Sure, let's break this problem down step by step.
1. Define the costs and budget:
- The cost of one rose is [tex]$3.
- The cost of one carnation is $[/tex]2.
- John's total budget is [tex]$35.
2. Formulate the inequality:
Let \( r \) represent the number of roses and \( c \) represent the number of carnations. The total amount John spends on roses and carnations must be less than or equal to his budget. This can be represented by the inequality:
\[
3r + 2c \leq 35
\]
3. Calculate the cost of the 8 carnations John buys:
John buys 8 carnations, so we calculate the total cost of carnations:
\[
\text{Cost of 8 carnations} = 8 \times 2 = 16 \text{ dollars}
\]
4. Determine the remaining budget after buying carnations:
Subtract the amount spent on carnations from the total budget:
\[
\text{Remaining budget} = 35 - 16 = 19 \text{ dollars}
\]
5. Calculate the maximum number of roses John can buy with the remaining budget:
Since each rose costs $[/tex]3, we divide the remaining budget by the cost of one rose:
[tex]\[
\text{Maximum number of roses} = \left\lfloor \frac{19}{3} \right\rfloor = 6 \text{ roses}
\][/tex]
Thus, after buying 8 carnations, John can buy a maximum of 6 roses with the remaining budget.
