Question

$\huge\underbrace\pink{Question}$

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) Rs. 2000

Note => Don't Spam I want correct answer ✅
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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

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

Answer 2

$\sf\:Explanation: \:$

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. (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 :

$\sf\:X = [x \: \: \: \: 30000 - x]$

Annual interest obtained is Rs.2000

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

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

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

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

$\sf\:5x + 210000 - 7x = 200000$

$\sf\: - 2x = - 10000$

$\sf\implies\: x = 5000$

Amount invested in the first bond= x = Rs.5000

Amount invested second bond= Rs (30000 - x) = Rs (30000 - 5000) = Rs. 25000

$\sf\:Final \: answer: \:$

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.