So I buy a $257 gift card at 3% off (no fee, no cashback). I then buy a $250 gift card with a $7 fee and get $21.995 off, I then use that $250 gift card to buy a $257 gift card at 3% off (no fee, no cashback). I then use that gift card to buy a $250 gift card with a $7 fee and $21.995 discount. I want the savings for the final card via the savings chain.

Let's break down the series of transactions to understand how the savings accumulate in the final $250 gift card purchase through the chain of savings.

1. **First Purchase - $257 Gift Card at 3% Off:**
   - The discount is $ 3\% $ of $257.
   - $ \text{Discount} = 257 \times 0.03 = 7.71 $.
   - So, the actual cost of the $257 gift card is $ 257 - 7.71 = 249.29 $.

2. **Second Purchase - $250 Gift Card with $7 Fee and $21.995 Discount:**
   - The cost would be $250 + $7 (fee) = $257, but you get a discount of $21.995.
   - The effective price of this $250 gift card is $ 257 - 21.995 = 235.005 $.

3. **Third Purchase - $257 Gift Card at 3% Off (using the second gift card):**
   - The discount here is $ 3\% $ of $257 again.
   - $ \text{Discount} = 257 \times 0.03 = 7.71 $.
   - So, the cost of the $257 gift card is $ 257 - 7.71 = 249.29 $, paid using the $250 gift card.

4. **Final Purchase - $250 Gift Card with $7 Fee and $21.995 Discount (using the third gift card):**
   - The cost for the final $250 gift card is the same as in step 2.
   - The cost is $257 with a discount of $21.995, so the effective price is $ 257 - 21.995 = 235.005 $.

### **Final Savings Calculation**

Since each $257 gift card purchase was made at a 3% discount, and each subsequent $250 gift card purchase includes a $7 fee but also a $21.995 discount, the total savings on the final $250 gift card is simply the cumulative effect of the discount from the chain.

In this case, the total savings from the discounts over the process result in an **effective savings of $21.995 on the final $250 gift card**.

### Do you need further details or have any questions?

Here are 5 follow-up questions:
1. How does the cycle affect total expenditure compared to savings?
2. What happens to savings if the fee or discount changes?
3. How would the process change with cashback offers included?
4. Could you repeat this indefinitely, or is there a limiting factor?
5. How would this approach scale with larger gift card amounts?

**Tip:** Keep track of any fees, as they can reduce the value of discounts or cashback over time!

Explore how to calculate cumulative savings when purchasing a series of gift cards with percentage discounts and fees. This detailed breakdown covers algebraic principles and discount formulas, suitable for Grades 10-12.

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