships: Establish the split between heater oil (\(40\%\)) and other oil (\(60\%\)) using simple variables.
3. Establish cost relationships: Define the cost per gallon difference - heater oil costs half as much per gallon as other oil.
4. Calculate total costs for each category: Multiply volume by cost per gallon for both heater oil and other oil to get total costs.
5. Determine the fraction: Find what portion the heater oil cost represents of the total factory oil cost.

Execution of Strategic Approach

1. Translate the problem requirements

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:

• The oil heater uses \(40\%\) of all the oil (by volume)
• The remaining \(60\%\) of oil is used for other purposes
• The oil heater uses cheaper oil - it costs half as much per gallon compared to the other oil
• We need to find: (Cost of heater oil) ÷ (Total cost of all oil)

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up volume relationships

Let's use simple numbers to make this concrete. Imagine the factory uses \(100\) gallons of oil total in a month.

This means:

• Oil heater uses: \(40\% \text{ of } 100 = 40\) gallons
• Other purposes use: \(60\% \text{ of } 100 = 60\) gallons
• Total: \(40 + 60 = 100\) gallons ✓

Using variables: If total oil volume = \(\mathrm{V}\), then:

• Heater oil volume = \(0.4\mathrm{V}\)
• Other oil volume = \(0.6\mathrm{V}\)

3. Establish cost relationships

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:

• Other oil price = \(\mathrm{P}\) per gallon
• Heater oil price = \(\frac{\mathrm{P}}{2}\) per gallon

4. Calculate total costs for each category

Now we multiply volume × price per gallon to get total costs.

Using our concrete example:

• Cost of heater oil = \(40 \text{ gallons} \times \$1/\text{gallon} = \$40\)
• Cost of other oil = \(60 \text{ gallons} \times \$2/\text{gallon} = \$120\)
• Total cost = \(\$40 + \$120 = \$160\)

Using variables:

• Cost of heater oil = \((0.4\mathrm{V}) \times (\frac{\mathrm{P}}{2}) = \frac{0.4\mathrm{VP}}{2} = 0.2\mathrm{VP}\)
• Cost of other oil = \((0.6\mathrm{V}) \times (\mathrm{P}) = 0.6\mathrm{VP}\)
• Total cost = \(0.2\mathrm{VP} + 0.6\mathrm{VP} = 0.8\mathrm{VP}\)

5. Determine the fraction

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!

4. Final Answer

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).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the cost relationship

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.

2. Confusing volume percentages with cost percentages

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.

3. Setting up the wrong fraction

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.

Errors while executing the approach

1. Arithmetic errors when calculating costs

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.

2. Incorrectly simplifying the final fraction

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.

Errors while selecting the answer

No likely faltering points

Alternate Solutions

Smart Numbers Approach

This problem involves percentage relationships and cost ratios that can be effectively solved using carefully chosen concrete numbers.

Step 1: Choose Smart Numbers for Oil Volume

Since we're dealing with \(40\%\) of oil going to the heater, let's choose a total oil volume that makes percentage calculations clean:

• Total oil used by factory = \(100\) gallons (chosen to make percentage calculations straightforward)
• Oil used by heater = \(40\% \text{ of } 100 = 40\) gallons
• Oil used by other equipment = \(60\% \text{ of } 100 = 60\) gallons

Step 2: Choose Smart Numbers for Cost per Gallon

The problem states that heater oil costs half as much per gallon as other oil. Let's choose:

• Cost per gallon of other oil = \(\$2\) (chosen as a clean number that's easy to halve)
• Cost per gallon of heater oil = \(\$1\) (half of \(\$2\))

Step 3: Calculate Total Costs

Now we can calculate the total cost for each type of oil:

• Cost of heater oil = \(40 \text{ gallons} \times \$1/\text{gallon} = \$40\)
• Cost of other oil = \(60 \text{ gallons} \times \$2/\text{gallon} = \$120\)
• Total cost of all oil = \(\$40 + \$120 = \$160\)

Step 4: Find the Fraction

The fraction of total cost represented by heater oil:

Fraction = Cost of heater oil / Total cost = \(\$40 / \$160 = \frac{1}{4}\)

Verification

Our smart numbers were chosen logically:

• \(100\) gallons total makes percentage calculations clean
• \(\$2\) per gallon for other oil allows for a clean half-price of \(\$1\) for heater oil
• All calculations result in whole numbers, making the arithmetic straightforward

Answer: C. \(\frac{1}{4}\)