Question

In a certain department store, which has four sizes of a specific shirt, there are 1/3 as many small shirts as medium shirts, and 1/2 as many large shirts as small shirts. If there are as many x-large shirts as large shirts, what percent of the shirts in the store are medium?

• A. 10%
• B. 25%
• C. 33%
• D. 50%
• E. 60%

Hint:

For problems involving multiple ratios, a powerful technique is to pick a concrete number for one of the quantities. Choose a number that is a multiple of all the denominators involved to keep the calculations with whole numbers, which is often faster and less error-prone.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves ratios between different quantities (shirt sizes). The goal is to find the proportion of one quantity (medium shirts) relative to the total. A good strategy is to express all quantities in terms of a single variable or to pick a convenient number to work with.
Step 2: Detailed Explanation:
Part 1: Set up the relationships
Let S, M, L, and XL be the number of small, medium, large, and x-large shirts, respectively.
From the problem statement:
1. \( S = \frac{1}{3} M \)
2. \( L = \frac{1}{2} S \)
3. \( XL = L \) Part 2: Express all sizes in terms of one size (M)
The relationships link from M to S, and then from S to L and XL. So, let's express everything in terms of M.
- We already have \( S = \frac{1}{3} M \).
- Now substitute this into the equation for L: \( L = \frac{1}{2} S = \frac{1}{2} \left(\frac{1}{3} M\right) = \frac{1}{6} M \).
- Since \( XL = L \), we also have \( XL = \frac{1}{6} M \). Part 3: Choose a convenient number for M to avoid fractions
The denominators are 3 and 6. The least common multiple is 6. Let's assume there are 6 medium shirts (M = 6).
- Number of Medium shirts: \( M = 6 \)
- Number of Small shirts: \( S = \frac{1}{3} \times 6 = 2 \)
- Number of Large shirts: \( L = \frac{1}{6} \times 6 = 1 \)
- Number of X-Large shirts: \( XL = L = 1 \) Part 4: Calculate the total number of shirts
\[ \text{Total} = M + S + L + XL = 6 + 2 + 1 + 1 = 10 \] Part 5: Calculate the percentage of medium shirts
\[ \text{Percentage of Medium} = \frac{\text{Number of Medium Shirts}}{\text{Total Number of Shirts}} \times 100% \] \[ \text{Percentage of Medium} = \frac{6}{10} \times 100% = 0.6 \times 100% = 60% \] Step 5: Final Answer
Medium shirts make up 60% of the total shirts in the store.