It takes a florist 3 hour and 15 minutes to make 3 small centerpieces and 3 large centerpieces. It takes 6 hours and 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 answers.
let the time taken to make large piece be x hrs
let the time taken to make the small piece by y hrs
3x + 3y = 3 1/4 = 13/4 (#1)
7x + 4y = 6 1/3 = 19/3 (#2)
#1 times 4: 12x+12y = 13
#2 times 3: 21x+12y=19
subtract them ...
9x = 6
x = 2/3 hour or 40 minutes
in #1
3(2/3) + 3y = 13/4
2 + 3y = 13/4
3y = 13/4 - 2 = 5/4
y = 5/12 hours or 25 minutes
To solve this problem, let's assign variables to represent the time it takes to make each small and large centerpiece.
Let's say it takes x hours to make a small centerpiece and y hours to make a large centerpiece.
From the given information, we can form the following equations:
Equation 1: 3x + 3y = 3 hours and 15 minutes
Equation 2: 4x + 7y = 6 hours and 20 minutes
Since we want to write the equations in terms of hours, we need to convert the times given in minutes to hours.
15 minutes is equal to 15/60 = 0.25 hours
20 minutes is equal to 20/60 = 0.33 hours
Now let's rewrite the equations:
Equation 1: 3x + 3y = 3.25 hours
Equation 2: 4x + 7y = 6.33 hours
Now we have a system of two equations with two unknowns.
To solve this system, we can use the method of substitution or elimination. Let's use the method of elimination.
First, we'll multiply Equation 1 by 4 and Equation 2 by 3 to get rid of the decimals:
Equation 1: 12x + 12y = 13 hours
Equation 2: 12x + 21y = 19 hours
Now we can subtract Equation 1 from Equation 2 to eliminate the x variable:
(12x + 21y) - (12x + 12y) = 19 - 13
9y = 6
y = 6/9
y = 0.67 hours
Now, substitute this value of y back into either Equation 1 or Equation 2 to solve for x. Let's substitute it into Equation 1:
3x + 3(0.67) = 3.25
3x + 2 = 3.25
3x = 3.25 - 2
3x = 1.25
x = 1.25/3
x = 0.42 hours
Therefore, it takes approximately 0.42 hours (or 25 minutes) to make each small centerpiece, and it takes approximately 0.67 hours (or 40 minutes) to make each large centerpiece.
Let's assume that it takes x hours to make one small centerpiece, and y hours to make one large centerpiece.
From the given information, we can set up the following system of equations:
Equation 1: 3x + 3y = 3 hours and 15 minutes
Equation 2: 4x + 7y = 6 hours and 20 minutes
To solve this system, we need to convert the time in minutes.
15 minutes is equivalent to 15/60 = 1/4 hour.
20 minutes is equivalent to 20/60 = 1/3 hour.
Now we can rewrite the equations:
Equation 1: 3x + 3y = 3 ΒΌ hours = 3.25 hours
Equation 2: 4x + 7y = 6 1/3 hours = 6.33 hours
Now we can solve the system of equations.
Step 1: Multiply Equation 1 by 4 to eliminate the x coefficient in Equation 2:
4(3x + 3y) = 4(3.25)
12x + 12y = 13
Equation 2: 4x + 7y = 6.33
Step 2: Multiply Equation 2 by 12 to eliminate the y coefficient in Equation 1:
12(4x + 7y) = 12(6.33)
48x + 84y = 75.96
Equation 1: 12x + 12y = 13
Step 3: Now we have the following system of equations:
12x + 12y = 13
48x + 84y = 75.96
Step 4: We can use the method of substitution to solve this system. Divide Equation 1 by 12 to simplify it:
x + y = 13/12
Step 5: Solve Equation 2 for x:
48x = 75.96 - 84y
x = (75.96 - 84y) / 48
x = (75.96 / 48) - (84y / 48)
x = 1.5825 - 1.75y
Step 6: Substitute the expression for x in Equation 1:
(1.5825 - 1.75y) + y = 13/12
1.5825 - 0.75y = 13/12
Now, we solve the equation above for y:
(1.5825 - 0.75y)(12) = 13
18.99 - 9y = 13
-9y = 13 - 18.99
-9y = -5.99
y = -5.99 / -9
y = 0.6656
Step 7: Substitute the value of y into Equation 1 to find x:
x + 0.6656 = 13/12
x = (13/12) - 0.6656
x = 1.0833
So, it takes approximately 1.0833 hours (or 1 hour and 5 minutes) to make each small centerpiece and approximately 0.6656 hours (or 39.94 minutes) to make each large centerpiece.