you are planning a cookout. you figure that you will need at least 5 packages of hot dogs and hamburgers in total. a package of hot hods costs $1.90 and a package of hamburgers costs $5.20. you can spend a maximum of $20 in total on the hot dogs and hamburgers.
Identify two possible combinations of packages of hot dogs and hamburgers you can buy.
To identify the possible combinations, we need to find the number of packages of hot dogs and hamburgers that satisfy two conditions: (1) the total number of packages is at least 5, and (2) the total cost is within the maximum budget of $20.
Let's start by assigning variables:
Let's say the number of packages of hot dogs is 'x', and the number of packages of hamburgers is 'y'.
From the given information, we can create a system of equations:
Equation 1: x + y ≥ 5 (to ensure at least 5 packages in total)
Equation 2: (1.90 * x) + (5.20 * y) ≤ 20 (to ensure the total cost doesn't exceed $20)
Now, we can find the possible combinations by solving this system of equations.
Combination 1:
Let's assume x = 3 (three packages of hot dogs).
Substituting this value into Equation 1: 3 + y ≥ 5
Solving for y: y ≥ 2 (minimum two packages of hamburgers are required)
Substituting x = 3 into Equation 2: (1.90 * 3) + (5.20 * y) ≤ 20
Simplifying: 5.70 + (5.20 * y) ≤ 20
Collecting like terms: 5.20 * y ≤ 14.30
Dividing both sides by 5.20: y ≤ 2.75
So, for Combination 1, we have:
Number of hot dog packages (x) = 3
Number of hamburger packages (y) ≥ 2
Cost with this combination: (1.90 * 3) + (5.20 * n), where n can be any valid value for y.
Combination 2:
Let's assume x = 4 (four packages of hot dogs).
Substituting this value into Equation 1: 4 + y ≥ 5
Solving for y: y ≥ 1 (minimum one package of hamburgers is required)
Substituting x = 4 into Equation 2: (1.90 * 4) + (5.20 * y) ≤ 20
Simplifying: 7.60 + (5.20 * y) ≤ 20
Collecting like terms: 5.20 * y ≤ 12.40
Dividing both sides by 5.20: y ≤ 2.38
So, for Combination 2, we have:
Number of hot dog packages (x) = 4
Number of hamburger packages (y) ≥ 1
Cost with this combination: (1.90 * 4) + (5.20 * n), where n can be any valid value for y.
Thus, the two possible combinations are:
Combination 1: 3 packages of hot dogs and at least 2 packages of hamburgers.
Combination 2: 4 packages of hot dogs and at least 1 package of hamburgers.