Best Practices
Needless to say, I have evolved some best practices over time. We will
take a look at them. I searched online for a problem to use as an
example. Below is the one I chose. It seemed
it could happen in real life with the right person, in the right vehicle, during the right season, in the right state.
The Radiator Problem: A
car radiator holds 10 quarts and is currently filled with a mixture
containing 30% antifreeze. How much must be drained off and replaced
with pure (100%) antifreeze to bring the mixture up to 50% antifreeze?
This is meant to be algebra with an X but I would start with
calculating the resulting percentage from a number of quarts replaced.
Going forward from the usual input to a result feels more natural than starting backwards and is instructive in any case.
The Aproach: It is important to lay things out in a sensible space for your
problem and the solution. I find it useful, especially with test questions I
run across, to spend
a minute thinking just about the units involved and their
relationships. Here we have quarts of mixtures, quarts of
pure antifreeze, and percentages of pure antifreeze in a mixture. A
"cars built in a day by Bob" seems weird until you think of it as a speed like MPH. This observation can bring a sense of relief when these units are used on a test.
 If there are key values you will control or
are often needed, put them near the top. Try to move down the
table, left to right, calculating new items in the order they are needed with final
results near the bottom. Don't worry about repeating yourself a bit.
Notice that the Quarts Replaced will show up twice here -- once in the
context of the number we want to enter and again in a section
that calculates the new mixture of antifreeze from the two
concentrations. These are very different usages justifying different
locations. Don't cramp things. Put in the constants, but postpone
going to the Trails screen until you have given the layout some thought. You can always change it.
Some Considerations: I
would encourage you to do the math following a few principles. 1) Make
a sincere and solid effort to avoid cell names. Use memories freely and
pick names that are more like short words for a value held rather than
abbreviatons which are difficult to remember. 2) Try to avoid
putting two steps over the same cell.
T/Maker is perfectly capable of handling that while allowing you to
change any of the steps involved. But I look at each table as something
like a
painting and don't want to paint on top of any part of it. Since I am
striving to do this in one trail for purely instructive
purposes, I have to be more
disciplined and tidy. That challenge can also be a source of
satisfaction if you want to try to test what's possible.
 Moving On To Trails:
This image displays the steps in the trail in my solution. Be sure to
always look at steps in a trail in their numbered order (1,2,3,etc.).
There are many other solutions including those with more trails and no
memories. Let me
spell out some of the principles here that I follow. For variables you
will use in a
number of steps, memories as mentioned above are a good choice. The names are not case sensitive, but you can mix case if that
helps you in some way. BTW, you can go back and insert new steps, delete
old ones, or make modifications in any step including changing its step number.
In this example, I have initially chose the Quarts Replaced and
Total Quarts to be immediately put in memories. I have also put in two
steps (3 & 4) to fill the Quarts column in the two line section devoted to
establishing the antifreeze contributed by each component. It is never too early to invoke the debugger. A strategy of a test even after small leaps of progress pays worthy dividends, because it is all in one area and fresh in your mind.
Steps 5 and 6
calculate the amounts of pure antifreeze contributed by
each mixture.
Sticking to my goal of only one trail, I denoted the locations of
some values by their column letter name and put the complete
calculation in one cell. This is an example of how the freedom provided
by trails versus formulas pays off nicely. Steps
7 through 10 finish the percentage calculation of the new mixture. Trial and error suggested replacing about 3 quarts which was
close enough for practical purposes. This philosophy should probably
not be used with prescription medicines. Please hang up and dial 911 if
you are using this example for such a situation right now.
 Built In Trial And Error: In the table to the
right, I made use of the one above which calculates a new percentage
for a number of quarts replaced. I inserted a step at the top to
initialize some values and some steps at the end to check the situation
and repeat the procedure as necessary.
Here's what's happening. The table starts off with a zero value for the
quarts replaced. It then runs through the calculations. I added a line
near the end of the table which holds the target percentage. The logic
compares the percentage achieved with that target. If the last
step moved the needle closer to the target, I added one teaspoon of
antifreeze to the amount to be replaced and repeated the calculations
again. I doubt I could even do any filling to an accuracy of one
teaspoon. Now if instead, I got further away from the target as opposed
to closer, I stopped the process. I don't think this is hard to
understand for someone familiar with the symbol/value syntax inluding
the placement operators # or = followed by the designated location when
not the adjacent white cell.
 Binary Search: Finally,
if I did intend to fill the radiator with a syringe... This final table
performs a binary search for a value as accurate as you like. It starts
with a high and low estimate, tries calculating with the average of the
two, then changes the high or low estimate depending on the result
until the difference between the two is less than a specified value.
This is called a "binary search" because it cuts the range where the
correct value may be in half with each pass. Because the process is stopped at some point, we can avoid Zeno's Paradox.
This
technique and the previous one can be used with a variety of problems
to find a precise solution. I doubt many would want to do that, but
there are a lot of possibilites for adventurers and explorers.
Another use of iterations: Risk / Reward Optimal Roulette Strategy Simulation
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Here's
a little painting project to figure out the square footage to paint so
the correct amount of paint can be purchased. I decided to measure the
perimeters of rooms to get the value for the full surface then subtract
areas for doors or windows or whatever that would not need paint. You
can see the one trail that is used throughout the table to convert
dimensions and numbers of them into square feet.
ble and monitor kilowatt hours used during various periods. It was only a few days bfore his visit I placed a
The
final trail spilled out the needed totals and calculated the overall
day rate. The video runs through a few applications for each trail with
the debugger and begins by cloning a row at the data screen level. In
the debugger the cell box associated in trail 2 for steps 7 though 12 step will
continue to jump around when the second trail is reapplied, but the math never uses the adjacent cell. It is of no consequence.
You
would need a lot of unobtainable numbers to do this perfectly, but a
simple table may help you decide. The choice is not about escaping
taxes. It's about when you pay them and what you pay them on. Get it
over with now (Roth). Or forget it now and pay later (Traditional).
Finally
we need to show what we get as a total amount when withdrawing portions
of it at the withdrawal tax rate. This only applies to the
Traditional IRA. The taxes for the Roth IRA were paid before the
deposits were made.
Additional Touches