Susan invited 13 of her friends for her birthday party and created return gift hampers comprising one each of $3, $4, and $5 gift certificates. One of her friends did not turn up and Susan decided to rework her gift hampers such that each of the 12 friends who turned up got $13 worth gift certificates. How many gift hampers did not contain $5 gift certificates in the new configuration?

To solve this, let's analyze the problem step-by-step:

1. **Initial Setup**:
   - Susan initially made 13 gift hampers for her friends who confirmed to attend the party.
   - Each gift hamper was composed of one $3, one $4, and one $5 gift certificate.
   - Therefore, each gift hamper initially contained:
     \[
     3 + 4 + 5 = 12 \text{ dollars}
     \]

2. **One Friend Did Not Turn Up**:
   - Only 12 friends attended the party.
   - Susan now needs to adjust the hampers so that each of the 12 friends gets a total of $13 in gift certificates.

3. **Total Value of All Certificates**:
   - Originally, 13 hampers were made, each worth $12, which means:
     \[
     13 \times 12 = 156 \text{ dollars}
     \]
   - This $156 must now be distributed among the 12 friends who actually attended the party.

4. **New Gift Value Per Friend**:
   - Susan decided that each friend who attended should receive $13.
   - Therefore, the total amount needed is:
     \[
     12 \times 13 = 156 \text{ dollars}
     \]
   - Since both the original total (156 dollars) and the new requirement (156 dollars) match, Susan has enough certificates.

5. **Finding the Configuration**:
   - We need each friend to receive $13 in certificates, but each original hamper was worth $12. To increase the value from $12 to $13 per hamper, Susan can adjust the composition of the hampers.
   - Since we need to increase by $1 per hamper, Susan can replace some $5 certificates with $4 certificates to reach a total of $13 without going over.

6. **Calculating Hampers Without $5 Certificates**:
   - If a hamper includes a $3, two $4 certificates, and no $5 certificate, it would be worth:
     \[
     3 + 4 + 4 = 13 \text{ dollars}
     \]
   - Let $ x $ represent the number of hampers with this configuration (no $5 certificate).

7. **Setting Up the Equation**:
   - Out of the 12 hampers, $ x $ hampers contain no $5 certificate, while the remaining $ 12 - x $ hampers retain the original configuration of $3, $4, and $5 certificates, totaling $12 each.
   - Since each hamper must be worth $13, we can set up the equation for the total value:
     \[
     13x + 12(12 - x) = 156
     \]
   - Simplifying,
     \[
     13x + 144 - 12x = 156
     \]
     \[
     x = 12
     \]

8. **Conclusion**:
   - Therefore, all 12 hampers needed to reach $13 by excluding the $5 certificate.

Thus, **all 12 gift hampers did not contain $5 gift certificates**.

Would you like a more detailed breakdown, or do you have any other questions?

### Related Questions
1. How would the solution change if Susan wanted each friend to receive $15 worth of certificates?
2. What if two friends did not attend the party? How would Susan distribute the certificates?
3. Can you create a general formula for different numbers of friends and certificate values?
4. If Susan had additional types of certificates (e.g., $1 and $2), how would this affect the solution?
5. What if the certificates had to total exactly $14? What changes would be needed?

**Tip:** When facing problems with fixed totals, try expressing unknowns as variables to set up equations easily.

Learn how to solve a real-world algebra problem involving the distribution of $3, $4, and $5 gift certificates among party guests. This problem demonstrates how to use linear equations to adjust gift hampers, ensuring each friend receives exactly $13 in certificates. Suitable for students in Grades 6-8.

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