Jeff bought flowers for his mom. He bought 24 flowers that were a combination of roses and tulips. The roses cost $3.00 each and the tulips cost $1.25 each. He spent a total of $58.00 on flowers. How many of each flower did he buy?
Answer (1)
Define 'r' as the number of roses and 't' as the number of tulips.
Set up the equations: r + t = 24 and 3 r + 1.25 t = 58 .
Solve for 'r' by substituting t = 24 − r into the second equation: r = 16 .
Find 't' using t = 24 − r : t = 8 . The final answer is 16 roses and 8 tulips .
Explanation
Problem Analysis Let's analyze the problem. We know that Jeff bought a total of 24 flowers, which were a combination of roses and tulips. Roses cost $3.00 each, and tulips cost $1.25 each. The total amount he spent was $58.00. Our goal is to find out how many of each type of flower he bought.
Setting up the Equations Let's define our variables. Let 'r' be the number of roses and 't' be the number of tulips. We can set up two equations based on the information given:
The total number of flowers: r + t = 24
The total cost of the flowers: 3 r + 1.25 t = 58
Solving for 't' Now, let's solve the system of equations. We can solve the first equation for 't':
t = 24 − r
Substitute this expression for 't' into the second equation:
3 r + 1.25 ( 24 − r ) = 58
Solving for 'r' Now, we solve for 'r':
3 r + 30 − 1.25 r = 58
Combine like terms:
1.75 r + 30 = 58
Subtract 30 from both sides:
1.75 r = 28
Divide by 1.75:
r = 1.75 28 = 16
Solving for 't' Now that we have the number of roses, we can find the number of tulips using the equation t = 24 − r :
t = 24 − 16 = 8
Final Answer So, Jeff bought 16 roses and 8 tulips.
Examples
This type of problem, involving systems of equations, is useful in many real-world scenarios. For example, a store manager might use it to determine how many of each item to order based on budget and space constraints. Similarly, a dietician could use it to plan a meal with specific nutritional requirements and calorie limits. Understanding how to set up and solve these equations helps in making informed decisions in various resource allocation problems.
