Question

If a certain toy store's revenue in November was \(\frac{2}{5}\) of its revenue in December and its revenue in January was \(\frac{1}{4}\) of its revenue in November, then the store's revenue in December was how many times the average (arithmetic mean) of its revenues in November and January?

• A. \(\frac{1}{4}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{2}{3}\)
• D. 2
• E. 4

Hint:

In problems involving ratios and relationships, choosing one variable as a "base" (like D in this case) and expressing all other variables in terms of that base simplifies the calculations significantly. The base variable will almost always cancel out in the final ratio.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The problem describes the relationship between a store's revenues in three different months. We need to express all revenues in terms of a single month's revenue, calculate the average of two of them, and then find the ratio of the third to this average.
Step 2: Key Formula or Approach:
Let N, D, and J represent the revenues for November, December, and January, respectively.
We are given: 1. \( N = \frac{2}{5} D \) 2. \( J = \frac{1}{4} N \) We need to find the value of the ratio: \(\frac{D}{\text{Average}(N, J)}\), which is \(\frac{D}{(N+J)/2}\).
Step 3: Detailed Explanation:
First, let's express all revenues in terms of a single variable, D, since D is the reference in the first relationship and the final ratio.
We already have \(N = \frac{2}{5} D\).
Now, let's express J in terms of D by substituting the expression for N into the second equation: \[ J = \frac{1}{4} N = \frac{1}{4} \left( \frac{2}{5} D \right) = \frac{2}{20} D = \frac{1}{10} D \] Next, calculate the average of the revenues in November and January: \[ \text{Average}(N, J) = \frac{N + J}{2} = \frac{\frac{2}{5} D + \frac{1}{10} D}{2} \] To add the fractions in the numerator, we find a common denominator (10): \[ \frac{2}{5}D = \frac{4}{10}D \] \[ \text{Average}(N, J) = \frac{\frac{4}{10} D + \frac{1}{10} D}{2} = \frac{\frac{5}{10} D}{2} = \frac{\frac{1}{2} D}{2} = \frac{1}{4} D \] Finally, we find the ratio of the December revenue to this average: \[ \text{Ratio} = \frac{D}{\text{Average}(N, J)} = \frac{D}{\frac{1}{4} D} \] The variable D cancels out: \[ \text{Ratio} = \frac{1}{\frac{1}{4}} = 1 \times 4 = 4 \] Step 4: Final Answer:
The store's revenue in December was 4 times the average of its revenues in November and January.