A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year and the second bond pays 7% per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds if the trust fund must obtain an annual total interest of (i) Rs 1800 and (ii) Rs 2000.

Given that Rs 30000 must be invested into two types of bonds with 5% and 7% interest rates.

Let Rs x be invested in bonds of the first type.

Thus, Rs (30000 – x) will be invested in the other type.

Hence, the amount invested in each type of the bonds can be represented in matrix form with each column corresponding to a different type of bond as -

(i) Annual interest obtained is Rs 1800.

We know the formula to calculate the interest on a principal of Rs P at a rate R% per annum for t years is given by,

Here, the time is one year and thus T = 1.

Hence, the interest obtained after one year can be expressed in matrix representation as -

⇒ 5x + 7(30000 – x) = 1800 × 100

⇒ 5x + 210000 – 7x = 180000

⇒ –2x = 180000 – 210000

⇒ –2x = –30000

∴ x = 15000

Amount invested in the first bond = x = Rs 15000

Amount invested in the second bond = 30000 – x

⇒ Amount invested in the second bond = 30000 – 15000

∴ Amount invested in the second bond = Rs 15000

Thus, the trust has to invest Rs 15000 each in both the bonds in order to obtain an annual interest of Rs 1800.

(ii) Annual interest obtained is Rs 2000.

As in the previous case, the interest obtained after one year can be expressed in matrix representation as -

⇒ 5x + 7(30000 – x) = 2000 × 100

⇒ 5x + 210000 – 7x = 200000

⇒ –2x = 200000 – 210000

⇒ –2x = –10000

∴ x = 5000

Amount invested in the first bond = x = Rs 5000

Amount invested in the second bond = 30000 – x

⇒ Amount invested in the second bond = 30000 – 5000

∴ Amount invested in the second bond = Rs 25000

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