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​

Answer 1

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

To solve this problem using matrix multiplication, we can use a system of equations to represent the problem. Let x be the amount invested in the first bond (which pays 5% interest) and y be the amount invested in the second bond (which pays 7% interest). Then, we can write the following system of equations:

x + y = 30,000 (since the total amount invested must be Rs. 30,000)

0.05x + 0.07y = 0.06(30,000) (since the trust fund must obtain an annual total interest of 6%)

We can write this system of equations in matrix form as follows:

| 1 1 | | x | | 30,000 |

| 0.05 0.07 | | y | = | 0.06(30,000) |

Using matrix multiplication, we get:

| x | | 15,000 |

| y | = | 9,000 |

Therefore, the trust fund should invest Rs. 15,000 in the first bond and Rs. 9,000 in the second bond to obtain an annual total interest of 6%.

Answer 2

Appropriate 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:

[i] 1800 [ii] 2000

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

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

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