It takes a florist 3 hours 15 minutes to make 3 small centerpieces and 3 large centerpieces. It takes 6 hours 20 minutes to make 4 small centerpieces and 7 large centerpieces. How... It takes a florist 3 hours 15 minutes to make 3 small centerpieces and 3 large centerpieces. It takes 6 hours 20 minutes to make 4 small centerpieces and 7 large centerpieces. How long does it take to make each small centerpiece and each large centerpiece? Write and solve a system of equations to find your answer.
Understand the Problem
The question describes a scenario where a florist makes small and large centerpieces. Two time durations are provided for different quantities of small and large centerpieces. The goal is to determine the time required to make one small centerpiece and one large centerpiece using a system of equations.
Answer
Small centerpiece: 25 minutes Large centerpiece: 40 minutes
Answer for screen readers
Small centerpiece: 25 minutes Large centerpiece: 40 minutes
Steps to Solve
- Convert time to minutes
Convert the hours and minutes into total minutes. $3 \text{ h } 15 \text{ min} = (3 \times 60) + 15 = 180 + 15 = 195 \text{ minutes}$ $6 \text{ h } 20 \text{ min} = (6 \times 60) + 20 = 360 + 20 = 380 \text{ minutes}$
- Define variables
Let $x$ be the time (in minutes) to make one small centerpiece. Let $y$ be the time (in minutes) to make one large centerpiece.
- Write the system of equations
From the problem description, we can write two equations: $3x + 3y = 195$ $4x + 7y = 380$
- Simplify the first equation
Divide the first equation by 3 to simplify: $x + y = 65$
- Solve for x in the simplified equation
Solve for $x$ in terms of $y$: $x = 65 - y$
- Substitute x into the second equation
Substitute $x$ in the second equation with the expression $65 - y$: $4(65 - y) + 7y = 380$
- Solve for y
Expand and solve for $y$: $260 - 4y + 7y = 380$ $3y = 380 - 260$ $3y = 120$ $y = \frac{120}{3}$ $y = 40$
- Solve for x
Substitute the value of $y$ back into the equation $x = 65 - y$: $x = 65 - 40$ $x = 25$
- State the time for each centerpiece
One small centerpiece takes 25 minutes. One large centerpiece takes 40 minutes.
Small centerpiece: 25 minutes Large centerpiece: 40 minutes
More Information
It is important to clearly define the variables to correctly set up the equations. Once the system of equations is set up correctly, several methods can be applied to solve it (e.g. substitution, elimination).
Tips
A common mistake involves errors in the arithmetic when solving the system of equations (e.g. during substitution or elimination). Another mistake could be failing to correctly convert the time from hours and minutes to minutes only, and proceeding with inconsistent units.
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