The charge for a single room at Hotel P is 25 percent less than the charge for a single room...

1. Translate the problem requirements: We need to understand the relative pricing relationships between three hotels (P, R, and G) and find what percent greater Hotel R's charge is compared to Hotel G's charge.
2. Set up the relationships using a strategic base variable: Choose Hotel P's charge as our base since both other hotels are described relative to it, avoiding fractions in our initial setup.
3. Express all hotel charges in terms of the base variable: Use the given percentage relationships to write Hotel R and Hotel G charges in terms of Hotel P's charge.
4. Calculate the percentage difference between Hotel R and Hotel G: Apply the standard percent greater formula using the simplified expressions.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us in plain English:

We have three hotels: P, R, and G. We're told:
• Hotel P charges 25% less than Hotel R
• Hotel P charges 10% less than Hotel G
• We need to find: Hotel R's charge is what percent greater than Hotel G's charge?

Think of it this way: if Hotel P is cheaper than both R and G, but it's much cheaper compared to R (25% less) than compared to G (10% less), then Hotel R must be more expensive than Hotel G. We need to figure out by how much.

Process Skill: TRANSLATE - Converting the percentage relationships into a clear mathematical understanding

2. Set up the relationships using a strategic base variable

Here's the smart move: instead of picking Hotel R or G as our starting point, let's use Hotel P's charge as our base. Why? Because both other hotels are described in relation to what Hotel P costs.

Let's say Hotel P charges $100 for a room (we can pick any number - the percentage relationships will work out the same).

Now we can work backwards from this $100 to find what R and G charge:
• If P charges 25% less than R, then P pays 75% of what R charges
• If P charges 10% less than G, then P pays 90% of what G charges

This approach keeps our numbers clean and avoids messy fractions right from the start.

3. Express all hotel charges in terms of the base variable

Starting with P = $100, let's find R and G:

For Hotel R:
P is 25% less than R, so \(\mathrm{P} = 75\% \text{ of } \mathrm{R}\)
\(\$100 = 0.75 \times \mathrm{R}\)
\(\mathrm{R} = \$100 \div 0.75 = \$100 \times \frac{4}{3} = \$133.33\)

For Hotel G:
P is 10% less than G, so \(\mathrm{P} = 90\% \text{ of } \mathrm{G}\)
\(\$100 = 0.90 \times \mathrm{G}\)
\(\mathrm{G} = \$100 \div 0.90 = \$100 \times \frac{10}{9} = \$111.11\)

So we have:
• Hotel P: $100
• Hotel R: $133.33
• Hotel G: $111.11

Let's verify this makes sense: R ($133.33) is indeed more expensive than G ($111.11), which matches our intuition.

4. Calculate the percentage difference between Hotel R and Hotel G

Now we need to find what percent greater R is compared to G.

The formula for "percent greater" is:
\(\text{Percent greater} = \frac{\text{Difference}}{\text{Base amount}} \times 100\%\)

Here:
• R charges $133.33
• G charges $111.11
• Difference = $133.33 - $111.11 = $22.22
• Base amount = G's charge = $111.11

\(\text{Percent greater} = \frac{\$22.22}{\$111.11} \times 100\%\)

Let's simplify this calculation:
\(\frac{\$22.22}{\$111.11} = 0.2 = 20\%\)

Therefore, Hotel R's charge is 20% greater than Hotel G's charge.

Final Answer

Hotel R charges 20% more than Hotel G.

The answer is B. 20%

Verification: We can double-check by working with exact fractions:
• \(\mathrm{R} = \frac{\$400}{3}, \mathrm{G} = \frac{\$1000}{9}\)
• \(\text{Difference} = \frac{\$400}{3} - \frac{\$1000}{9} = \frac{\$200}{9}\)
• \(\text{Percent greater} = \frac{\$200/9}{\$1000/9} \times 100\% = \frac{200}{1000} \times 100\% = 20\%\) ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the percentage relationships: Students often confuse "25% less than R" with "25% of R". When P is 25% less than R, it means \(\mathrm{P} = 0.75\mathrm{R}\), not \(\mathrm{P} = 0.25\mathrm{R}\). This fundamental misunderstanding of percentage language leads to completely incorrect relationships.

2. Setting up relationships in the wrong direction: Students may incorrectly interpret "P is 25% less than R" as "R is 25% less than P" and set up the equation as \(\mathrm{R} = 0.75\mathrm{P}\) instead of \(\mathrm{P} = 0.75\mathrm{R}\). This reversal of the relationship direction creates wrong mathematical foundations.

3. Choosing an inefficient base variable: Many students try to assign variables directly to R or G charges, leading to complex fractional relationships. Not recognizing that using P as the base (since both R and G are described relative to P) simplifies calculations significantly.

Errors while executing the approach

1. Arithmetic errors when converting percentages to decimals: Students may incorrectly calculate \(100 \div 0.75\) or \(100 \div 0.90\), especially when dealing with division by decimals. Some might forget to multiply by reciprocals \(\frac{4}{3}\) for 0.75 or \(\frac{10}{9}\) for 0.90) and make computational mistakes.

2. Using wrong base for percentage calculation: When calculating "what percent greater R is than G", students often use R as the base instead of G. They might calculate \(\frac{\mathrm{R}-\mathrm{G}}{\mathrm{R}} \times 100\%\) instead of the correct \(\frac{\mathrm{R}-\mathrm{G}}{\mathrm{G}} \times 100\%\), leading to incorrect percentage comparisons.

3. Rounding errors compounding: Students working with rounded decimals ($133.33, $111.11) throughout may accumulate rounding errors that lead to slight variations in the final percentage, potentially causing them to select a close but incorrect answer choice.

Errors while selecting the answer

1. Selecting the percentage difference instead of percentage greater: Students might calculate the simple difference between R and G as a percentage of their average or as a percentage point difference, rather than calculating how much greater R is compared to G's base value.

2. Confusing the direction of comparison: Students who correctly calculate that one hotel is 20% different from another might select the answer without carefully checking whether they calculated "R greater than G" or "G greater than R", especially if they mixed up their variable assignments during solving.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a strategic smart number

Let's set Hotel P's charge = $90. This number is chosen because it will eliminate fractions when we calculate the other hotel charges based on the given percentage relationships.

Step 2: Calculate Hotel G's charge

Since Hotel P's charge is 10% less than Hotel G's charge:
\(\text{Hotel P} = 0.90 \times \text{Hotel G}\)
\(90 = 0.90 \times \text{Hotel G}\)
\(\text{Hotel G} = 90 \div 0.90 = \$100\)

Step 3: Calculate Hotel R's charge

Since Hotel P's charge is 25% less than Hotel R's charge:
\(\text{Hotel P} = 0.75 \times \text{Hotel R}\)
\(90 = 0.75 \times \text{Hotel R}\)
\(\text{Hotel R} = 90 \div 0.75 = \$120\)

Step 4: Find what percent greater Hotel R is than Hotel G

\(\text{Percent greater} = \frac{\text{Hotel R} - \text{Hotel G}}{\text{Hotel G}} \times 100\%\)
\(\text{Percent greater} = \frac{120 - 100}{100} \times 100\%\)
\(\text{Percent greater} = \frac{20}{100} \times 100\% = 20\%\)

Verification

Our relationships check out:
- Hotel P ($90) is 25% less than Hotel R ($120): \(90 = 120 \times 0.75\) ✓
- Hotel P ($90) is 10% less than Hotel G ($100): \(90 = 100 \times 0.90\) ✓

Therefore, Hotel R's charge is 20% greater than Hotel G's charge.

Answer: B. 20%