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One night a certain hotel rented \(\frac{3}{4}\) of its rooms, including \(\frac{2}{3}\) of their air conditioned rooms. If \(\frac{3}{5}\) of its rooms were air conditioned, what percent of the rooms that were not rented were air conditioned?
Let's break down what we're actually looking for in plain English. We have a hotel with some rooms. Some rooms got rented that night, others didn't. Some rooms have air conditioning, others don't. The question is asking: "Of all the rooms that were NOT rented, what percentage of those empty rooms had air conditioning?"
To find this, we need to figure out:
Process Skill: TRANSLATE - Converting the complex fractional relationships into a clear question about empty rooms with AC
Instead of getting lost in fractions, let's pick a number of total rooms that makes all our fractions work out to whole numbers. We have fractions with denominators 4, 3, and 5, so let's use a total that's divisible by all of these.
The least common multiple of 4, 3, and 5 is 60. So let's say our hotel has 60 total rooms.
This smart choice means:
Process Skill: SIMPLIFY - Choosing convenient numbers to avoid messy fraction arithmetic
Now let's figure out the actual numbers:
Total rooms: 60
Rooms with air conditioning: \(\frac{3}{5} \times 60 = 36\) rooms
Rooms that were rented: \(\frac{3}{4} \times 60 = 45\) rooms
Air conditioned rooms that were rented: \(\frac{2}{3} \times 36 = 24\) rooms
Let's double-check this makes sense: We rented 45 rooms total, and 24 of those rented rooms had AC. That seems reasonable.
Now we can find what we need:
Rooms that were NOT rented: \(60 - 45 = 15\) rooms
Air conditioned rooms that were NOT rented: \(36 - 24 = 12\) rooms
Think about it this way: We have 36 air conditioned rooms total. We rented 24 of them. So the remaining 12 air conditioned rooms must be among the empty rooms.
We have 15 empty rooms total, and 12 of them have air conditioning.
Percentage = (Air conditioned empty rooms ÷ Total empty rooms) × \(100\%\)
Percentage = \((12 \div 15) \times 100\% = 0.8 \times 100\% = 80\%\)
The answer is E. 80
To verify: \(80\%\) of the non-rented rooms were air conditioned. This means that most of the empty rooms had AC, which makes sense given that \(\frac{2}{3}\) of the AC rooms got rented (leaving \(\frac{1}{3}\) of them empty) while \(\frac{3}{4}\) of all rooms got rented (leaving \(\frac{1}{4}\) of all rooms empty).
1. Misinterpreting what the question is asking for
Students often confuse what percentage they need to calculate. The question asks for "what percent of the rooms that were NOT rented were air conditioned" but students might mistakenly try to find what percent of air conditioned rooms were not rented, or what percent of total rooms were both not rented AND air conditioned. This confusion about the denominator (not rented rooms vs. air conditioned rooms vs. total rooms) leads to completely wrong setups.
2. Getting overwhelmed by multiple fractions and overlapping categories
With three different fractions (\(\frac{3}{4}\) rented, \(\frac{2}{3}\) of AC rooms rented, \(\frac{3}{5}\) rooms are AC), students often struggle to see how these relate to each other. They might try to add or subtract fractions directly without understanding that these fractions refer to different base quantities, leading to nonsensical calculations like \(\frac{3}{4} + \frac{2}{3} - \frac{3}{5}\).
3. Not recognizing the need to find overlapping groups systematically
Students may fail to realize they need to organize the rooms into clear categories (rented AC, rented non-AC, not rented AC, not rented non-AC). Instead, they might jump straight into calculations without a clear framework, missing the crucial step of finding how many AC rooms were NOT rented.
1. Arithmetic errors when working with the chosen total
Even when students correctly choose 60 as their total rooms, they often make calculation mistakes: \(\frac{3}{5} \times 60 = 36\) becomes 35, or \(\frac{2}{3} \times 36 = 24\) becomes 26. These small arithmetic errors cascade through the problem and lead to wrong final percentages.
2. Incorrectly calculating the number of non-rented air conditioned rooms
Students frequently struggle with the key calculation: if there are 36 total AC rooms and 24 AC rooms were rented, then 12 AC rooms were NOT rented. Common mistakes include subtracting in the wrong order (24 - 36) or using the wrong numbers entirely (subtracting from total rooms instead of total AC rooms).
3. Setting up the wrong fraction for the final percentage
When calculating the final percentage, students often flip the fraction, calculating (total non-rented rooms ÷ non-rented AC rooms) instead of (non-rented AC rooms ÷ total non-rented rooms). This gives them \(\frac{15}{12} = 125\%\) instead of \(\frac{12}{15} = 80\%\), leading them to think they made an error when they actually set up the wrong ratio.
1. Converting decimal to percentage incorrectly
When students correctly calculate \(\frac{12}{15} = 0.8\), they sometimes forget to multiply by 100 to convert to percentage, selecting 0.8 if it were an option, or they might multiply by 10 instead of 100, getting \(8\%\) and looking for the closest answer choice.
2. Second-guessing the high percentage result
Since \(80\%\) seems like a high percentage, students often doubt their answer and pick a more "moderate" option like \(40\%\). They don't trust that it's reasonable for most empty rooms to have AC, especially when the problem setup actually supports this outcome (fewer AC rooms were rented proportionally).
Step 1: Choose a convenient total number of rooms
Since we're dealing with fractions \(\frac{3}{4}\), \(\frac{2}{3}\), and \(\frac{3}{5}\), we need a total that makes all calculations result in whole numbers. The least common multiple of the denominators 4, 3, and 5 is 60.
Let's use 60 total rooms.
Step 2: Calculate room counts for each category
Step 3: Find non-rented air conditioned rooms
Step 4: Calculate the percentage
We want: (Non-rented air conditioned rooms) ÷ (Total non-rented rooms) × \(100\%\)
Percentage = \((12 \div 15) \times 100\% = \left(\frac{4}{5}\right) \times 100\% = 80\%\)
Answer: E. 80
The smart numbers approach works perfectly here because the LCM of our denominators gives us a logical, systematic way to choose our total, making all subsequent calculations clean and avoiding fractions entirely.