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Home » An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, (i) And,
An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, x+y+z=5000, \ldots (i) \begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow \\ 6 x+7 y+8 z=35800 \ldots (ii) \end{array} And, \frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70 \Rightarrow 6 x+7 y-8 z=7000 . \ldots \text { (iii) }
CBSE | Class 12 | Excercise 8A | Maths | RS Aggarwal | System of Linear Equations
An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, x+y+z=5000, \ldots (i) \begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow \\ 6 x+7 y+8 z=35800 \ldots (ii) \end{array} And, \frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70 \Rightarrow 6 x+7 y-8 z=7000 . \ldots \text { (iii) }

Solution:

Suppose the investments are \mathrm{x} \mathrm{x}, Fy and \mathrm{F} \mathrm{z}, respectively.
Therefore, x+y+z=5000
\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \\ 6 x+7 y+8 z=35800 \end{array}
And, \frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70
6 x+7 y-8 z=7000
Now converting in the matrix form,
\begin{array}{l} A X=B \\ {\left[\begin{array}{ccc} 1 & 1 & 1 \\ 6 & 7 & 8 \\ 6 & 7 & -8 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 5000 \\ 35800 \\ 7000 \end{array}\right]} \\ R_{3}-R_{2} \\ {\left[\begin{array}{ccc} 1 & 1 & 1 \\ 6 & 7 & 8 \\ 0 & 0 & -16 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 5000 \\ 35800 \\ -28800 \end{array}\right]} \\ R_{2}-6 R_{1} \\ {\left[\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & -16 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 5000 \\ 5800 \\ -28800 \end{array}\right]} \end{array}
Again converting into the equations we get
\begin{array}{l} x+y+z=5000 \\ Y+2 z=5800 \\ -16 z=-28800 \\ Z=1800 \\ Y+2 \times 1800=5800 \\ Y=5800-3600 \\ Y=2200 x+2200+1800=5000 \\ X=5000-4000 \\ X=1000 \end{array}
Amount of 1000,2200,1800 were invested in the investments of 6 \%, 7 \%, 8 \% respectively.

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