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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 Rs. 1,800.
Solution
It is given that Rs. 30,000 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. (30,000 − x) will be invested in the other type.
Hence, the amount invested in each type of bond can be represented in matrix form, with each column corresponding to a different type of bond as follows:
X = [x 30,000 − x]
annual interest obtained is Rs. 1800. We know, Interest = `"PTR"/100`
Here, the time is one year and thus T = 1.
Hence, the interest obtained after one year can be expressed in matrix representation as follows:
[x 30,000 − x]`[(5/100),(7/100)]` = [1800]
`=> [x xx 5/100 + (30000 - x) xx 7/100] = [1800]`
`=> (5x)/100 + (7(30000 - x))/100` = 1800
⇒ 5x + 210000 − 7x = 180000
⇒ - 2x = - 30000
∴ x = 15000
Amount invested in the first bond = x = Rs. 15000
⇒ Amount invested in the second bond = Rs. (30000 − x) = Rs. (30000 − 15000) = Rs. 15000
∴ The trust has to invest Rs. 15000 each in both the bonds in order to obtain an annual interest rate of Rs. 1800.
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