Question

Paul has $50,000 to invest. His intent is to earn 13% interest on his investment. He can invest part of his money at 8% interest and part at 16% interest. How much does Paul need to invest in each option to make a total 13% return on his $50,000?8% interest $ 16% interest $

Answer 1

Let x be the amount invest at 8%

Let y be the amount invest at 16%

Paul has $50,000 to invest:

`x+y=50,000`

His intent is to earn 13% interest on his investment. He can invest part of his money at 8% interest and part at 16% interest.

`\begin{gathered} 50,000(0.13)=x(0.08)+y(0.16) \\ \\ 6,500=0.08x+0.16y \end{gathered}`

Use the next system of equations to find x and y:

`\begin{gathered} x+y=50,000 \\ 6,500=0.08x+0.16y \end{gathered}`

1. Solve x in the first equation:

`x=50,000-y`

2. Substitute the x in the second equation by the value you get in the previous step:

`6,500=0.08(50,000-y)+0.16y`

3. Solve y:

`\begin{gathered} 6,500=4,000-0.08y+0.16y \\ 6,500=4,000+0.08y \\ 6,500-4,000=0.08y \\ 2,500=0.08y \\ (2,500)/(0.08)=y \\ \\ y=31,250 \end{gathered}`

4. Use the value of y to solve x:

`\begin{gathered} x=50,000-y \\ x=50,000-31,250 \\ x=18,750 \end{gathered}`

Solution for the system:

x=18,750

y=31,250

Answer: Paul needs to invers8% interest $18,75016% interest $31,250