Answer:
The answer is "rate 12%, 14%, 16% and value 4000, 10000, and 7000".
Step-by-step explanation:
Let the value is x, y, and z. so,
The value of x amount will be borrowed at 12% [tex]= \frac{12}{100} \times x= 0.12x[/tex]
The value of y amount will be borrowed at 14%[tex]= \frac{14}{100} \times x= 0.14x[/tex]
The value of z amount will be borrowed at 16% [tex]= \frac{16}{100} \times x= 0.16x[/tex]
If the total amount= 21000
[tex]\to x+y+z=21000[/tex]
If the total interest= 3000
calculating the total interest:
[tex]\to 0.12x +0.14y +0.16z = 3000[/tex]
The overall amount borrowing at 12% and 14% was half as much as the amount borrowed at 16%.
[tex]\to x + y = 2z\\\\\to x + y - 2z = 0[/tex]
by calculating the value we have 3 equations that are:
[tex]\to x+y+z=21000\\\\\to 0.12.x +0.14.y +0.16z = 3000\\\\\to x + y - 2z= 0\\[/tex]
calculate the matrix by above-given value:
[tex]\left[\begin{array}{ccc}1&1&1\\0.12&0.14&0.16\\1&1&-2\end{array}\right][/tex][tex]\left[\begin{array}{c}x&y&z\\ \end{array}\right] =\left[\begin{array}{c}21000&3000&0\\ \end{array}\right][/tex]
by solving the above matrix we get:
[tex]\left[\begin{array}{c}x&y&z\\ \end{array}\right] =\left[\begin{array}{c}4000&10000&7000\\ \end{array}\right][/tex]
[tex]\to \$ 4000 \text{ lent at} 12 \% \\\to \$ 10000 \text{ lent at} 14 \% \\\to \$ 7000 \text{ lent at} 16\%\\\\[/tex]
