An investment firm recommends that a client invest in bonds rated AAA, A, and B. The average yield on AAA bonds is $5 \%$, on A bonds $7 \%$, and on B bonds $12 \%$. The client wants to invest twice as much in AAA bonds as in B bonds. How much should be invested in each type of bond if the total investment is $\$ 26,000$, and the investor wants an annual return of $\$ 1,880$ on the three investments?
The client should invest $\$ \square$ in AAA bonds, $\$ \square$ in $\mathrm{A}$ bonds, and $\$ \square$ in $\mathrm{B}$ bonds.
Answer
Final Answer: The client should invest \(\boxed{\$ 12,000}\) in AAA bonds, \(\boxed{\$ 8,000}\) in A bonds, and \(\boxed{\$ 6,000}\) in B bonds.
Step 1 ：Let's denote the amount of money invested in AAA, A, and B bonds as x, y, and z respectively.
Step 2 ：We have three equations based on the problem: \(x + y + z = 26000\) (total investment), \(0.05x + 0.07y + 0.12z = 1880\) (total return), and \(x = 2z\) (the client wants to invest twice as much in AAA bonds as in B bonds).
Step 3 ：Solving this system of equations, we find that \(x = 12000\), \(y = 8000\), and \(z = 6000\).
Step 4 ：Final Answer: The client should invest \(\boxed{\$ 12,000}\) in AAA bonds, \(\boxed{\$ 8,000}\) in A bonds, and \(\boxed{\$ 6,000}\) in B bonds.
