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
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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

Answer:

Given:

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. To Prove:

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.Instruction Frame the matrices.

Calculationlf the person invests x amount in the 1st bond and the remaining in 2nd bond then,

Interest in 1st bond=5

Interest in 2nd bond=7

Thus, matrix equation will be 30000

[ 0.05 0.07]|

The 1st matrix denotes the interests and the 2nd matrix denotes the amounts.

If we multiply them then we will get the total interest. And the 1st matrix has 1row 2 columns and the 2nd matrix has 2 rows, 1column thus, they are multipliable.Instruction Solve (i).

Calculation

[ 0.05x+0.07(30000 - x) 1800

[ 0.05

0.07

30000

= 0.05x2100 - 0.07x 1800 =

-0.02x=-300

300

x = 0.02 = 15000 30000 - x = = 15000

Thus, he should invest 15000 in 1st bond and 15000 in 2nd bond to get this interest. [ 0.05

0.07

x

30000

0.05x+0.07(30000 - x) 2000 1=

0.05x2100 - 0.07x= 2100

-0.02x = -100

100 x = = 5000

0.02 30000 - x = = 25000

Thus, he should invest 5000 in 1st bond and 25000 in 2nd bond to get this interest.

[ 0.05 0.07 30000 x

0.05x +0.07(30000 − x) =2000 2000

0.05x+2100 -

0.07x= 2100

-0.02x = -100

100 0.02 X = = 5000 30000 x = 25000Final answer

(i) He should invest Rs.15000 in 1st bond and Rs.15000 in 2nd bond to get this interest.

(ii) He should invest Rs.5000 in 1st bond and Rs.25000 in 2nd bond to get this interest.