To determine the minimum amount the student needs to save every month, we first need to calculate how much the student is responsible for after accounting for contributions from the grandparents and the athletic scholarship.

Step 1: Calculate the grandparents' contribution

Since the grandparents will pay half of the total cost, their contribution will be:

$\frac{19,700}{2} = 9,850$

Step 2: Subtract the scholarship amount

The student will also receive an athletic scholarship of $5,000. So, the total amount the student needs to cover will be:

$19,700 - (9,850 + 5,000) = 19,700 - 14,850 = 4,850$

Step 3: Determine the monthly savings required

The student needs to save $4,850 over 12 months. To find the amount to save each month, we divide the remaining cost by 12:

$\frac{4,850}{12} \approx 404.17$

Thus, the minimum amount the student will need to save every month is approximately $404.17.

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Would you like further details or clarification on any part of this solution?

Here are 5 related questions for further exploration:

1. How would the monthly savings change if the grandparents paid 60% instead of 50%?
2. What if the total cost increased by 5% next year? How much more would the student need to save monthly?
3. How much more would the student need to save if the scholarship amount decreased by $1,000?
4. What if the student had 18 months to save instead of 12? How would that affect the monthly amount?
5. How would borrowing a loan for $2,000 change the savings plan?

Tip: When budgeting for a large goal, consider saving a bit more each month to cover unexpected expenses or price increases.