## Mixed Flows

Many time value of money problems that we face involve neither a single cash flow nor a single annuity. Instead, we may encounter a mixed (or uneven) pattern of cash flows.

The first step in solving the question above, or any similar problem, is to draw a time line, position the cash flows, and draw arrows indicating the direction and position to which you are going to adjust the flows. Second, make the necessary calculations as indicated by your diagram. (You may think that drawing a picture of what needs to be done is somewhat "childlike." However, consider that most successful home builders work from blueprints - why shouldn't you?)

Figure illustrates that mixed flow problems can always be solved by adjusting each flow individually and then summing the results. This is time-consuming, but it works.

Often we can recognize certain patterns within mixed cash flows that allow us to take some calculation shortcuts. Thus the problem that we have been working on could be solved in a number of alternative ways. One such alternative is shown in Figure 3.9. Notice how our twostep procedure continues to lead us to the correct solution:

• Step 1: Draw a time line, position cash flows, and draw arrows to indicate direction and position of adjustments.

• Step 2: Perform calculations as indicated by your diagram.

A wide variety of mixed (uneven) cash-flow problems could be illustrated. To appreciate this variety and to master the skills necessary to determine solutions, be sure to do the problems at the end of this chapter. Don't be too bothered if you make some mistakes at first. Time value of money problems can be tricky. Mastering this material is a little bit like learning to ride a bicycle. You expect to fall and get bruised a bit until you pick up the necessary skills.

But practice makes perfect.