Q11. - Algebra 2

Question

For Exercises 10 and 11, use the following information.

Jarrod Wright has $4000$ he would like to invest so that he can earn some interest on it. He has discovered that he could put it in a savings account paying $6.5\%$ interest annually, or in a certificate of deposit with an annual rate of $8\%$. He wants his interest for the year to be $297.50$, because earning more than this would put him into a higher tax bracket.

How much should be put in a savings account, and how much should he put in the certificate of deposit?

Step-by-Step Solution

Answer

The deposition of money related to the savings is $2227$ and for the deposit is $1773$. 

Step 1- Determine the value of s

Use the system of equations, 

$6.5s-8d=297.50$

$s+d=4000$

Apply the Cramer’s rule for the value of s. 

$s=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute $6.5$ in place of a, 1 in place of c, $-8$ in place of b, 1 in

place of d, $297.50$ in place of e and 4000 in place of f into$s=\frac{\left|\begin{array}{cc}e& b\\ f& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

and perform simplification of determinants to write the value of s.

$\begin{array}{c}s=\frac{\left|\begin{array}{cc}297.5& -8\\ 4000& 1\end{array}\right|}{\left|\begin{array}{cc}6.5& -8\\ 1& 1\end{array}\right|}\\ =\frac{297.5\cdot 1-\left(-8\right)\cdot 4000}{7\cdot 5-5\cdot 5}\\ =\frac{\frac{64595}{2}}{\frac{29}{2}}\\ \approx 2227\end{array}$

 

Step 2 – Determine the value of d

Apply the Cramer’s rule for the value of d. 

$d=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$

Substitute 7 in place of a, 5 in place of c, 8 in place of b, 1 in place of d, 

$297.50$ in place of e and 4000 in place of f into $d=\frac{\left|\begin{array}{cc}a& e\\ c& f\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}$and perform

 simplification of determinants to write the value of d.

$\begin{array}{c}d=\frac{\left|\begin{array}{cc}6.5& 297.5\\ 1& 4000\end{array}\right|}{\left|\begin{array}{cc}6.5& -8\\ 1& 1\end{array}\right|}\\ =\frac{6.5\cdot 4000-297.5\cdot 1}{7\cdot 5-5\cdot 5}\\ =\frac{\frac{51405}{2}}{\frac{29}{2}}\\ \approx 1773\end{array}$

Step 3- Write the solution

The obtained value of s is 2227 and d is 1773 which mean that the deposition of money related to the savings is $2227$ and for the deposit is $1773$.