Sure! Let's go through this problem step by step to find out how many weeks the "Math is Awesome" Foundation can give out scholarships before it runs out of money.
1. Establishing the Basics:
- The total amount of money available initially is \$1,000,000.
- The scholarship amount increases by \$1,000 each week. This means:
- In the first week, \$1,000 is given out.
- In the second week, \$2,000 is given out.
- In the third week, \$3,000 is given out.
- And so on.
2. Understanding the Problem:
- We need to determine the number of weeks the Foundation can continue this incrementing scholarship amount before the funds run out.
3. Calculating the Weekly Scholarships and Deducting from the Total:
- Each week, the scholarship amount is just 1,000 multiplied by the week number.
- Week 1: Scholarship amount = \$1,000
- Week 2: Scholarship amount = \$2,000
- Week 3: Scholarship amount = \$3,000
- The total amount deducted at the end of each week is cumulative:
- After week 1: Remaining money = \[tex]$1,000,000 - \$[/tex]1,000 = \$999,000
- After week 2: Remaining money = \[tex]$999,000 - \$[/tex]2,000 = \$997,000
- After week 3: Remaining money = \[tex]$997,000 - \$[/tex]3,000 = \$994,000
- This pattern continues each week.
4. Condition to Stop:
- The Foundation will stop giving out scholarships when it does not have enough money to fulfill the next week's scholarship amount.
- This happens when the total remaining money is less than the scholarship amount for the upcoming week.
5. Achieving the Result:
- Through the calculations, we determine that the Foundation is able to give out scholarships for a total of 44 weeks.
6. Summary of Results:
- After 44 weeks, the Foundation has \$10,000 remaining.
- The scholarship given out in the 44th week amounts to \$44,000.
- Attempting to give out a scholarship in the 45th week would require \[tex]$45,000, which the Foundation cannot afford with only \$[/tex]10,000 remaining.
Thus, the Foundation can give out scholarships for 44 weeks before the balance is too low to give out the full scholarship for the next week.
