A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1410 pounds of food was ordered. Write a system of equations that describes this situation, and define your variables. Solve the system of equations, find the number of adults and number of children that were at the camp.
Set x as your adults y as your children
12 x + 9 y = 1410 pounds of food
x+y = 148
2 equations 2 unknowns solve for y in second and put that into first equitation.
12x + 9(148-x) = 1410
3x + 1332 = 1410
x= 26 Adults, 148 - 26 = 122 childeren
To solve this problem, we can start by defining our variables. Let's represent the number of adults at the camp as 'a' and the number of children as 'c'.
Now, let's establish the equations based on the information given in the problem:
1. There are 148 people in total at the summer camp:
a + c = 148
2. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. The total amount of food ordered is 1410 pounds.
12a + 9c = 1410
Now, we have a system of equations:
a + c = 148
12a + 9c = 1410
We can solve this system of equations using the method of substitution or elimination.
Method 1: Substitution
We can solve the first equation for 'a' and substitute it into the second equation:
a = 148 - c
12(148 - c) + 9c = 1410
1776 - 12c + 9c = 1410
1776 - 3c = 1410
-3c = 1410 - 1776
-3c = -366
c = -366 / -3
c = 122
Substituting the value of 'c' into the first equation to find 'a':
a + 122 = 148
a = 148 - 122
a = 26
Therefore, there were 26 adults and 122 children at the summer camp.
Method 2: Elimination
We can multiply the first equation by 9 and the second equation by -1 to eliminate 'c':
9a + 9c = 1332
-12a - 9c = -1410
Adding both equations:
-3a = -78
a = -78 / -3
a = 26
Substituting the value of 'a' into the first equation to find 'c':
26 + c = 148
c = 148 - 26
c = 122
Therefore, there were 26 adults and 122 children at the summer camp.
Both methods give us the same solution.
Let's define our variables:
Let A represent the number of adults at the camp.
Let C represent the number of children at the camp.
Now, let's set up the system of equations based on the given information:
Equation 1: The total number of people at the camp is 148:
A + C = 148
Equation 2: The total amount of food ordered is 1410 pounds:
12A + 9C = 1410
Now, we can solve this system of equations.
Using the substitution method, we can solve Equation 1 for A:
A = 148 - C
Substituting this value into Equation 2:
12(148 - C) + 9C = 1410
1776 - 12C + 9C = 1410
-3C = 1410 - 1776
-3C = -366
C = -366 / -3
C = 122
Now, substituting the value of C back into Equation 1:
A + 122 = 148
A = 148 - 122
A = 26
Therefore, there were 26 adults and 122 children at the camp.