- Translate the problem requirements: We need to understand that November revenue = \(\frac{2}{5} \times \mathrm{December\ revenue}\), January revenue = \(\frac{1}{4} \times \mathrm{November\ revenue}\), and we're finding how many times December revenue is compared to the average of November and January revenues.
- Set up relationships using a smart base value: Choose December revenue as our base since other revenues are fractions of it, avoiding complex fraction calculations.
- Express all revenues in terms of the base: Calculate November and January revenues as multiples of December revenue to maintain simple arithmetic.
- Calculate the target average: Find the arithmetic mean of November and January revenues, keeping everything in terms of December revenue.
- Determine the ratio: Compare December revenue to the calculated average to find how many times larger it is.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what the problem is telling us in plain English:
- In November, the store made \(\frac{2}{5}\) of what it made in December
- In January, the store made \(\frac{1}{4}\) of what it made in November
- We want to find: How many times bigger is December's revenue compared to the average of November and January revenues?
Think of it this way: if December was a really good month, then November was much smaller (only \(\frac{2}{5}\) of December), and January was even smaller (only \(\frac{1}{4}\) of the already small November). We're comparing the big December number to the average of two much smaller numbers.
Process Skill: TRANSLATE - Converting the fraction relationships into clear mathematical understanding
2. Set up relationships using a smart base value
To make our arithmetic simple, let's say December's revenue was $20. Why $20? Because when we multiply $20 by fractions like \(\frac{2}{5}\), we'll get nice whole numbers that are easy to work with.
So our starting point is:
This choice eliminates messy fraction calculations and keeps everything concrete and manageable.
3. Express all revenues in terms of the base
Now let's find the other months' revenues using our $20 December base:
November revenue:
- November = \(\frac{2}{5} \times \mathrm{December}\)
- November = \(\frac{2}{5} \times \$20 = \$8\)
January revenue:
- January = \(\frac{1}{4} \times \mathrm{November}\)
- January = \(\frac{1}{4} \times \$8 = \$2\)
So our three months look like:
- November: $8
- December: $20
- January: $2
4. Calculate the target average
We need the arithmetic mean (average) of November and January revenues:
Average = \(\frac{\mathrm{November} + \mathrm{January}}{2}\)
Average = \(\frac{\$8 + \$2}{2}\)
Average = \(\frac{\$10}{2} = \$5\)
So the average of November and January revenues is $5.
5. Determine the ratio
Finally, we find how many times December revenue compares to this average:
Ratio = \(\frac{\mathrm{December\ revenue}}{\mathrm{Average\ of\ (November\ and\ January)}}\)
Ratio = \(\frac{\$20}{\$5} = 4\)
This means December's revenue was exactly 4 times the average of November and January revenues.
Final Answer
The store's revenue in December was 4 times the average of its revenues in November and January.
The answer is (E) 4.
Verification: We can verify this makes sense - December was the strongest month at $20, while November ($8) and January ($2) were much weaker, averaging only $5. It's reasonable that the strong month would be 4 times the average of the two weaker months.
Common Faltering Points
Errors while devising the approach
- Misinterpreting the final question: Students often get confused about what exactly is being asked. The question asks "December was how many times the average of November and January" but students might set up the ratio backwards, calculating "average divided by December" instead of "December divided by average."
- Setting up incorrect variable relationships: Students may struggle with the chain of relationships, especially with January being defined in terms of November (which is already defined in terms of December). They might incorrectly think January = \(\frac{1}{4} \times \mathrm{December}\) instead of January = \(\frac{1}{4} \times \mathrm{November}\).
- Choosing poor variable values: While any value works mathematically, students who choose decimal values like $1 for December will end up with messy fractions (November = $0.40, January = $0.10) that increase calculation errors and make verification difficult.
Errors while executing the approach
- Fraction multiplication errors: When calculating November = \(\frac{2}{5} \times 20\) or January = \(\frac{1}{4} \times 8\), students commonly make arithmetic mistakes, especially if they chose inconvenient starting values that result in non-whole numbers.
- Average calculation mistakes: Students might forget to divide by 2 when finding the average of November and January, or they might incorrectly include December in the average calculation even though the question only asks for the average of November and January.
Errors while selecting the answer
- Inverted ratio selection: Even after correctly calculating that December = $20 and Average = $5, students might select the reciprocal ratio (\(\frac{1}{4}\) instead of 4) because they flip the division at the last moment.
- Selecting intermediate calculations: Students might mistakenly select values they calculated during the process, such as the average value (5) or one of the monthly revenues, rather than the final ratio comparison that the question actually asks for.
Alternate Solutions
Smart Numbers Approach
This problem is well-suited for the smart numbers technique because we have proportional relationships between different months' revenues. We can choose a convenient value for one month's revenue and calculate the others accordingly.
Step 1: Choose a Smart Number for December Revenue
Since November revenue is \(\frac{2}{5}\) of December revenue, let's choose December revenue = $20. This number works well because:
- It's divisible by 5, making the calculation of November revenue clean
- It will lead to integer values for our calculations
Step 2: Calculate November Revenue
November revenue = \(\frac{2}{5} \times \mathrm{December\ revenue}\)
November revenue = \(\frac{2}{5} \times \$20 = \$8\)
Step 3: Calculate January Revenue
January revenue = \(\frac{1}{4} \times \mathrm{November\ revenue}\)
January revenue = \(\frac{1}{4} \times \$8 = \$2\)
Step 4: Calculate the Average of November and January Revenues
Average = \(\frac{\mathrm{November} + \mathrm{January}}{2}\)
Average = \(\frac{\$8 + \$2}{2} = \frac{\$10}{2} = \$5\)
Step 5: Determine How Many Times December Revenue is Compared to the Average
Ratio = \(\frac{\mathrm{December\ revenue}}{\mathrm{Average}}\)
Ratio = \(\frac{\$20}{\$5} = 4\)
Verification
Let's verify with our concrete numbers:
- December: $20
- November: $8 (which is indeed \(\frac{2}{5}\) of $20)
- January: $2 (which is indeed \(\frac{1}{4}\) of $8)
- Average of November and January: $5
- December revenue is 4 times the average
Answer: (E) 4
