Question

A trust has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year , and second bond pays 7% interest per year . Using matrix multiplication , determine how to divide Rs 30000 among the two bonds . If the trust fund must obtain an annual total interest of 1) Rs 1800 2) 2000

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

Amount invested in first type of bond be Rs x

Amount invested in second type of bond be Rs (30000 - x)

Given that, The first bond pays 5 % interest per annum and second one pays 7% interest per annum.

So, matrix form of the above data is as

$\sf \: A = \bigg[ \begin{matrix}x& 30000 - x \end{matrix} \bigg]$

$B = \left[\begin{array}{c} \dfrac{5}{100} \\ \\ \dfrac{7}{100} \end{array}\right]$

Given that,

$\sf \: Interest\:received \: = \: Rs \: 1800$

$\sf \: AB \: = \:[ 1800]$

$\sf \: \bigg[ \begin{matrix}x& 30000 - x \end{matrix} \bigg]\left[\begin{array}{c} \dfrac{5}{100} \\ \\ \dfrac{7}{100} \end{array}\right] = [ 1800]$

$\sf\: \left[x \times \dfrac{5}{100} + (30000 - x) \times \dfrac{7}{100} \right] = [ 1800]$

$\sf\: \left[ \dfrac{5x + 210000 - 7x}{100} \right] = [ 1800]$

$\sf\: \left[ \dfrac{210000 - 2x}{100} \right] = [ 1800]$

$\sf \: 210000 - 2x = 180000$

$\sf \: 2x = 210000 - 180000$

$\sf \: 2x = 30000$

$\implies\sf \: x = 15000$

Thus,

$\implies\sf \: Investment\:in\:first\:bond= Rs \: 15000$

and

$\implies\sf \: Investment\:in\:second\:bond = Rs \: 15000$

Given that,

$\sf \: Interest\:received \: = \: Rs \: 2000$

$\sf \: AB \: = \:[ 2000]$

$\sf \: \bigg[ \begin{matrix}x& 30000 - x \end{matrix} \bigg]\left[\begin{array}{c} \dfrac{5}{100} \\ \\ \dfrac{7}{100} \end{array}\right] = [ 2000]$

$\sf\: \left[x \times \dfrac{5}{100} + (30000 - x) \times \dfrac{7}{100} \right] = [ 2000]$

$\sf\: \left[ \dfrac{5x + 210000 - 7x}{100} \right] = [ 2000]$

$\sf\: \left[ \dfrac{210000 - 2x}{100} \right] = [ 2000]$

$\sf \: 210000 - 2x = 200000$

$\sf \: 2x = 210000 - 200000$

$\sf \: 2x = 10000$

$\implies\sf \: x = 5000$

Thus,

$\implies\sf \: Investment\:in\:first\:bond= Rs \: 5000$

and

$\implies\sf \: Investment\:in\:second\:bond = Rs \: 25000$