A garden center sells a certain grass seed in 5-pound bags at $13.85 per bag, 10-pound bags at $20.43 per...

1. Translate the problem requirements: Find the minimum cost to purchase between 65-80 pounds of grass seed using three bag sizes: 5-pound (\(\$13.85\)), 10-pound (\(\$20.43\)), and 25-pound (\(\$32.25\)) bags.
2. Calculate cost efficiency for each bag type: Determine price per pound for each bag size to understand which offers the best value.
3. Identify the optimal base strategy: Since we need at least 65 pounds, determine how to use the most cost-effective bags as the foundation.
4. Systematically test feasible combinations: Generate all possible combinations that fall within the 65-80 pound range and calculate their total costs.
5. Select the minimum cost option: Compare all valid combinations to identify the least expensive solution.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to find in everyday terms. We have a customer who wants to buy grass seed, and they need between 65 and 80 pounds total. Think of this like shopping for groceries - you need a certain amount, but you want to spend as little as possible.

The garden center offers three different bag sizes:

• Small bags: 5 pounds for \(\$13.85\) each
• Medium bags: 10 pounds for \(\$20.43\) each
• Large bags: 25 pounds for \(\$32.25\) each

Our goal is to find the combination of these bags that gives us at least 65 pounds but no more than 80 pounds, while spending the least amount of money.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical requirements

2. Calculate cost efficiency for each bag type

To make smart shopping decisions, let's figure out how much we pay per pound for each bag size. This is like comparing unit prices at the grocery store.

For the 5-pound bags: \(\$13.85 ÷ 5 \text{ pounds} = \$2.77 \text{ per pound}\)
For the 10-pound bags: \(\$20.43 ÷ 10 \text{ pounds} = \$2.043 \text{ per pound}\)
For the 25-pound bags: \(\$32.25 ÷ 25 \text{ pounds} = \$1.29 \text{ per pound}\)

This tells us something important: the bigger bags give us much better value! The 25-pound bags cost only \(\$1.29\) per pound, while the small 5-pound bags cost more than double that at \(\$2.77\) per pound.

So our strategy should be: use as many large 25-pound bags as possible, then fill in the remaining weight with the next best option.

3. Identify the optimal base strategy

Since we need at least 65 pounds and the 25-pound bags are cheapest per pound, let's see how many we can use as our foundation.

If we buy 2 large bags: \(2 × 25 = 50 \text{ pounds}\)
We still need: \(65 - 50 = 15 \text{ more pounds minimum}\)

If we buy 3 large bags: \(3 × 25 = 75 \text{ pounds}\)
This already gives us 75 pounds, which is within our 65-80 pound range!

But wait - we should check if 2 large bags plus some smaller bags might be cheaper, even though the large bags have better per-pound pricing. Sometimes the exact quantities needed can make a difference.

4. Systematically test feasible combinations

Let's check all realistic combinations that get us between 65-80 pounds:

Option A: 3 large bags (75 pounds)
Cost: \(3 × \$32.25 = \$96.75\)
Weight: 75 pounds ✓ (within 65-80 range)

Option B: 2 large bags + smaller bags
2 large bags = 50 pounds, need 15-30 more pounds

• 2 large + 2 medium bags: \(50 + 20 = 70 \text{ pounds}\)
  Cost: \(2(\$32.25) + 2(\$20.43) = \$64.50 + \$40.86 = \$105.36\)
• 2 large + 1 medium + 1 small: \(50 + 10 + 5 = 65 \text{ pounds}\)
  Cost: \(2(\$32.25) + \$20.43 + \$13.85 = \$64.50 + \$34.28 = \$98.78\)
• 2 large + 3 small bags: \(50 + 15 = 65 \text{ pounds}\)
  Cost: \(2(\$32.25) + 3(\$13.85) = \$64.50 + \$41.55 = \$106.05\)

Option C: 1 large bag + smaller bags
1 large bag = 25 pounds, need 40-55 more pounds
This would require many smaller bags and would be expensive based on our per-pound analysis.

Process Skill: CONSIDER ALL CASES - Systematically checking all feasible combinations prevents missing the optimal solution

5. Select the minimum cost option

Comparing our realistic options:

• 3 large bags (75 lbs): \(\$96.75\)
• 2 large + 2 medium (70 lbs): \(\$105.36\)
• 2 large + 1 medium + 1 small (65 lbs): \(\$98.78\)
• Other combinations cost even more

The winner is 3 large bags for \(\$96.75\).

This makes sense because the large bags offered the best value per pound, and buying exactly 3 of them gives us 75 pounds (well within our 65-80 range) at the lowest total cost.

Final Answer

The least possible cost is \(\$96.75\), which corresponds to answer choice B.

This is achieved by purchasing 3 bags of the 25-pound size, giving the customer exactly 75 pounds of grass seed for \(\$96.75\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the weight constraint range
Students often focus only on the minimum requirement of "at least 65 pounds" and forget about the upper limit of "no more than 80 pounds." They might try combinations that exceed 80 pounds, thinking more is always better for bulk purchasing. This leads them to consider options like 4 large bags (100 pounds) which violates the constraint.

2. Assuming the cheapest per-pound option is always optimal
After calculating that 25-pound bags have the lowest cost per pound (\(\$1.29\)), students might immediately conclude that using only large bags is the answer without checking if mixing bag sizes could result in a lower total cost for the specific weight range needed.

3. Incomplete systematic approach
Students may check only a few obvious combinations (like all large bags or all small bags) rather than systematically exploring realistic mixed combinations. They might miss checking combinations like "2 large + 1 medium + 1 small" which could potentially be optimal.

Errors while executing the approach

1. Arithmetic errors in cost calculations
When calculating total costs for different combinations, students frequently make addition or multiplication errors. For example, when computing \(2(\$32.25) + \$20.43 + \$13.85\), they might incorrectly calculate \(\$64.50 + \$34.28\) as \(\$98.87\) instead of \(\$98.78\).

2. Errors in per-pound cost calculations
Students may make division errors when calculating cost per pound, such as computing \(\$20.43 ÷ 10 = \$2.43\) instead of \(\$2.043\), which could lead them to incorrectly rank the cost-effectiveness of different bag sizes.

3. Missing feasible combinations during enumeration
When systematically checking combinations, students might skip valid options or double-count others. They may forget to check combinations like "2 large + 2 medium" or incorrectly calculate the total weight for certain combinations.

Errors while selecting the answer

1. Selecting a combination that meets requirements but isn't the minimum cost
After finding several valid combinations (like 75 lbs for \(\$96.75\) and 65 lbs for \(\$98.78\)), students might choose the one that exactly meets the minimum weight requirement (65 lbs) thinking it should be cheapest, rather than selecting the actual lowest cost option.

2. Choosing the wrong answer choice due to calculation errors
Even if students follow the correct approach and identify that 3 large bags is optimal, they might select answer choice A (\(\$94.03\)) or C (\(\$98.78\)) if they made arithmetic errors earlier in their calculations, rather than the correct answer B (\(\$96.75\)).