For a sales promotion, a dog food producer produces packages with 40 percent more dog food than the regular-size package...

1. Translate the problem requirements: We need to find how much less expensive the price per ounce becomes when we get 40% more dog food for the same total price as the regular package
2. Set up the comparison framework: Establish baseline values for the regular package (amount and price) to create a clear reference point
3. Calculate the new price per unit: Determine the price per ounce for the larger package that contains 40% more food but costs the same total amount
4. Find the percentage decrease: Compare the two price-per-ounce values to find the percent reduction, keeping in mind the answer choices suggest approximation is acceptable

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in everyday terms. Imagine you usually buy a regular bag of dog food that contains a certain amount and costs a certain price. Now the company is running a promotion where they give you a bigger bag with 40% more dog food, but they're charging you the same total price as the regular bag.

The question is asking: since you're getting more food for the same total money, how much cheaper is each ounce of dog food in the promotional bag compared to the regular bag?

To find this, we need to compare the "price per ounce" of both packages. The price per ounce is simply the total price divided by the total amount of food.

Process Skill: TRANSLATE - Converting the promotional offer into a clear mathematical comparison

2. Set up the comparison framework

To make this concrete and easy to work with, let's use simple numbers. Let's say:

- Regular package contains: 10 ounces of dog food
- Regular package costs: $10
- So the regular price per ounce = \(\$10 ÷ 10 \mathrm{ounces} = \$1 \mathrm{per\ ounce}\)

Now for the promotional package:

- It has 40% more food than regular = \(10 + (40\% \mathrm{of\ } 10) = 10 + 4 = 14 \mathrm{ounces}\)
- It costs the same as regular = $10
- So the promotional price per ounce = \(\$10 ÷ 14 \mathrm{ounces} ≈ \$0.714 \mathrm{per\ ounce}\)

Using concrete numbers like 10 and $10 makes our arithmetic much simpler while giving us the exact same percentage result.

3. Calculate the new price per unit

Now let's be more precise with our promotional package calculation:
Price per ounce for promotional package = \(\$10 ÷ 14 \mathrm{ounces}\)

To make this easier to work with, let's express this as a fraction:
\(\frac{\$10}{14} = \frac{\$5}{7} \mathrm{per\ ounce}\)

To convert this to a decimal: \(\frac{\$5}{7} ≈ \$0.714 \mathrm{per\ ounce}\)

So we have:
- Regular package: $1.00 per ounce
- Promotional package: $0.714 per ounce

4. Find the percentage decrease

To find what percent less the promotional price is compared to the regular price:

The difference in price per ounce = \(\$1.00 - \$0.714 = \$0.286\)

The percentage decrease = (Difference ÷ Original price) × 100%
Percentage decrease = \((\$0.286 ÷ \$1.00) × 100\% = 28.6\%\)

Let's verify this with exact fractions:

- Regular price per ounce: $1
- Promotional price per ounce: \(\frac{\$5}{7}\)
- Difference: \(\$1 - \frac{\$5}{7} = \frac{\$7}{7} - \frac{\$5}{7} = \frac{\$2}{7}\)
- Percentage decrease: \((\frac{\$2}{7} ÷ \$1) × 100\% = \frac{\$2}{7} × 100\% ≈ 28.57\%\)

Rounding to the nearest whole percent, this is approximately 29%.

4. Final Answer

The price per ounce of the larger package is approximately 29% less than the regular package.

Looking at our answer choices, this matches choice B: 29%.

To verify this makes sense: when you get 40% more product for the same price, the price per unit should decrease significantly, and 29% seems reasonable - it's substantial but not as large as the 40% increase in quantity, which is exactly what we'd expect mathematically.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "40% more" means

Students often confuse "40% more than the original" with "40% of the original." If the regular package has 10 ounces, 40% more means \(10 + (40\% \mathrm{of\ } 10) = 14 \mathrm{ounces}\), not just \(40\% \mathrm{of\ } 10 = 4 \mathrm{ounces}\). This fundamental misinterpretation would lead to incorrect calculations throughout the problem.

2. Focusing on total price difference instead of price per unit

Since both packages cost the same total price, students might think there's no price difference at all. The key insight is that we need to compare the "price per ounce" (unit price), not the total price. Missing this concept means missing the entire point of the problem.

3. Confusion about the reference point for percentage calculation

Students might set up the percentage calculation incorrectly by using the promotional package price as the reference instead of the regular package price. The question asks "what percent less is the price per ounce of the larger package compared to the regular-size package," so the regular package price should be the denominator.

Errors while executing the approach

1. Arithmetic errors when dividing by 14

Calculating \(\$10 ÷ 14 \mathrm{ounces}\) requires careful arithmetic. Students often struggle with this division, either getting the decimal wrong ($0.714...) or making errors when converting to the fraction \(\frac{\$5}{7}\). These calculation mistakes carry through to the final percentage.

2. Percentage decrease formula errors

The percentage decrease formula is \(\frac{\mathrm{(Original - New)}}{\mathrm{Original}} × 100\%\). Students frequently mix up the numerator and denominator, or forget to multiply by 100%, or use the wrong values in the formula. For example, using \(\frac{\mathrm{(New - Original)}}{\mathrm{New}}\) instead of \(\frac{\mathrm{(Original - New)}}{\mathrm{Original}}\).

3. Fraction arithmetic mistakes

When working with the exact calculation using fractions (\(\$1 - \frac{\$5}{7} = \frac{\$2}{7}\)), students often make errors in fraction subtraction or in converting \(\frac{\$2}{7}\) to a percentage. The calculation \((\frac{\$2}{7}) × 100\% ≈ 28.57\%\) requires careful handling of fractions.

Errors while selecting the answer

1. Selecting 40% because it matches the given increase

Seeing "40% more dog food" in the problem, students might instinctively choose 40% as the answer without doing the calculation. This is a classic trap - the percentage increase in quantity does not equal the percentage decrease in unit price.

2. Rounding errors or selecting the unrounded answer

Since the exact calculation gives 28.57%, students might look for this exact value among the choices or round incorrectly. The key is recognizing that 28.57% rounds to approximately 29%, making choice B correct, not looking for an exact decimal match.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient concrete values
Let's assign smart numbers that make calculations easy. Since we're dealing with percentages and need to find price per ounce, let's use:
• Regular package size: 100 ounces (makes percentage calculations clean)
• Regular package price: $100 (makes unit price calculations simple)

Step 2: Calculate the larger package specifications
• Larger package size: \(100 + (40\% \mathrm{of\ } 100) = 100 + 40 = 140 \mathrm{ounces}\)
• Larger package price: $100 (same as regular package per the problem)

Step 3: Calculate price per ounce for each package
• Regular package: \(\$100 ÷ 100 \mathrm{ounces} = \$1.00 \mathrm{per\ ounce}\)
• Larger package: \(\$100 ÷ 140 \mathrm{ounces} = \$0.714 \mathrm{per\ ounce}\) (approximately)

Step 4: Find the percentage decrease
Percentage decrease = (Original price per ounce - New price per ounce) ÷ Original price per ounce × 100%
= \((\$1.00 - \$0.714) ÷ \$1.00 × 100\%\)
= \(\$0.286 ÷ \$1.00 × 100\%\)
= \(28.6\% ≈ 29\%\)

Answer: B. 29%

Why these smart numbers work: Using 100 for both the package size and price creates clean calculations since percentages become whole numbers, and the price per ounce calculation becomes straightforward division that leads directly to the percentage change calculation.