Q. 156 - Intermediate Algebra

Question

A trust fund worth \($25,000$ is invested in two different portfolios. This year, one portfolio is expected to earn $5.25$% interest and the other is expected to earn $4$%. Plans are for the total interest on the fund to be \)$1150$ in one year. How much money should be invested at each rate?

Step-by-Step Solution

Answer

The trust should invest $\$12,000$ in the portfolio with $5.25\%$ and $\$13,000$ in the portfolio with $4\%$ interest.

Step 1. Given Information

We are given that the total amount invested is $\$25,000$ and the total interest on the fund to be $$1150$ in one year.

Step 2. Assumptions and Formation of equations

Let $x$ and $y$ be the amount invested in one portfolio with $5.25\%$ interest and other portfolio with $4\%$ interest respectively.

Total amount invested is $\$25,000$, so

$x+y=25000 -\left(1\right)$

The other equation representing the interest in one year will be,

$$\begin{gathered} 5.25\%x+4\$y=1150 \\ 0.0525x+0.04y=1150 -\left(2\right) \end{gathered}$$

Step 3. Solving the equations.

The first equation can be written as,

$x=25000-y -\left(3\right)$

Now, putting the value of $x$ in second equation, we get

$$\begin{gathered} 0.0525x+0.04y=1150 \\ 0.0525(25000-y)+0.04y=1150 \\ 1312.5-0.0525y+0.04y=1150 \\ -0.0125y=-162.5 \end{gathered}$$

Dividing by $-0.0125$, we get

$$\begin{gathered} y=\frac{-162.5}{-0.0125} \\ y=13,000 \end{gathered}$$

Now, putting the value of $y$ in third equation, we get

$$\begin{gathered} x=25000-13000 \\ x=12,000 \end{gathered}$$

Hence the trust should invest $\$12,000$ in the portfolio with $5.25\%$ and $\$13,000$ in the portfolio with $4\%$ interest.

Step 4. Checking the solution

Checking the solution by putting the value of $x,y$ in the equations, we get

$$\begin{gathered} x+y=25000 \\ 12000+13000=25000 \\ 25000=25000 \\ 0.0525x+0.04y=1150 \\ 0.0525\left(12000\right)+0.04\left(13000\right)=1150 \\ 630+520=1150 \\ 1150=1150 \end{gathered}$$

This is true, hence the solution is correct.