An apple orchard charges a fixed amount for the first four bushels of apples purchased and an extra amount for each additional bushel purchased. If the cost of 11 bushels of apples is $33 and the cost of 19 bushels is $55, what is the fixed charge for the first four bushels of apples?
fc + (11-4)*b = fc + 7*b = 33
fc + (19-4)*b = fc + 15*b = 55
where fc is the fixed charge of the first 4 bushels, and b is the price per bushel of each additional bushel over 4
Use algebra to solve these two equations for b and fc:
fc + 7*b = 33
fc + 15*b = 55
let
cost = m(n-4) + b , where n is the number of bushels, b is the extra charge for the first 4 bushels and m is the cost per bushel
so for 11 bushels, cost is 33
33 = 7m + b
so for 19 bushels, cost = 55
55 = 15m + b
subtract them
22 = 8m
m = 22/8 = 2.75
sub into 33 = 7m+b
33 = 19.25 + b
b = 13.75
the fixed cost is $13.75
check:
our equation would be
cost = 2.75(n-4) + 13.75
if n = 11, cost = 2.75(7) + 13.75 = 33
if n=19, cost = 2.75(15) + 13.75 = 55 , YEAH
To solve this problem, we'll need to set up a system of equations. Let's denote the fixed charge for the first four bushels of apples as "x" and the extra charge for each additional bushel as "y".
From the given information, we can establish two equations:
Equation 1: x + 7y = 33
- This equation represents the cost of 11 bushels of apples at $33. Since the fixed charge for the first four bushels is "x", the cost for the additional 7 bushels is 7y.
Equation 2: x + 15y = 55
- This equation represents the cost of 19 bushels of apples at $55. Similar to Equation 1, the fixed charge for the first four bushels is "x", and the cost for the additional 15 bushels is 15y.
Now, we have a system of equations:
x + 7y = 33 (Equation 1)
x + 15y = 55 (Equation 2)
To solve this system, we can use the method of substitution or elimination.
Let's solve using the method of elimination:
Multiply Equation 1 by 2 and Equation 2 by -1:
2(x + 7y) = 2(33) => 2x + 14y = 66 (Equation 3)
-1(x + 15y) = -1(55) => -x - 15y = -55 (Equation 4)
Adding Equation 3 and Equation 4:
(2x + 14y) + (-x - 15y) = 66 + (-55)
x - y = 11
Now we have a new equation:
x - y = 11 (Equation 5)
To find the fixed charge for the first four bushels of apples, we need to isolate "x" in Equation 5.
Add y to both sides:
x = 11 + y
We don't have enough information to determine the values of x and y separately. However, we can find the fixed charge for the first four bushels by substituting the values of x and y in either Equation 1 or Equation 2.
Let's use Equation 1:
x + 7y = 33
Substitute x with (11 + y):
(11 + y) + 7y = 33
11 + 8y = 33
8y = 33 - 11
8y = 22
y = 22/8
y = 2.75
Now that we have the value of y, we can substitute it back into Equation 5 to find x:
x - 2.75 = 11
x = 11 + 2.75
x = 13.75
Therefore, the fixed charge for the first four bushels of apples is $13.75.