In a factory there are three machines, A, B, and C. When all three machines are working, they produce 287 bolts per hour. When only machines A and C are working, they produce 197 bolts per hour. When only machines A and B are working, they produce 202 bolts per hour. How many bolts can each machine produce per hour?
Let's assign variables to the number of bolts each machine produces per hour. Let's say machine A produces x bolts per hour, machine B produces y bolts per hour, and machine C produces z bolts per hour.
We have the following information from the problem:
When all three machines are working, they produce 287 bolts per hour:
x + y + z = 287 ................(1)
When only machines A and C are working, they produce 197 bolts per hour:
x + z = 197 ....................(2)
When only machines A and B are working, they produce 202 bolts per hour:
x + y = 202 ....................(3)
We now have a system of three equations with three variables, which we can solve to find the values of x, y, and z.
Subtracting equation (2) from equation (1), we get:
(x + y + z) - (x + z) = 287 - 197
x + y + z - x - z = 90
y = 90 ......................(4)
Subtracting equation (3) from equation (1), we get:
(x + y + z) - (x + y) = 287 - 202
x + y + z - x - y = 85
z = 85 ......................(5)
Substituting the values of y = 90 and z = 85 into equation (1), we get:
x + 90 + 85 = 287
x + 175 = 287
x = 287 - 175
x = 112 .....................(6)
So, machine A produces 112 bolts per hour, machine B produces 90 bolts per hour, and machine C produces 85 bolts per hour.
Let's assume the number of bolts machine A can produce per hour is x, the number of bolts machine B can produce per hour is y, and the number of bolts machine C can produce per hour is z.
According to the information given, we have the following equations:
Equation 1: x + y + z = 287 (when all three machines are working)
Equation 2: x + z = 197 (when only machines A and C are working)
Equation 3: x + y = 202 (when only machines A and B are working)
To solve these equations, we can use substitution or elimination method.
Using substitution method:
From Equation 2, we can express z in terms of x: z = 197 - x.
Substituting this value into Equations 1 and 3, we get:
Equation 1: x + y + 197 - x = 287
Equation 3: x + y = 202
Simplifying Equation 1, we have:
y + 197 = 287
y = 287 - 197
y = 90
Substituting y = 90 into Equation 3, we have:
x + 90 = 202
x = 202 - 90
x = 112
Now we can substitute the values of x and y into Equation 1 to find z:
112 + 90 + z = 287
z = 287 - (112 + 90)
z = 287 - 202
z = 85
Therefore, machine A can produce 112 bolts per hour, machine B can produce 90 bolts per hour, and machine C can produce 85 bolts per hour.
To find out how many bolts each machine can produce per hour, we can set up a system of equations using the given information.
Let's assume that machine A can produce x bolts per hour, machine B can produce y bolts per hour, and machine C can produce z bolts per hour.
From the first piece of information, when all three machines are working, they produce 287 bolts per hour, we can write the equation:
x + y + z = 287 ----(1)
From the second piece of information, when only machines A and C are working, they produce 197 bolts per hour, we can write the equation:
x + z = 197 ----(2)
From the third piece of information, when only machines A and B are working, they produce 202 bolts per hour, we can write the equation:
x + y = 202 ----(3)
Now we have a system of three equations with three variables. We can solve this system using various methods such as substitution or elimination.
First, let's solve equation (2) for z:
z = 197 - x
Substitute this value of z in equation (1):
x + y + (197 - x) = 287
Simplifying:
y + 197 = 287 - x
Rearranging:
x + y = 287 - 197
x + y = 90 ----(4)
Now we have two equations with two variables: equations (3) and (4).
We can solve this system by subtracting equation (4) from equation (3):
(x + y) - (x + y) = 202 - 90
0 = 112
This equation gives a contradiction, which means the system of equations is inconsistent. Therefore, the given information is not possible.
It seems there is an error in the given information or there might be some missing data that would allow us to find the solution.