Here is the given question:
A toy manufacturer produces two types of dolls; a basic version doll $A$ and a deluxe version doll $B$. Each doll of type $B$ takes twice as long to produce as one doll of type $A$. The company have time to make a maximum of $2000$ dolls of type $A$ per day, the supply of plastic is sufficient to produce $1500$ dolls per day and each type requires equal amount of it. The deluxe version, i.e type $B$ requires a fancy dress of which there are only $600$ available per day. If the company makes a profit of $₹ 3$ and $₹ 5$ per doll, respectively, on doll $A$ and doll $B$; how many of type should be produced per day in order to maximize the profit?
Now, in the solution the equation involving the time required and available is given as:
$x + 2y ≤ 2000$ (where $x$ dolls of type $A$ and $y$ dolls of type $B$ are produced per day to maximize the profit.)
But shouldn't the equation be $2x + y ≤ 2000$ OR $x + (1/2)y ≤ 2000$ ?
Please explain how to interpret the problem correctly.
In my logic, since type B dolls takes twice the time to produce compared to doll $A$, only half of them could be produced in a given time as compared to $A$.
Best Answer
I suppose that, implicitly, $x$ is the quantity of doll $A$ which is produced, and $y$ is the quantity of doll $B$ which is produced (on a given day). Suppose that one of each doll type is made. Then, in terms of time units it takes to make doll $A$, you have one time unit from the one doll $A$ produced and two time units from the one doll $B$ that is produced -- that is, it takes $x + 2y$ time units (with $x = 1$ and $y = 1$ in our example). Does this make sense?