It takes a florist 3h 15min to make 3 small centerpieces and 3 large cebterpieces. It takes 6h 20min to make 4 small centerpieces abd 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.
Let's assume it takes x hours to make a small centerpiece and y hours to make a large centerpiece.
From the first statement, we can create the following equation:
3x + 3y = 3 hours 15 minutes
Converting the minutes to hours, we have:
3x + 3y = 3.25 hours
From the second statement, we can create another equation:
4x + 7y = 6 hours 20 minutes
Converting the minutes to hours, we have:
4x + 7y = 6.33 hours
Now, we have a system of equations:
3x + 3y = 3.25
4x + 7y = 6.33
To solve this system, we can use the method of substitution or elimination. Let's use the elimination method:
Multiply the first equation by 4 and the second equation by 3 to eliminate x:
12x + 12y = 13
12x + 21y = 18.99
Subtract the first equation from the second equation:
(12x + 21y) - (12x + 12y) = 18.99 - 13
9y = 5.99
y = 5.99 / 9
y ≈ 0.665
Substitute the value of y back into the first equation:
3x + 3(0.665) = 3.25
3x + 1.995 ≈ 3.25
3x ≈ 1.255
x ≈ 1.255 / 3
x ≈ 0.418
Therefore, it takes approximately 0.418 hours (or 25 minutes) to make each small centerpiece and 0.665 hours (or 39.9 minutes) to make each large centerpiece.
To find the time it takes to make each small centerpiece and each large centerpiece, we can set up a system of equations.
Let's assume it takes x hours to make a small centerpiece and y hours to make a large centerpiece.
From the given information, we have two equations:
Equation 1: 3x + 3y = 3 hours and 15 minutes. We can convert 3 hours and 15 minutes to hours by dividing 15 by 60, which gives us 0.25 hours. So, the first equation becomes 3x + 3y = 3.25.
Equation 2: 4x + 7y = 6 hours and 20 minutes. Again, we convert 6 hours and 20 minutes to hours by dividing 20 by 60, which gives us 0.33 hours. Therefore, the second equation becomes 4x + 7y = 6.33.
Now, we have a system of equations:
3x + 3y = 3.25,
4x + 7y = 6.33.
To solve this system of equations, we can use the method of substitution or elimination.
Let's solve it using the method of substitution:
First, rearrange Equation 1 to solve for x:
3x = 3.25 - 3y,
x = (3.25 - 3y) / 3.
Now, substitute this value for x in Equation 2:
4((3.25 - 3y) / 3) + 7y = 6.33.
Simplify the equation:
(13 - 12y) / 3 + 7y = 6.33.
Multiply every term by 3 to eliminate the fractions:
13 - 12y + 21y = 18.99.
Combine like terms:
9y + 13 = 18.99.
Subtract 13 from both sides:
9y = 18.99 - 13,
9y = 5.99.
Finally, divide both sides by 9 to solve for y:
y = 5.99 / 9,
y ≈ 0.67.
Now, substitute this value for y back into Equation 1 to find x:
3x + 3(0.67) = 3.25.
3x + 2.01 = 3.25.
3x = 3.25 - 2.01.
3x = 1.24.
x = 1.24 / 3,
x ≈ 0.41.
Therefore, it takes approximately 0.41 hours (or 24.6 minutes) to make each small centerpiece and 0.67 hours (or 40.2 minutes) to make each large centerpiece.
if the times required are x and y for small and large, respectively,
3x+3y = 3 1/4 = 13/4
4x+7y = 6 1/3 = 19/3
clearing fractions, we have
12x+12y = 13
12x+21y = 19
subtract top from bottom to get
9y = 6
y = 2/3
so,
small takes 25 min
large takes 40 min