Math: ounces to pounds and ounces

The tutor shows a conversion from early Math 10 Pre-Calc.

Herein, I use the abbreviations oz for ounce and lb for pound.

Example: Convert 1013 ounces to pounds and ounces.

Solution:

  1. Realize that 16 oz=1 lb.
  2. Divide 1013 by 16 to get how many whole pounds you have.
  3. On a calculator, you’ll get the answer 63.3125. Keep the 63, but discard the decimal.
  4. The 63 means 63 pounds. You just need to find the number of ounces left over.
  5. Multiply 63 by 16; you’ll get 1008.
  6. The number of ounces left over is 1013-1008, or 5 ounces.
  7. Realize the meaning of the answer: 1013 oz is 63 lb, 5 oz.
  8. Check your answer: 63×16+5=1013:)

There are other ways to do this conversion, but this way seems the neatest that doesn’t use the fraction button.

I’ll be discussing unit conversions in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Financial math: can time be negative?

The tutor answers a question that brought someone to his site.

These days, Oracle Tutoring gets between 5000 and 6000 distinct visitors per month. Looking over the raw log entries of how some visitors arrived, I saw an inquiry yesterday that I’m sad to say may have gone unsatisfied. The literal query:

financial math can time be negative?

It’s a great question, but I don’t think I’ve covered it specifically – until now. In honour of that brave inquirer, here’s my response:

Let’s consider the compound interest formula

A=P(1+i)t

where

A=accumulated amount after time t

P=principal amount (amount today)

i=annual interest rate

t=time in years

In this context, negative t can represent years previous.

Example 1: Using negative time, find the amount that would have been invested three years ago, at 3.2% compounded annually, to be worth 5000 today.

Solution: In this case, A will represent the amount needed back then to give 5000 now; P will be 5000, the amount today. The interest rate 3.2% must be written in decimal form 0.032:

A=5000(1+0.032)-3

Entering the expression straight into a forward-entry calculator gives

A=4549.16

Using negative time, we have back-valued today’s principal of 5000 to what it would have been, in that account, three years ago.

I love raw, straightforward questions like that. HTH:)

Source:

Tan, Soo Tang. Applied Finite Mathematics. Boston: PWS-KENT, 1990.

Thanks to w3schools.com

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Solar energy: price per Watt

The tutor continues from his last post about the value of solar energy dollars.

In my last post I discussed the Wattage one might expect from solar energy for $100. Based on a graph from Scientific American, the equation for Watts per $100, W, at time t (with 1980 as year 0 and 2008 as year 28) turns out to be

    \[W=4.2e^{0.0787t}\]

Extrapolating, one can plug in 35 for t to get the predicted Watts per $100 today:

    \[W=4.2e^{0.0787(35)}=66Watts for \  \$100\]

In fact, just talking about the hardware itself, one can get more than 100W for $100 in 2015 (see sunelec.com, for example.) However, for a typical residential application the price per Watt might be more like $4.50, installed. Thus, the Watts available for use might be more like 22 per $100.

Interestingly, taking the average of the “theoretical” 100W for $100, along with the “practical” 22W per $100, one arrives at 61W per $100, which is fairly close to the 66W predicted by the graph.

I’ll be talking more about solar energy for households in coming posts:)

Source not already mentioned:

pv-magazine.com

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Financial math: ordinary annuity versus annuity due

The tutor continues exploring annuities.

 
In my previous post I introduced the idea of an annuity with an example. That example had each payment at month’s-end; hence, it was an ordinary annuity.

With an annuity due, the payment is received at the beginning of the month.

Using, once again, the TI-83 Plus TVM solver, we’ll rework yesterday’s example as if it is an annuity due; i.e., each payment comes at month’s beginning:

Recall that the annuity was purchased for $185 000, offered interest of 4.5% (compounded monthly), and was to be paid monthly over 15 years. We’ll set up the TI-83 Plus TVM solver (see my post here for how to use it) with the following parameters:

N=15×12
I%=4.5
PV=-185000 (negative because it’s being paid out)
PMT:leave this one blank for now
FV=0 (at the end of the 15 years, there’s nothing left)
P/Y=12 (12 payments per year)
C/Y=12 (monthly compounding periods)
PMT:BEGIN (Payment at beginning of month: annuity due)

The PMT entry on the fourth line we leave blank at first; for the one at the bottom, we choose BEGIN.

After all the parameters are entered, we return to the PMT on the fourth line and key in ALPHA ENTER. Hopefully, the value 1409.95 appears.

Notice that, when this was an ordinary annuity (once again, see yesterday’s post), the payment was $1415.24.

There is still more to say about annuities, and of course how to find their payments on other makes of financial calculator. I’ll be offering more coverage in future posts:)

Source:

Tan, S.T. Applied Finite Mathematics. Boston: PWS-KENT Publishing Company, 1990.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Financial math: annuities, part I

Financial math awaits many students.  The tutor introduces annuities.

From the point of view of a typical consumer, an annuity is the reverse of a loan.  The consumer becomes the lender, while the institution pays them back, often with monthly payments.

Consider the following example:

Rhonda has personal resources of $185 000.  She buys an annuity that will pay her a fixed sum each month over the next 15 years.  What monthly payment can she expect if the interest rate being offered is 4.5% (compounded monthly)?

Solution:  For this question we’ll use the TI-83 Plus TVM solver (see my post here for how to use it:)

Set up the TVM solver by entering the following values:
N=15×12
I%=4.5
PV=-185000 (negative because she’s paying it to the institution)
FV=0 (At the end, there’s nothing left.)
P/Y=12
C/Y=12
PMT:END (Found at bottom of screen; means each payment comes at month’s end.)

Now, go to PMT on the fourth line and press ALPHA ENTER.

Hopefully, you’ll receive the value 1415.24. This value is positive because she will receive it. Apparently, Rhonda can anticipate receipt of $1415.24 per month for the next 15 years if she purchases the annuity described above.

Of course, the institution that sells the annuity must hope that by investing Rhonda’s money, they can do better than the 4.5% they are offering her. Why they might believe this, and some ideas about how they might do so, I’ll discuss in future posts:)

Source:

Tan, S.T. Applied Finite Mathematics. Boston: Kent Publishing Company, 1990.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Financial math: interest rate conversions on the HP-10B

A lifetime ago, the tutor studied financial planning.  That’s when he got acquainted with the HP-10B.

Of the financial calculators I know, the Hewlett-Packard HP-10B is potentially the easiest to use.

One feature of particular value to anyone in financial math is interest rate conversions.  Let’s look at how they’re done on the HP-10B.

Example 1:  Find the effective annual interest rate if the nominal rate is 3.1% compounded monthly.

Solution:  On the HP-10B, there is an orange key on the left side, second from the bottom.  You press it to access the orange function printed above a given key.

In this case, we know the nominal rate is 3.1%. We type in 3.1. Next, we press the orange key, then the I/YR key (above which is printed NOM%).

To tell the calculator the compounding is monthly, we type in 12, then press the orange key, followed by the PMT key – which, you’ll see, has P/YR printed above.

Now, we press the orange key again, then the PV key. Above PV, you’ll see EFF%.

The answer 3.14 appears across the screen. Apparently, the nominal rate of 3.1% compounded monthly is the effective rate of 3.14% compounded annually.

Now, let’s imagine you know the effective annual rate, but you want to convert to a nominal rate.

Example 2: Find the nominal rate, with weekly compounding, of 4.2% compounded annually.

Solution: This time, we know the effective rate of 4.2% compounded annually. We type in 4.2, then press the orange key, then the PV key. Next, we type in 52, the orange key again, then the PMT key. Finally, we press the orange key, then the I/YR key. The answer 4.12 appears. Apparently, the rate of 4.2% compounded annually is equivalent to a nominal rate of 4.12% compounded weekly.

I’ll be covering more features of the HP-10B in future posts. HTH:)

Source:

HP-10B Business Calculator Owner’s Manual. Corvallis, OR: Hewlett-Packard, 1988.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Consumer Education: the Consumer Price Index

When the tutor was in high school, a course called Consumer Education was mandatory.

The beautiful, energetic high school students of today will, sooner or later, be workers. They will gain one kind of freedom at the expense of another.

Even at age 18, I never paid attention to prices.  By age 22, I did.  A university student sharing a 1BR apartment, I was very poor.  I finished my degree at age 25, always working a minimum wage job on the side.  Being young was fun in ways, but making a living was sobering.

Statistics Canada has a measure called the Consumer Price Index or CPI .  For many, its immediate use is tracking the cost of living.

The scale works as follows:  the base cost of 100 represents the year 2002.  Each year (or month) thence is given a number that compares its cost of living to that of 2002.  For BC, the November 2014 CPI was 118.8 (statcan.gc.ca), meaning the cost of living had increased by 18.8% since 2002.

Of course, the cost of living is only half the story.  To begin to know how much harder, or easier, it is to make a living, we need to compare the increase in cost of living with the increase in wages.

Among the people most affected by changing prices are those earning minimum wage. In 2002, the BC minimum wage was $8/hr (hrvoice.org); now, it’s $10.25 (globalnews.ca).  The percent increase is (10.25-8)/8 = 0.28 or 28%.

Although living’s not easy on minumum wage, it seems the wage itself has gone up by 28% since 2002, while the cost of living has increased by 18.8% over that period. That’s for BC.

There is endless material for discussion in the world of home economics. Look for more posts about it coming soon:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.