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# Mr. Anil wants to invest at most Rs.60000 in Fixed Deposit (F.D.) and Public Provident Fund (P.P.F.). He wants to invest at least Rs.20000 in F.D. and at least Rs.15000 in P.P.F. The rate of interest on F.D. is 8% p.a. and that on P.P.F. is 10% p.a. formulate the above problem as L.P.P. to determine maximum yearly income.  Hint: Assume the amounts invested in Fixed Deposit and Public Provident Fund to be some variables. Consider the condition of minimum investment in both. And then with the help of rates of interest formulate the equation for total yearly income subject to the above conditions.

Complete Step-by-Step solution:
Let the money invested in Fixed Deposit = $x$ and money invested in P.P.F. = $y$
As the amount to be invested is at most Rs. 60000.
Based on the above conditions, constraints on $x$ and $y$ is:
$\Rightarrow$Constraint $x + y \leqslant 60000$
According to the question, Mr. Anil wants to invest at least Rs. 20000 in F.D. So, we have:
$\Rightarrow x \geqslant 20000$
Further, he wants to invest at least Rs. 15000 in P.P.F.
$\Rightarrow y \geqslant 15000$
Let the total yearly income be Rs.$z$.
From the question, the rate of interest on F.D. is 8% p.a. and that on P.P.F. is 10% p.a. So, his total yearly income will be:
$\Rightarrow z =$(8% of $x$) + (10% of $y$)
$\Rightarrow z = \dfrac{8}{{100}} \times x + \dfrac{{10}}{{100}} \times y \\ \Rightarrow z = 0.08x + 0.1y \\$
Hence L.P.P. is Maximize: $z = 0.08x + 0.1y$ subject to conditions $x + y \leqslant 60000$, $x \geqslant 20000$ and $y \geqslant 15000$.

Note: As long as we don’t know the exact amount of investment in any one of the above situations, we can’t find the exact value of his total yearly income. His income is subjected to variations in the amounts of investment in FD and PPF.
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