The word problem that follows can be classified as an investment problem because it is one involving income from an investment. The income can be in the from of interest, and in that case we use the formula

\[ I = P \cdot R\]

where $\mathbf{I}$ dollars is the annual interest earned when $\mathbf{P}$ dollars is invested at a rate $\mathbf{R}$ per year. The rate is usually given a percent; thus if the rate is 8 percent, then $\mathbf{R=0.08}$.

Investment Problem

A main invested part of USD 15,000 at 12 percent and the remainder at 8 percent. If his annual income from the two investments is USD 1,456, how much does he have invested at each rate?

Solution

Because we want to find the number of dollars invested at each rate, we make the following definitions:

        $\mathbf{x:}$  the number of dollars invested at 12 percent
 $\mathbf{15,000 - x:}$  the number of dollars invested at 8 percent

From the Interest Formula given above, we can derive a table like below.

Make a Table

Because the annual income from two investment is USD 1,456, the sum of the entries in the last column on the table is 1,456; therefore we have the equation:

    \[ 0.12x + 0.08(15,000 –x) = 1456 \]
           \[ 0.12 + 1200 – 0.08x = 1456 \]
                  \[ 0.04x = 256 \]
                 \[ x = 6400 \]  
\[ AND \]

\[ 15,000 – x = 15,000 – 6400 \]
                                 \[ = 8600 \]

Thus the man has USD 6,400 invested at 12 percent and USD 8,600 at 8 percent.

Check

The number of dollars in the annual interest from the USD 6,400 invested at 12% is $\mathbf{0.12(6400) = 768}$ and from the USD 8,600 at 8% is $\mathbf{0.08(8600) = 688}$; and $768 + 688 = 1456$.