A manufacturer makes two products A and B, each of which is processed in two departments, production and finishing. Each A takes 3 hours to produce and six hours to finish, whereas each B takes 5 hours to produce and 2 hours to finish. How many units of A and B can be produced and finished if exactly 24 hours are available in each department and all hours must be used?
**I know the answer is 8 but i don't know how to do it and create the equation to get to that answer. **
Let the number of A's be x
let the number of B's be y
for the production: 3x + 5y = 24
for the finishing : 6x + 2y = 24
double the first and subtract the 2nd
8y = 24
y = 3
sub back into the 1st:
3x + 15 = 24
x = 3
So they must make 3 A's and 3 B's
check: to product A takes 9 hours, to produce B takes 15 hours for a total of 24 in production.
to finish A takes 18 hrs, and it takes 6 hrs for B for a total of 24 hrs to finish.
my answer checks out.
I have not clue how or from where you got 8.
8 what? it makes no sense.
To find the number of units of A and B that can be produced and finished, let's create equations for the available time in each department and the time required to produce and finish each unit.
Let's denote:
- x as the number of units of A produced and finished
- y as the number of units of B produced and finished
The time required to produce and finish each unit of A is given as follows:
- Production time for each unit of A: 3 hours
- Finishing time for each unit of A: 6 hours
The time required to produce and finish each unit of B is given as follows:
- Production time for each unit of B: 5 hours
- Finishing time for each unit of B: 2 hours
We have the following constraints:
1. The total production time available in both departments is 24 hours.
2. The total finishing time available in both departments is 24 hours.
3. All hours must be used, so the sum of production time and finishing time for both A and B must equal 24 hours.
Now, let's set up equations based on the given constraints:
1. Production time equation:
3x + 5y = 24
This equation represents the total production time required for A and B based on the number of units produced.
2. Finishing time equation:
6x + 2y = 24
This equation represents the total finishing time required for A and B based on the number of units produced.
Now, we have two equations with two unknowns (x and y). We can solve this system of equations to find the values of x and y.
Multiplying the production time equation by 2 and the finishing time equation by 3 to eliminate y, we have:
6x + 10y = 48
18x + 6y = 72
Now, let's solve the system of equations. Subtracting the second equation from the first:
(6x + 10y) - (18x + 6y) = 48 - 72
-12x + 4y = -24
Dividing both sides by -4:
3x - y = 6 [Equation 3]
Now, we have two equations:
3x - y = 6 [Equation 3]
6x + 2y = 24 [Equation 4]
Next, we can solve the system of equations by substitution, elimination, or other methods. In this case, let's solve by elimination:
Multiply Equation 3 by 2:
6x - 2y = 12 [Equation 5]
Adding Equation 4 and Equation 5:
(6x + 2y) + (6x - 2y) = 24 + 12
12x = 36
Dividing both sides by 12:
x = 3
Now, substitute the value of x into Equation 3:
3(3) - y = 6
9 - y = 6
Subtracting 9 from both sides:
-y = -3
Multiplying both sides by -1:
y = 3
Therefore, the number of units of A (x) is 3, and the number of units of B (y) is 3.
In conclusion, 3 units of A and 3 units of B can be produced and finished in exactly 24 hours in each department, using all the available time.