Let's break down the problem step by step.

Step 1: Initial conditions

• The number of daffodils initially is $2t + 1$.
• The number of roses is 14.

Step 2: After adding 5 more daffodils

• The new number of daffodils becomes $2t + 1 + 5 = 2t + 6$.
• The number of roses remains 14.

Step 3: Given ratio

• The new ratio of daffodils to roses is $\frac{11}{7}$.

This gives the equation: $\frac{2t + 6}{14} = \frac{11}{7}$

Step 4: Solve for $t$

Cross-multiply to eliminate the fraction: $7(2t + 6) = 11 \times 14$ Simplifying both sides: $7(2t + 6) = 154$ $14t + 42 = 154$

Subtract 42 from both sides: $14t = 112$

Divide by 14: $t = 8$

Final Answer:

The value of $t$ is $\boxed{8}$.

Do you have any questions, or would you like further details on any step?

Here are some related questions to consider:

1. How would the problem change if the ratio was different, such as $3:2$?
2. What if 10 daffodils were added instead of 5?
3. How can we generalize this problem for other flower types and ratios?
4. What happens if the number of roses also changes over time?
5. How could we represent this situation graphically to illustrate the change?

Tip: Always check the dimensions of both sides of an equation when working with ratios to ensure consistency.