Question

Two factory plants are making tv panels. yesterday, plant A produce 8000 panels. three percent of the panels from plant A and 10% of the panels from plant B were defective. How many panels did plant B produce, if the overall percentage of defective panels from the two plants was 8%?

Answer 1

We can express the defective panels (D) multypling the production (P) by the defective rate (f):

`D=f\cdot P`

This can be applied to the total production as well as the production in each plant:

`\begin{gathered} D_a=f_a\cdot P_a \\ D_b=f_b\cdot P_b \\ D=f\cdot P \end{gathered}`

Also, we know that the total production P is equal to the sum of the production of both plants:

`P=P_a+P_b`

We can find the production of Plant B by stating that the sum of the defective panels of each plant is equal to the total defective panels:

`\begin{gathered} D_a+D_b=D \\ f_aP_a+f_bP_b=fP \\ f_aP_a+f_bP_b=f(P_a+P_b) \\ f_aP_a+f_bP_b=fP_a+fP_b \\ f_bP_b-fP_b=fP_a-f_aP_a \\ (f_b-f)\cdot P_b=(f-f_a)P_a \\ P_b=(f-f_a)/(f_b-f)\cdot P_a \end{gathered}`

We know have an expression for the production of Plant B in function of the defective rates and the production of Plant A.

We replace with the values and solve for Pb:

`\begin{gathered} P_b=(f-f_a)/(f_b-f)\cdot P_a \\ P_b=(0.08-0.03)/(0.10-0.08)\cdot8000 \\ P_b=(0.05)/(0.02)\cdot8000 \\ P_b=2.5\cdot8000 \\ P_b=20000 \end{gathered}`

Answer: The production of plant B is 20000 panels.