Answer
The investment for short-term bond is $30000$ dollars, the investment for intermediate-term bond is $30000$ dollars, and the investment for long-term bond is $40000$ dollars,
Work Step by Step
Step 1. Identify the quantities given in the Exercise: investment $P=100000$ dollars, interest of short term bond $r_1=4\%$, interest of intermediate term bond $r_2=5\%$, and interest of long term bond $r_3=6\%$, amount to be invested in short or intermediate term bonds $P_1=P_2$, final interest $r=5.1\%$
Step 2: Establish equations based on the information given: Interest to be gained with short term bond: $I_1=P_1r_1=0.04P_1$, Interest to be gained with intermediate term bond: $I_2=P_2r_2=0.05P_1$, Interest to be gained with long term bond: $I_3=P_3r_3=0.06(100000-2P_1)$,
Step 3. The total interests $I=Pr=100000\times0.051=5100$ which equals to the sum of the above individual interests, so we have $I_1+I_2+I_3=I$ and $0.04P_1+0.05P_1+0.06(100000-2P_1)=5100$
Step 4. Solve the above equation by combining like terms to get $0.03P_1=900$, thus $P_1=30000$ dollars, $P_2=P_1=30000$ dollars, and $P_3=100000-2P_1=40000$ dollars.
Step 5. We conclude that the investment for short-term bond is $30000$ dollars, the investment for intermediate-term bond is $30000$ dollars, and the investment for long-term bond is $40000$ dollars,
