Question

An amount of Rs 5000 is put into three investments at the rate of interest of 6%,7% and 8% per annum respectively. The total annual income is? 350. If the combined income from the first two investments is Rs 70 more than the income from the third .Find the amount of each investment ?

note_ swap will be deleted with user acc​

Answer 1

Answer:

Originally known as Bytown, Ottawa was once a very small lumber town, very isolated as well. In 1857, when Queen Victoria chose Ottawa to be the new capital of the United Province of Canada, many people in more established cities such as Montreal, Toronto, Kingston, or Quebec were very surprised by her decision.

solly

btw ello sis

how r u

after long time!

Answer 2

Answer:

Given:

An amount of rs 5000 is put into three investments at the rate of interests of 6.7%, 7.7% and 8% per annum respectively.

The total annual income is rs 350.

To find:

If the combined income from the first two investments is rs 70 more than the income from the third, find the amount of each investment by matrix method.

Solution:

Let the investments be x, y and z.

x = 6.7% × 5000 = 6.7/100 × 5000 = 335

y = 7.7% × 5000 = 7.7/100 × 5000 = 385

z = 8% × 5000 = 8/100 × 5000 = 400

6.7x + 7.7y + 8z = 100

x + y + z = 350

x + y - z = 70

AX = B

$\begin{gathered}\left[\begin{array}{ccc}6.7&7.7&8\\1&1&1\\1&1&-1\end{array}\right] \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}100\\ 350\\ 70\end{bmatrix}\end{gathered}$

⎣⎢⎡6.7117.71181−1⎦⎥⎤

⎣⎢⎡xyz⎦⎥⎤

=⎣⎢⎡10035070⎦⎥⎤

$\begin{gathered}\Delta =\begin{bmatrix}6.7&7.7&8\\ 1&1&1\\ 1&1&-1\end{bmatrix}\\=6.7\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}-7.7\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}+8\cdot \det \begin{pmatrix}1&1\\ 1&1\end{pmatrix}\\=6.7\left(-2\right)-7.7\left(-2\right)+8\cdot \:0\\\Delta = 2\end{gathered}$

Δ= ⎣⎢⎡6.7117.71181−1⎦⎥⎤

=6.7⋅det( 111−1 )−7.7⋅det( 111−1 )+8⋅det( 1111 )

=6.7(−2)−7.7(−2)+8⋅0

Δ=2

$\begin{gathered}\Delta_1 =\begin{bmatrix}100&7.7&8\\ 350&1&1\\ 70&1&-1\end{bmatrix}\\=100\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}-7.7\cdot \det \begin{pmatrix}350&1\\ 70&-1\end{pmatrix}+8\cdot \det \begin{pmatrix}350&1\\ 70&1\end{pmatrix}\\=100\left(-2\right)-7.7\left(-420\right)+8\cdot \:280\ \\\Delta_1 = 5274\end{gathered}$

Δ 1 = ⎣⎢⎡100350707.71181−1⎦⎥⎤

=100⋅det( 11 1−1 )−7.7⋅det( 350701−1 )+8⋅det( 350711 )

=100(−2)−7.7(−420)+8⋅280

Δ 1 =5274

$\begin{gathered}\Delta_2 =\begin{bmatrix}6.7&100&8\\ \:\:1&350&1\\ \:\:1&70&-1\end{bmatrix}\\=6.7\cdot \det \begin{pmatrix}350&1\\ 70&-1\end{pmatrix}-100\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}+8\cdot \det \begin{pmatrix}1&350\\ 1&70\end{pmatrix}\\=6.7\left(-420\right)-100\left(-2\right)+8\left(-280\right)\ \\\Delta_2 = -4854\end{gathered}$

Δ 2 = ⎣⎢⎡6.7111003507081−1⎦⎥⎤

=6.7⋅det( 350701−1 )−100⋅det( 111−1 )+8⋅det( 1135070 )

=6.7(−420)−100(−2)+8(−280)

Δ 2 =−4854

$\begin{gathered}\Delta_3 =\begin{bmatrix}6.7&7.7&100\\ 1&1&350\\ 1&1&70\end{bmatrix}\\=6.7\cdot \det \begin{pmatrix}1&350\\ 1&70\end{pmatrix}-7.7\cdot \det \begin{pmatrix}1&350\\ 1&70\end{pmatrix}+100\cdot \det \begin{pmatrix}1&1\\ 1&1\end{pmatrix}\\=6.7\left(-280\right)-7.7\left(-280\right)+100\cdot \:0\ \\\Delta_3 = 280\end{gathered}$

Δ 3 = ⎣⎢⎡6.7117.71110035070⎦⎢⎤

=6.7⋅det( 1135070−7.7⋅det( 1135070+100⋅det( 1111 )

=6.7(−280)−7.7(−280)+100⋅0

Δ 3 =280

Now let us consider,

Hence the investments are as follows:

x = Δ1/Δ = 5274/2 = 2637

y = Δ2/Δ = -4854/2 = -2427

z = Δ3/Δ = 280/2 = 140

Verification:

The combined income from the first two investments is rs 70 more than the income from the third.

x + y = z + 70

2637 + (-2427) = 140 + 70

210 = 210