A trust fund has Rs 30,000 that must be invested in two different types of bonds.

Question:

A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

(a) Rs 1,800

(b) Rs 2,000

Solution:

(a) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).

It is given that the first bond pays 5% interest per year and the second bond pays 7% interest per year.

Therefore, in order to obtain an annual total interest of Rs 1800, we have:

$\left[\begin{array}{ll}x & (30000-x)\end{array}\right]\left[\begin{array}{c}\frac{5}{100} \\ \frac{7}{100}\end{array}\right]=1800 \quad\left[\right.$ S.I. for 1 year $\left.=\frac{\text { Principal } \times \text { Rate }}{100}\right]$

$\Rightarrow \frac{5 x}{100}+\frac{7(30000-x)}{100}=1800$

$\Rightarrow 5 x+210000-7 x=180000$

$\Rightarrow 210000-2 x=180000$

$\Rightarrow 2 x=210000-180000$

$\Rightarrow 2 x=30000$

$\Rightarrow x=15000$

Thus, in order to obtain an annual total interest of Rs 1800, the trust fund should invest Rs 15000 in the first bond and the remaining Rs 15000 in the second bond.

(b) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).

Therefore, in order to obtain an annual total interest of Rs 2000, we have:

$\left[\begin{array}{ll}x & (30000-x)\end{array}\right]\left[\begin{array}{c}\frac{5}{100} \\ \frac{7}{100}\end{array}\right]=2000$

$\Rightarrow \frac{5 x}{100}+\frac{7(30000-x)}{100}=2000$

$\Rightarrow 5 x+210000-7 x=200000$

$\Rightarrow 210000-2 x=200000$

$\Rightarrow 2 x=210000-200000$

$\Rightarrow 2 x=10000$

$\Rightarrow x=5000$

Thus, in order to obtain an annual total interest of Rs 2000, the trust fund should invest Rs 5000 in the first bond and the remaining Rs 25000 in the second bond.