To find the concentration of the new mixture, we need to calculate the total amount of sugar in the solution and divide it by the total volume of the mixture.

Step-by-Step Solution:

1. Calculate the amount of sugar in each solution:

   • For the 2% sugar solution: $\text{Amount of sugar} = 8 \, \text{qt} \times \frac{2}{100} = 0.16 \, \text{qt}$

   • For the 51% sugar solution: $\text{Amount of sugar} = 6 \, \text{qt} \times \frac{51}{100} = 3.06 \, \text{qt}$

2. Calculate the total amount of sugar in the mixture:

   $\text{Total sugar} = 0.16 \, \text{qt} + 3.06 \, \text{qt} = 3.22 \, \text{qt}$

3. Calculate the total volume of the mixture:

   $\text{Total volume} = 8 \, \text{qt} + 6 \, \text{qt} = 14 \, \text{qt}$

4. Calculate the concentration of the new mixture:

   $\text{Concentration} = \left( \frac{\text{Total sugar}}{\text{Total volume}} \right) \times 100 = \left( \frac{3.22 \, \text{qt}}{14 \, \text{qt}} \right) \times 100$

   $\text{Concentration} = \left(0.23\right) \times 100 = 23\%$

Conclusion:

The concentration of the new mixture is 23%.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would the concentration change if we used different volumes of each solution?
2. What if we mixed a 10% sugar solution instead of a 51% one?
3. How can you find the volume of a specific concentration needed to achieve a target mixture concentration?
4. How would you create a 10% solution by mixing a 2% and a 51% solution?
5. How does dilution affect the concentration of a solution?

Tip: Always ensure the units match when performing mixture calculations to avoid errors.