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In this puzzle, you need to find two mystery numbers based on the product and sum given in each row. These numbers are derived from the multiplication facts from 1 to 12, and they could be either positive or negative.

Let me guide you step-by-step through each row, solving for the smaller and greater numbers using basic algebra.

For each row, let’s denote the mystery numbers as \(x\) (smaller) and \(y\) (greater), and we have the following two equations:

1. \( x \times y = \text{Product} \)
2. \( x + y = \text{Sum} \)

We can solve these equations by substitution or factoring.

### Row A) Product = 84, Sum = 19
We need two numbers whose product is 84 and sum is 19. The pair that works is 7 and 12 because:
- \( 7 \times 12 = 84 \)
- \( 7 + 12 = 19 \)

Thus, the smaller number is 7 and the greater number is 12.

### Row B) Product = -72, Sum = 6
We need two numbers whose product is -72 and sum is 6. The correct pair is -6 and 12 because:
- \( -6 \times 12 = -72 \)
- \( -6 + 12 = 6 \)

Thus, the smaller number is -6 and the greater number is 12.

### Row C) Product = -27, Sum = -6
We need two numbers whose product is -27 and sum is -6. The correct pair is -9 and 3 because:
- \( -9 \times 3 = -27 \)
- \( -9 + 3 = -6 \)

Thus, the smaller number is -9 and the greater number is 3.

### Row D) Product = 54, Sum = -15
We need two numbers whose product is 54 and sum is -15. The correct pair is -9 and -6 because:
- \( -9 \times -6 = 54 \)
- \( -9 + (-6) = -15 \)

Thus, the smaller number is -9 and the greater number is -6.

### Row E) Product = 40, Sum = 13
We need two numbers whose product is 40 and sum is 13. The correct pair is 5 and 8 because:
- \( 5 \times 8 = 40 \)
- \( 5 + 8 = 13 \)

Thus, the smaller number is 5 and the greater number is 8.

### Row F) Product = -32, Sum = 4
We need two numbers whose product is -32 and sum is 4. The correct pair is -4 and 8 because:
- \( -4 \times 8 = -32 \)
- \( -4 + 8 = 4 \)

Thus, the smaller number is -4 and the greater number is 8.

### Row G) Product = -24, Sum = -5
We need two numbers whose product is -24 and sum is -5. The correct pair is -8 and 3 because:
- \( -8 \times 3 = -24 \)
- \( -8 + 3 = -5 \)

Thus, the smaller number is -8 and the greater number is 3.

### Row H) Product = 96, Sum = -20
We need two numbers whose product is 96 and sum is -20. The correct pair is -12 and -8 because:
- \( -12 \times -8 = 96 \)
- \( -12 + (-8) = -20 \)

Thus, the smaller number is -12 and the greater number is -8.

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### Final Answers:
A) Smaller: 7, Greater: 12  
B) Smaller: -6, Greater: 12  
C) Smaller: -9, Greater: 3  
D) Smaller: -9, Greater: -6  
E) Smaller: 5, Greater: 8  
F) Smaller: -4, Greater: 8  
G) Smaller: -8, Greater: 3  
H) Smaller: -12, Greater: -8

Would you like a more detailed explanation for any of the rows or methods used? Let me know if you have any questions!

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Here are 5 related questions you could explore:
1. How do you factor quadratic equations to find pairs of numbers?
2. What are the common methods to solve systems of equations?
3. How does changing the sign of a product or sum affect the solution?
4. How can you apply these techniques to solve real-world problems?
5. Can this method be used for numbers larger than 12?

**Tip**: Always check your product and sum calculations when dealing with negative numbers, as signs are crucial to finding the correct pair!

Find two mystery numbers based on given products and sums using the multiplication facts table. Mystery numbers can be positive or negative.

Learn how to find mystery numbers using the multiplication facts from 1 to 12. This problem involves solving systems of equations where the product and sum of two numbers are provided. Suitable for Grades 6-8.

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