A restaurant manager wants to purchase 200
sets of dishes. One design costs $15 per set, while another costs $45 per set. If she wants to use her entire budget of $5700, how many of each design should be ordered?
____ for $15 set
____ for $45 set
number of cheaper sets ---x
number of other sets ----- 200-x
15x + 45(200-x) = 5700
solve for x
Let's use the following variables to solve the problem:
Let x represent the number of sets of dishes costing $15.
Let y represent the number of sets of dishes costing $45.
According to the problem, the restaurant manager wants to purchase 200 sets in total, so we have the first equation:
x + y = 200
We also know that the budget is $5700, and that the cost per set of the $15 design is $15. Therefore, the cost of the $15 designs will be 15x. In a similar manner, we can state that the cost of the $45 designs will be 45y. This gives us the second equation:
15x + 45y = 5700
Now we can solve these two equations simultaneously using any method, such as substitution, elimination, or graphing.
One approach is to solve for x in the first equation:
x = 200 - y
Substituting this expression for x in the second equation, we get:
15(200 - y) + 45y = 5700
Expanding and simplifying:
3000 - 15y + 45y = 5700
Combining like terms:
30y = 2700
Dividing both sides by 30:
y = 90
Now we can substitute this value of y back into the first equation to solve for x:
x + 90 = 200
x = 200 - 90
x = 110
Therefore, the restaurant manager should order 110 sets of the $15 design and 90 sets of the $45 design.
110 for $15 set
90 for $45 set
To determine the number of sets of each design the restaurant manager should order, let's use a system of equations.
Let's assume the number of sets of the $15 design as "x" and the number of sets of the $45 design as "y".
We know that the restaurant manager wants to purchase a total of 200 sets, so our first equation is:
x + y = 200
Next, we know that the total cost of the dishes should equal the manager's budget of $5700. The cost of x sets of the $15 design is 15x and the cost of y sets of the $45 design is 45y. So our second equation is:
15x + 45y = 5700
Now, we can solve this system of equations to find the values of x and y.
First, let's solve the first equation for x:
x = 200 - y
Now, substitute this value of x into the second equation:
15(200 - y) + 45y = 5700
Simplify and solve for y:
3000 - 15y + 45y = 5700
30y = 2700
y = 90
Now, substitute the value of y back into the first equation to find x:
x + 90 = 200
x = 110
Therefore, the restaurant manager should order 110 sets of the $15 design and 90 sets of the $45 design.