Let Anil invests Rs x and Rs y in saving certificate (SC) and National saving bond (NSB) respectively.
Since, the rate of interest on SC is 8% annual and on NSB is 10% annual. So, interest on Rs x of SC is \(\frac{8x}{100}\) and Rs y of NSB is \(\frac{10x}{100}\) per annum.
Let Z be total interest earned so,
Given he wants to invest Rs 12000 is total
x + y ≤ 12000
According to the rules he has to invest at least Rs 2000 in SC and at least Rs 4000 in NSB.
x ≥ 2000
y ≥ 4000
Hence the mathematical formulation of LPP is to find x and y which
Maximizes Z
The region represented by x ≥ 2000: line x = 2000 is parallel to the y - axis and passes through (2000, 0).
The region not containing the origin represents x ≥ 2000
As (0, 0) doesn’t satisfy the inequation x ≥ 2000
The region represented by y ≥ 4000: line y = 4000 is parallel to the x - axis and passes through (0, 4000).
The region not containing the origin represents y ≥ 4000
As (0, 0) doesn’t satisfy the inequation y ≥ 4000
Region represented by x + y ≤ 12000: line x + y = 12000 meets axes at A(12000, 0) and B(0, 12000) respectively. The region which contains the origin represents the solution set of x + y ≤ 12000
as (0, 0) satisfies the inequality x + y ≤ 12000.
Region x, y ≥ 0 is represented by the first quadrant.
The corner points are E(2000, 10000), C(2000, 4000), D(8000, 4000).
The values of Z at these corner points are as follows:
The maximum value of Z is Rs 1160 which is attained at E(2000, 10000).
Thus the maximum earning is Rs1160 obtained when Rs 2000 were invested in SC and Rs 10000 in NSB.