An investment firm recommends that a client invest in bonds rated AAA, A, and B. The average yield on AAA bonds is \( 4 \% \), on A bonds \( 6 \% \), and on B bonds \( 11 \% \). 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 \( \$ 25,000 \), and the investor wants an annual return of \( \$ 1,560 \) on the three investments? The client should invest \( \$ \square \) in AAA bonds, \( \$ \square \) in A bonds, and \( \$ \square \) in B bonds.

Let's denote the investment amounts in AAA, A, and B bonds as \( x \), \( y \), and \( z \) respectively. We know from the problem that the total investment is: \[ x + y + z = 25,000 \] Additionally, the client wants to invest twice as much in AAA bonds as in B bonds: \[ x = 2z \] The annual return from the investments should equal $1,560. The returns can be expressed as: \[ 0.04x + 0.06y + 0.11z = 1,560 \] Now, we substitute \( x = 2z \) into the equations. The first equation becomes: \[ 2z + y + z = 25,000 \] \[ 3z + y = 25,000 \] \[ y = 25,000 - 3z \] Next, we substitute \( x = 2z \) and \( y = 25,000 - 3z \) into the return equation: \[ 0.04(2z) + 0.06(25,000 - 3z) + 0.11z = 1,560 \] This simplifies to: \[ 0.08z + 1,500 - 0.18z + 0.11z = 1,560 \] \[ 0.08z - 0.18z + 0.11z + 1,500 = 1,560 \] \[ 0.01z + 1,500 = 1,560 \] \[ 0.01z = 60 \] \[ z = 6,000 \] Now, substituting back to find \( x \) and \( y \): \[ x = 2z = 2(6,000) = 12,000 \] \[ y = 25,000 - 3z = 25,000 - 18,000 = 7,000 \] Therefore, the client should invest: - \( \$ 12,000 \) in AAA bonds - \( \$ 7,000 \) in A bonds - \( \$ 6,000 \) in B bonds. The final results are: The client should invest \( \$ 12,000 \) in AAA bonds, \( \$ 7,000 \) in A bonds, and \( \$ 6,000 \) in B bonds.