Additional Math Pages & Resources

Friday, October 30, 2009

How is your 401(k) doing today?

What does this number 401(k) mean anyway?

401(k) is a section of the US Internal Revenue tax code. It defines a retirement savings plan which corporations may offer to their employees. Employers may contribute matching funds to the account. Income taxes that would normally be due on the salary put into the account (plus any employer contributions or earnings) are deferred until withdrawals are made, years later. The 401(k) scheme was set up to encourage people to save for their retirement, and to give employers a cheaper option than a fully-funded pension plan.

The Employee Benefit Research Institute (EBRI) and the Investment Company Institute (ICI) track the funds invested in 401(k) accounts. They collect figures with the cooperation of insurance companies, investment firms and mutual funds. Their latest 2009 report was published just a few days ago and contains information on about half of the total participants in 401(k) plans.
  • How much money are we talking about? One trillion dollars. 
  • How many people are contributing? Twenty-four million.  
  • What is the average balance? Forty-one thousand dollars.
  • Great. So everyone has $40,000 for retirement, right? Wrong.
  • What is the median account balance? Twelve-thousand six hundred fifty dollars.

You mean half the 401(k) retirement accounts don't have enough money in them to buy a tiny little red Chevy Aveo?  That's right.


Let's look at the definitions for average and median.

Average is calculated using division. We take all the money (dividend) and divide it by the number of accounts (divisor) to get a result (quotient). This is the average balance.

Median is found by ranking. We take all the individual savers and their account balances, and rank them in order from smallest balance to largest. Then we count along that line of savers until we get to the middle person. The balance of the person in the middle is the median balance.

If you are a visual person, here are some bar graphs for you to look at. Click the graphs to get a larger version that you can more easily study.

The yellow and cream bars show the average and median for ALL savers. The light and dark green bars show average and median for people who have had their accounts for MORE THAN 5 years.

The question mark asks what will happen at the end of this year? Will the numbers go down or up?

Over the past decade, the 401(k) total balance has fluctuated between growing and shrinking. Here's a chart showing you how the money pot has changed size each year. Remember, these numbers do not show you earnings in individual accounts, they show how big the total is.

The overall total is a result of young people coming in, contributions and investment earnings, minus old people leaving the plan, withdrawing money, taking loans, investment losses, etc.

Here's the final chart. It shows the distribution of account balances. The first tall bar means 47% of the savers in the country have less than $10,000 in their accounts. Although the median age of that group is around 30, there are plenty of older people with less than 10K in their accounts!

The final column shows that 12% of the savers in the country have more than $100,000 in their accounts. The median age in this group is in the late fifties. So what does this chart tell us?

85% of savers have less than $85,000.
75% of savers have less than $50,000.
50% of savers have less than $13,000.
That's the median, but not the bottom line!

Only one out of every six dollars of retirement assets 
(pension, after-tax savings, real estate, collectibles, etc.)   
are held in these 401(k) retirement accounts.

Thursday, October 29, 2009

How Good is Good Enough?

This is a proverbial question. A suitable answer requires wisdom and discretion, rather than possession of the correct facts. This is NOT a question that math can always help you to answer. Can you spare a moment to think about this issue?

I used to get into heated discussions with Christie and Tom. We all worked at a publishing company. Both of them said, in various ways,

It takes me longer than others to do the work because I care about it more than they do. My additional efforts make a better product. 

or occasionally, 

I can't type any faster because I can't think as fast as I am typing, anyway!

To which I always wanted to say You're Fired! but was compelled instead to reply,

That may be how it seems to you, but
(a) we can't pay you more for working slowly

(b) your work isn't better than others who work faster.

As you can readily see, this is a no-win argument. Red faces, hurt feelings and missed deadlines.  How could they be wrong when they were so sincere? Can there ever be too much quality? Too much concern for the product?

Yes. I think so. One editorial group I consulted for was best described with this statement,  

We'd rather do it right, than do it at all.

To which you might exclaim, as the owner of that company did, What!?

Their intense concern for verifying details, improving art, and making indexes meant nearly every book was past its deadline, over budget, and too big. The editors never felt that ready-to-go-to-camera thrill.

The word tolerance covers situations like this. It means accepting less than perfect. Giving reasonable leeway for imperfections. Allowing for normal variation.

Here's a statement about a famous watchmaker:

As far as I know he has only delivered one watch, after all these years. This is because he is such a perfectionist. He cannot let go until he is completely satisfied with even the tiniest detail. His emotional involvement is exceptional. He does not stop until he gets it right for his standards. This watch is a mirror of his inner constitution; it is imbued with his soul. It will be a true master's piece – nothing less! Just imagine the immense degree of freedom he affords for himself to achieve just that!

Yet he doesn't have the satisfaction of seeing his watches on dozens of wrists.

I have a watch designed by another Swiss genius with a tolerance-free approach. They operate perfectly when simulated on a CAD system. However, when realized in metal, the builders have to allow some slack or the watch won't work under all variations of temperature, position, etc. Tolerances that are too small sometimes cause trouble.

Look at the watch in the first picture at the top of this post. Here's its second hand. See the little pipe sticking downwards? The left one is as designed, but sometimes it won't hold on to its shaft. If the watch was bumped or the stopwatch feature was operated sharply, the hand would come loose and fall off. The new, longer design on the right corrects that problem.

You could argue that (for reliability, safety, etc.) every component could be a little larger, thicker, heavier - but eventually you would have a very expensive pocket watch, not a wrist watch. That rationale has resulted in our vehicles getting longer, wider, heavier and more gas-consuming. And in the end, we all lose.

The winning strategy is this: The best employees (companies and products) find an acceptable balance of quality and speed, and continue to improve over time.

This watch company must have a formula that works. I have happily purchased 3 watches from them.

Wednesday, October 28, 2009

ARPU is not good for you

What is ARPU? A new drug? A chemical used in coffee cups?  A virus?

No, ARPU means Average Revenue Per User (per month). It's how mobile / cell phone service providers count what they collect from you. Cable TV and other subscription service companies use the same type of calculation.

Can math help us understand our phone bills? Let's give it a shot.

ARPU = Total revenues from base plans, extra minutes, data charges, messenging charges, ring tones, etc. divided by AAU (Average Active Users = Users this month + Users last month divided by 2).

Companies also report Voice ARPU, Non-Voice ARPU, Data ARPU, etc.

Before we go on, there are a few considerations.
  • On a family plan, a user is each SIM card or phone ID, not the person who pays the bill
  • Average numbers are skewed to the low end, as you can't use less than 0 minutes a month, nor more than about 5000 minutes (many unlimited plans are limited to far fewer than the 44,000 minutes in a month)
  • ARPU excludes one-time activation or cancellation charges
  • What they keep differs from what you pay, as seen in the fine print: Our surcharges (Fed. Univ. Svc. of 10.2% of interstate & int'l telecom charges, 7¢ Regulatory & 70¢ Administrative charge per line per month) are not taxes; government taxes and our surcharges could add 4%–35% to your bill. Offers & coverage, varying by service, not available everywhere. 
OK, now to the real numbers. Here are some Voice ARPU numbers I have found (all statistics are per month in US dollars):
  • The average US user pays $63 while US ARPU is about $35-40 (these can't both be true)
  • Half of all users talk less than 200 minutes a month
  • AT&T 2008 non-iPhone ARPU was $52; the iPhone ARPU was $90+
  • Current European ARPU ranges from $22-50
  • The highest ARPU in the world is in Turmenistan, at $83, and lowest is Bangladesh at $3.30
  • Across the world, Voice ARPU is declining at about 2-5% per year
  • Virgin Mobile's prepaid phone card 2009 ARPU is $19
  • About 65% of the users in the world have pre-paid, no-contract phones
What about data?
  • A minute of 2-way voice generates the same revenue as a single, millisecond SMS message
  • An iPhone user pays twice as much as a voice-only user, but consumes 50 times the bandwidth
  • 2007 Data ARPU was about $9 in the US, $7 in Europe, and $4.25 in the rest of the world
Average levels of churn (switching carriers) range from 20-40% depending on the carrier. That's why at the beginning we saw the AAU calculation! The subscriber numbers fluctuate wildly every month.

If carriers reduce prices, improve features, and offer better incentives, they can reduce that expensive churn, but that also reduces ARPU. They're always looking for the right mix between making the most profit and making you so mad you switch!

PS - No, math can't help figure out your phone bill. It's intentionally impervious to investigation!

Tuesday, October 27, 2009

An Intersection of Unions?

No, I am not talking about the United Auto Workers and the International Longshore and Warehouse Union getting together for a meeting, but instead about one of my favorite math topics.

The Union and Intersection of Sets (as demonstrated by Venn Diagrams)

Venn Diagrams show the relationships between groups of things. Although people had toyed with the concepts before, John Venn popularized these diagrams. In 1880 he published a paper expounding on their use, but he called them Eulerian Circles. The idea is to visually show relationships between sets, using overlapping circles.

I often get calls from parents confused about the terms union and intersection of sets. The UNION is the total of ALL members of two or more sets, and the INTERSECTION is the set of ONLY the members that belong to both.

Here's a simple diagram showing 4 regions. This is called an Order-two diagram.

1 is outside the circle and the rectangle
2 is inside the circle and the rectangle
3 is inside the circle
4 is inside the rectangle

The answer is 2.

Here's a problem from Excel Math that uses a Venn diagram to illustrate the relationship of drinking coffee, tea and water. This is an Order-three diagram.

The second set of circles (to the right) shows how there are 8 possible regions.

If you want to show more relationships, you need to have different shapes because circles won't do it. I came up with a new list of drinkers based on the preferences of people in our office. I also included people who drink soda all the time, including in the morning. The rectangles show the possible combinations.

I created this diagram using a very nice web-based Venn diagram generator from Chris Seidel.

The following diagram came from a Venn diagram generator created by Juan Carlos Oliveros. It uses ovals to display the same relationships between soda, coffee, tea and water drinkers.

The meaning and the relationships are the same in both of these last 2 diagrams, but the physical representations differ.

Here's a diagram that can represent 5 sets. It was created by Anthony William Fairbank Edwards (whew!), who somehow got the idea of projecting some of these shapes on a flat surface after plotting them on a sphere. Notice how the green line looks like the seam on a tennis ball?

You will have to come up with your own water-cooler survey, and plug the people in yourself!

Monday, October 26, 2009

Why do ink cartridges cost so much?

People often complain It costs so much to print! A new cartridge costs $$$ and they only last a few hundred pages!

Does it really cost a lot to print? Do cartridges really run out quickly? Let's use some math.

Inkjet printers are the least expensive color printers you can buy. They spray dots of ink on the pages to create text and graphics. Black and white dots are generally larger than color dots. They blend together on the paper to give us a color image. Although cheap to buy, these printers can be expensive to use.

Ink is mostly water, but that doesn't mean it's cheap or simple. I went to the HP website and took a look at their Material Safety Data Sheets. Those gave me the rough proportions of ingredients in an inkjet cartridge. This chart shows cyan's ingredients but generally water's about 2/3 of the total volume of ink.

Printer manufacturers use a very small fraction of a liter (known as a picoliter) to measure the size of ink droplets. How large is a picoliter?
  1. milliliter = 1/1,000 of a liter
  2. microliter 1/1,000,000 of a liter
  3. nanoliter 1/1,000,000,000 of a liter
  4. picoliter 1/1,000,000,000,000 of a liter

I pulled a color cartridge off our shelf. It cost us about $40 and contains 69 ml of ink. How many picoliters is that? Drop three zeros and multiply 1 x 69 to get:

69,000,000,000  or 69 billion picoliters. The HP site says each color drop contains 4 picoliters of ink, so that means:

69,000,000,000 ÷ 4 = 17.25 billion droplets.

Distribution of the ink on the page can vary dramatically. HP assumes 5% of the surface area of the paper gets ink. I found some users who estimate 15% coverage for B/W printing, and 50% for color (consumers tend to print lots of family photos in color).

So 17.25 billion divided by ( N [number of droplets possible on a page] x C [% of coverage] x M [mix of color inks to get the color you want]) = P (number of pages)

Hold that thought.

Besides what comes out on the page, there are many other ways your ink can disappear. And I don't mean just the pages where you have made an obvious error, crumpled up the paper and thrown it out.
  • Cleaning - the printer pumps a little bit of ink through the print nozzles to keep them clean. It  does this before and after each print job.
  • Shelf Life - ink is mostly water. Some just disappears, evaporates, or the ink dries up.
  • Calibration / color adjustment - you can run calibration pages yourself, but even if you don't the printer is doing calibration and self-adjustment on a regular basis.
  • Unbalanced color use - not all the ink in each of the multiple colors will be used evenly. If you never print some colors, that ink could dry up or (in a multi-color cartridge) be thrown out.
  • Residual Loss - as HP says, you can't get all the toothpaste out of a tube and you can't get all the ink out of a cartridge.
  • Blended colors -  printers use color inks together with black to get the best solid black tones. You have to select gray-scale options in your software to prevent this from happening.
  • High Quality - there are many levels of quality or print resolution. The higher ones take more ink.
  • Overlap Design - the best printers are designed to give good coverage on the paper, and take into account the seepage or wicking of the color along the paper fibers. Here are HP patent drawings showing various ways they arrange the droplets. (These are greatly enlarged.)

I think this is more complicated than ROCKET SCIENCE. With rockets, we only have to hit the moon or Mars, and we have decades and trillions of dollars.

A $100 inkjet printer has to print pictures of your cute child, adorable aunt or precocious pet in seconds, for pennies.

                                                               Typical family photos

Try to calculate the proportions of each ink color to print these three pictures!

Ignoring the benchmarks and calculations, users have tracked many thousands of pages to find ink alone costs at least $.25-35 per page (color) and $ .08 - .10 (B/W).

Happily a tip from my friend Chris led me to a Dutch group who created a font that takes less ink, and saves you money!

Friday, October 23, 2009

Math, math, MATH, mATh

Today we are just going to do math. C'mon, it's easy. Let's go:

14 ÷ 7 = 2

35 x 3 = 105

1 + 2 + 3 = 6

Q. If I give the Staples lady 10.00 for a 8.99 package of labels, how much will she give to me?
A. She will give me $1.01 and my address labels.

√64 = __

How much of this can you take? Are these mindless drills? Do you drop over asleep in 20 seconds?

I don't. I can do problems like this all day. I like to solve problems that I can solve correctly. I'm not afraid of failure but I do like success.

We have some customers who say:

If the work is easy, it's not worth doing. If you succeed right away, it's been a waste of time doing the work. If it isn't hard, it can't be any good. If the kids aren't struggling, they aren't learning.

Why do any math at all that's easy? Why do we using spiraling instruction in our curriculum? Give them lots of chances to solve similar problems, over one or more years?

Well, because lots of other things in any kid's day are hard. Like catching the bus on time, choosing what to have for lunch, surviving the criticism of friends, striking out in the softball game, remembering to take books home (or back to school), getting a flu shot.

As adults we sometimes forget how traumatic these little things can be. We don't want math to be the pinnacle of pain for kids. We need to give them a chance to know they are right and they solved the problem correctly.

Math is not exactly like crossing the street, but there are lots of ways to do it correctly or incorrectly. I love this poster - I got it in Switzerland about 25 years ago. Can you see how many ways NOT to cross the street?

A related question is this - are answers always either right or wrong? Could there be shades of grey? (gray?) Here's an example:
  • Was Lincoln the first president of the United States? No Correct
  • Was Lincoln a president of the United States? Yes Correct
  • Was Lincoln the thirteenth president of the United States? I don't know True but not the answer
One thing I have learned in doing Excel Math is that answers have to be right. If I phrase the question correctly you can arrive at THE RIGHT ANSWER. 
  • Name one president and two other men Abraham Lincoln, Madonna, my dog Spot 
          (1 right; 2 wrong. Is the whole thing wrong? I'd say yes, you didn't read the question.)

If I don't phrase the questions correctly, and kids arrive at two different answers, I get harassed  by customers. Fair enough. To any single, precisely-worded problem there should be one correct answer.

But for the question What is the best way to teach? there is no one right answer. We use spiraling and repetition because we don't know in what order the light goes on for you.

Will you get the problem the first time? Possibly not if you have math right after lunch on a hot day. Will you get the problem after seeing it the second time, or after doing a similar but parallel problem in another context? Or after doing it 5 times over a period of 5 months? Or seeing it phrased in another way?

We think you have a better chance through spiraling than if you study a concept for a week and move along to something else.

C'mon now, try it again.

What is the square root of 64? 8

Thursday, October 22, 2009

What do you mean, ABRASIVE?

Have you ever known an abrasive person? A person who rubs you the wrong way? Seems to be constantly wearing you down?

Like yesterday's post on density, today we look at a term used for personalities and for physical characteristics that we can describe mathematically.

Abrasives are used to rub things so they become smoother, usually in preparation for a finish coating or paint. If you buy sandpaper, it is described with a grit number. I know 80 is very coarse and 400 is fine. But I had no idea what the numbers really mean.

One place I checked said:
"Grit is expressed in the form of numbers on back of the paper. These numbers range from 24 all the way up to 1,000, though in practice most homeowners will never use grits at the top or bottom of the scale. These numbers are industry designations and do not correlate to the physical world."

Really? Do not correlate to the physical world? I doubt it; it probably means that the person who thought up the scale had other things in mind that are not obvious to us. Let's keep looking.

Grit numbers refer to the size of the sharp particles used to abrade a surface. Like most mature industries, various standards have developed to describe abrasives. The most common in the United States are:

CAMI - Coated Abrasive Manufacturers Institute; now part of the Unified Abrasives Manufacturers' Association

FEPA - Federation of European Producers of Abrasives; the same as the ISO 6344 standard for describing abrasives.

Here's a table of a few common sizes of grits. You can download a complete table at the Woodturner's organization site.

The size of the particles is measured in micro-meters; you can see by looking at the sizes that only the 120 grit is close to describing the micro-meter size of the particles! Perhaps there's just a tiny correlation to the physical world...


  Particles (┬Ám)
Extra Coarse

Very Fine
Extra fine
Super Fine
Ultra fine

Aluminum Oxide is commonly used on wood. The pieces used as the abrasive tend to break up when used, thus renewing the sharp edges and continuing to work longer.

Garnet uses tiny bits of stone used for the abrasive. It wears faster than oxide and produces a smoother surface (for the same grit number).

Silicon Carbide is best for smoothing metals, plastic, and fiberglass.

Ceramic is the hardest of common abrasives and is used on rough grades. It can also shape and level rough wood and is often used on belt sander belts.

Emery is used on metal, and is usually attached to a cloth backing rather than paper.

Sand becomes an abrasive when shot out of a nozzle onto a rough surface. The process is called "sand-blasting" and can be done with all sorts of abrasive media, including different sorts of sand, glass beads, walnut shells, water, etc.
You can buy abrasives on paper, cloth, plastic, a belt, in a block, on a stick, or in a bag. The choice depends on how you will use it.

There are standards for describing sandpaper backing, its strength, thickness, waterproofness, how it attaches to a tool, etc. Paper is described from A (Lightest) to F (Heaviest) then the letters refer to cloth J (Lightest) on up to X (Heaviest).

After all that, then you still have to decide the shape - do you want sheets, rolls, discs, or other shapes.

Abrasives are even embedded into rubber blocks, which gives you something to hold onto, and allows for easier sanding on uneven surfaces. 

These abrasives can be used to prepare everything from the side of your house to your fingernails.

And if you use them frequently, they will wear you down as well as the surface you are sanding!

Wednesday, October 21, 2009

How can you be SO DENSE!

Density means being "thick" (as in - you didn't do what your brother / sister / parent expected!). But it also means more than that.

Mathematically speaking, density is equal to mass ÷ volume.  It's stated in units like kilos per cubic meter, or grams per cubic centimeter. When talking about populations, it is expressed as people per square mile or kilometer.

Density is a subject that interests me. Remember the blog about the Thunderbolt Darjeeling tea leaves and how tea of the same weight can result in different-sized packages? That's an example of relative density.

Well, in playing with Wolfram Alpha, a web search/calculation engine, I was able to uncover some interesting facts about density - let's see if we can use it to answer a few questions:

Question: What are the relative densities of water and diamonds?

Answer: Diamonds are 3.5 times as heavy as water.

Question: What are the relative densities of water and gold?

Answer: Gold weighs 19.3 times as much as water. Platinum weighs 21.1 times as much as water.

Question: What are the relative densities of air and helium (for those party ballons)?

Answer: Helium is one-fifth the weight of an equivalent amount of air (at the same altitude/pressure).

Now how about population density? If your neighborhood seems crowded, you could move to the Falkland Islands, where the density is one person per square mile. How does that compare with the rest of the world?

Fact: Macau, an island off the Chinese mainland near Hong Kong, has the highest population density in the world. About 44,000 people per square mile. Despite the crowding, the citizens of Macau also have the longest life-expectancy in the world - 84.4 years.

Fact: Greenland is the least-dense population, with only .07 people per square mile. And their life expectancy is 70.1 years.

Fact: Swaziland in Africa has 172 people per square mile, but sadly their life expectancy is only 31.9 years.

Wolfram Alpha can do many other cool things, like building a population density map. What do you think of it?

Moving on...

Question: What is the population density of San Diego county?

Facts:  San Diego County has a population of 3 million, spread over 4200 square miles of land area.

Answer: 3,000,000 ÷ 4200 = 714 people per square mile. And life expectancy of 78 years.

Of course, if you get bored with density, Wolfram Alpha lets you calculate strange things of any sort:

Question: What is the median age of people living today?

Answer: Approximately 27.5 years old. It ranges from 45.5 years in Monaco to 15 years in Uganda.

Fact:  I am older than the median age in ANY country, and more than twice as old as the global median age - but probably still have a few decades to go - I hope!

Tuesday, October 20, 2009

Walmart Fractions

Reading the newspaper yesterday,  I saw a statement that made me think twice:

Walmart ... is recruiting a new team of managers and buyers that will focus exclusively on the smaller [Neighborhood Market] stores, which, with 150 locations, currently represent only a fraction of Walmart’s total US sales.

I saw the word fraction. Sounds like math to me. Is this statement true? How can we check?

I know he is talking about sales revenue, but what if we compare square footage? Are Neighborhood Markets a fraction of the total Walmart square footage? How big a fraction?

Luckily the Walmart website outlines their various store options:

Walmart now has 2705 Supercenters and each store averages 187,000 square feet. That's 2705 x 187,000 = 505,835,000 square feet. Five hundred six million.

The second category is called Discount Stores. There are 833 of these stores, and they average 107,000 square feet. I think 833 x 107,000 = 89,131,000 square feet. Eighty-nine million.

Sam's Club also appears on the Walmart website, so I will include its square footage too. There are 605 locations with an average of 132,000 square feet each. That's 605 x 132,000 = 79,860,000 square feet. Eighty million.

Let's take a quick detour through the distribution centers. Although you can't shop there, these centers are necessary to support the stores. Walmart has more than 40 general distribution centers and about 70 specialty distribution centers. On average they seem to be about 1 million square feet, so that's 110 x 1,000,000 = 110,000,000. One hundred ten million.

(I suspect the distribution centers might help support neighborhood markets, too - but it's not clear.)

Adding the three types of stores and the distribution centers (while ignoring  parking lots, room the trucks take up on the highways, and other esoteric data) gives us Seven hundred eighty-five million square feet of floor space.

There are now 151 markets, with an average size of 42,000 square feet. That's 151 x 42,000 = 6,342,000 square feet. Six million.

Walmart is testing a new store format similar to Tesco Fresh and Easy stores. They have 4 of these in Arizona, with an average of 15,000 square feet per store. That's not even enough to make a rounding error on the big store totals, but we'll calculate the number anyway, since we can do it in our heads. I find that 4 x 15,000 = 60,000. Six percent of one million, or one-sixteenth of a million.

Adding these two types of stores gives us Six million square feet of floor space.

Now we can do some math and reach a conclusion.

6 million ÷ 785 million = .00764

Are the neighborhood markets just a fraction of the total Walmart? Yes, they are. Here's a pie chart to show the comparison.

Neighborhood Markets are less than one percent, or in (simplified and rounded) fractional terms, they are 1/130th of the total Walmart presence.

Monday, October 19, 2009

Is there Math in a Pomegranate?

You know pomegranates, right? Red fruits from the Middle East (and California) with lots of tiny red juicy seeds enclosed in a tough skin. The juice stains your hands and clothes. Punica granatum is the horticultural name of this plant.

Is there math in a pomegranate as well as tiny seeds? Well, why not? Let's take a look.

Pomegranate trees grow 20-30 feet in height. Here's one next to a house that is three stories tall (plus basement).

You can't tell from this photo but I estimate that this tree is about 18 feet tall, based on my experience reaching up to get fruit off the top branches. If I am 5 feet 10 inches tall and the picker is 12 feet long, how high can I reach? Don't forget my arm's length.

The average pomegranate has about 100 calories, but you might burn that many just trying to get the seeds out, then washing your hands, and clothes!
October has been pomegranate harvest time in Southern California for exactly 240 years, since the mission-founding Fathers brought them here from Spain and the Middle East.

In what year did they bring them?

Today we went off to our friends' house to pick and seed the pomegranates. They have 4 trees which produce over 1000 pomegranates a year.

In Jewish tradition, pomegranates were said to contain 613 seeds, and each of those represented a good deed that an observant Jew could perform.

I have to admit that even though we opened an entire wheelbarrow of pomegranates, we didn't count the seeds from a single one. But I am sure we got more than 613! 

Did I mention that pomegrante juice stains, both your hands and your clothing? (Yes)

Don't worry about the hands, because soap and water will take care of the staining. Clothing tends to have a more permanent color problem...

We made sure we wore old clothes so it didn't matter if they turned reddish-purple.

We had such fun doing this we didn't think about counting either the number of seeds or fruit. But I did some research later.

I found a site where people counted the seeds in more than 200 pomegranates. The range went from 165-1370 per fruit. The average average was indeed 613! But they could have counted more or less, or fruits from other countries, etc. etc. and come up with different number.

So we could safely say 500 or more seeds in a pomegranate.

Country Sample Size min #seeds max #seeds avg #seeds min weight (g) max weight (g) avg weight (g)
US 150 286 1370 680 145 1065 421
Singapore 2 339 579 459 250 400 325
Iran 37 165 1263 338 200 660 354
Spain 2 580 837 709 280 330 305
Turkey 12 267 971 576 226 1001 603
Brazil 3 647 1054 809 568 658 599

All 206 165 1370 613 145 1065 420

Now that we have some data, we could plot it, slice it and dice it.  It would be fun math to do so. But we won't.

Because it's more fun to make the juice. You throw the seeds into the ACME 6000 JUICER (No, I didn't steal it from Wiley Coyote, that's really the name of this machine!) which spins them at the a fantastic 3600 revolutions per minute. The seeds stay inside and juice comes out the back. A new juicer costs about $200 but our friends got it much cheaper at a garage sale.

After a straining through a filter, the juice goes into jars to become jelly, or a healthy fruit drink.

If the large jugs hold a gallon and the smaller containers hold 20 ounces each, and we got about 20 quarts of juice today, how many ounces of juice did we get?

Each container is full of anti-oxidants and vitamins and stuff.

(See how boring it is to say " container full of stuff" - don't you want the numbers?)

Of course, the pomegranate has been a favorite subject of painters for millenia (thousands of years).

Still Life, with Pomegranate

And that's how we get math out of a pomegranate.

Friday, October 16, 2009

Want to Buy Some Money?

Sounds funny, doesn't it? Buying Money. But people do it all the time.

How? you say, and why? And at what price?

It's not always easy to count the cost or know the price. But since we are math fans, we'll consider that as we look at some of the many ways we buy money.

Suppose you are having a yard sale. You know there are lots of small items, and you will need change. You go down to the ATM, then remember it only gives you $20 bills. So you walk into the bank, hand the cashier 5 $20 bills, and ask for $100 in small bills and a few rolls of quarters.

In this kind of situation you should not have to pay extra for the money you are buying.

You have a coin collection. You want an uncirculated coin, or a mint set. Or a rare bill. You go to a coin collector's shop, or you go to the website of the US Mint . If you want paper money, you go to the Bureau of Engraving and Printing's Money Factory.

Most of the items they sell are in pristine condition, perhaps specially-manufactured for collectors, and so they cost more than the face value of the coin or bill. For example, a new, un-circulated roll of 25 $1 coins costs $35.

Nowadays you can't easily get really large bills, like this thousand-dollar bill. The widespread use of checks and now electronic payment methods reduced the need for large bills.

If you are going on vacation in a foreign country, you might want to have some of their paper money on hand when you arrive. If it's a commonly-visited country, you can buy the money from your local bank, or from a company that specializes in "foreign exchange". They can mail you the money, or if you do wait until the last minute, you could buy it at a small booth in most international airports.

In addition to taking physical money, you might also need to send money to someone overseas. In that case you need to have it delivered to them directly or to their bank account. The most common way of doing that is called "wiring the funds" where your bank exchanges money with their bank (for a fee).  Other ways include using Western Union, or Paypal, or another exchange service.

Any time you purchase foreign money you have to pay at least one or more fees. There's normally a  convenience charge for giving you the money, and there is an exchange rate margin where they buy the money for less than they sell it to you, and pocket the difference. The smaller the amount you change, the greater the percentage of the charges will be. In many cases if you come home with a small amount of foreign currency, you will not be able to sell it. So consider it a souvenir, or send it back to someone in the country who can spend it. Or leave it on the airplane in the UNESCO envelope.

It's always more fun to get real money when you are a kid. Checks aren't very nice, because you have to give them to your parents, or go to the bank and try to get cash. As a parent or grandparent, it's also more fun to get lots of brand-new bills, put them in envelopes, and hand them out. You can also get rolls of coins which are great fun too. The best place to get brand new money is at a big bank, or the Mint or Money Factory.

In case you are interested, the Mint is having a sale (sounds funny, doesn't it?) on dollar coins. You can buy up to $500 each of newly-released Presidential dollar coins (Washington, Adams, Jefferson, etc.) for the face value, and the mint will ship them free, directly to your house in a box. What fun! A box of money...

If you are a business, you might have cash registers or vending machines, and collect lots of coins. In that case you want to change those coins into paper money for your cash register, or credit in your bank account.

Banks don't like to accept huge piles of coins, so a business might need a coin-counting machine to sort the coins, count them, and put them into rolls. The bank will credit your account for the rolls only if you do business with them regularly.

In this case, you are just like the business, only you don't have a trustworthy record of correctly filling rolls of coins. If you roll the coins, the bank might just open them all and charge you for counting them. So you might be better off letting them do it in the first place.

Alternatively you could use a coin-counting machine like you see in the front of the grocery store. They charge a lot for the convenience, though - a few hundred dollars  changed in the machine will cost you as much in fees as a small counting machine.

This isn't exactly the same thing as buying money, but it is possible to generate money by collecting bottles or cans and exchanging them for cash at a recycling yard. Containers which require a deposit are "as good as cash" to those who find them.

Of course there are lots of other things to trade for money. Like WORK.

It is important to remember that the two ways NOT to get money are

1. Make it yourself (unless you are a government)!
2. Take it without asking permission (unless you are a government)?

Thursday, October 15, 2009

On a scale from one to ten,

  • how well did we handle your bounced check?
  • were you happy with how we fixed your car?
  • how much heartburn did you have when we audited your tax return?
Have you ever wondered how this "on a scale of" business started? No, I didn't think so. Neither did I.

But I was at the doctor recently, and they asked,

How much does it hurt? On a scale from 1 to 10, tell us what it feels like.  I said 6, at this moment.

But that's not very helpful, at least from my perspective as the sufferer! Compared to what? Getting run over by a truck, gored by a raging bull, or bitten by a snake?

So in the interest of MATH and my blogs on how to measure odd things, I decided to review pain scales. After looking at dozens, here's my favorite:

This is the Wong-Baker graphical scale. Click for a more elaborate version.

It's easy to understand and the graphics are funny enough that they don't make me hurt more or get bitter. Of course it's only for kids, not adults.

Here's a scale developed just to measure pain from various kinds of bees and wasps:
  • 1.0 Sweat bee: Light, ephemeral, almost fruity. A tiny spark has singed a single hair on your arm.
  • 1.2 Fire ant: Sharp, sudden, mildly alarming. A shock from walking across a carpet and touching a light switch.
  • 1.8 Bullhorn ant: A rare, piercing, elevated sort of pain. Someone has fired a staple into your cheek.
  • 2.0 Bald-faced hornet: Rich, hearty, slightly crunchy. Like getting your hand mashed in a drawer.
  • 2.0 Yellowjacket: Hot and smoky, almost irreverent. Extinguishing a cigar on your tongue.
  • 2.5 Honey bee: Like a matchhead that flips off and burns on your skin.
  • 3.0 Harvester ant: Bold and unrelenting. Using a drill to excavate your in-grown toenail.
  • 3.0 Paper wasp: Caustic and burning. Bitter aftertaste. Like spilling a beaker of hydrochloric acid on a paper cut.
  • 4.0 Pepsis wasp: Blinding, fierce, shockingly electric. A hair drier has been dropped into your bubble bath.
  • 4.+ Bullet ant: Pure, intense, brilliant pain. Like fire-walking over flaming charcoal with a rusty nail in your heel.
OUCH! Just reading the descriptions hurts me. Apparently this index was developed through personal "research" conducted unintentionally by Dr. Justin Schmidt, an entomologist. When asked What is the most interesting insect venom, and why? he replied

Three come to mind. Bullet ant because it causes such intense pain and it lasts so long and is not "diluted" out of the pain-inducing concentration in the local sting area. Second is the tarantula hawk because its pain-inducing component(s) cause such immediate pain, yet the pain is gone within a couple of minutes.  Third is the harvester ant venom because it seems to directly affect neuromuscular ... and other receptors (unique among insect venoms) and is so incredibly toxic.

I can't comment on most of these dangerous critters, as I stay as far away from them as possible. (But notice the detailed definitions of pain he provides.)

However, I have been stung at least 100 times by honey bees, one day getting about 20 stings in a single attack. It's an occupational hazard of a beekeeper.

Here are some photos of me with my hive of honey bees.

Of course with a big thick bee suit you rarely get stung, but there are always small openings the bees explore, and up-the-pant-leg stings or in-the-bonnet stings are particularly exciting!

Perhaps now you know why I have transitioned from bees to a worm factory - the worms don't sting!

On a scale of 1 to 10 for most dangerous, worms rate a 1.


    Wednesday, October 14, 2009

    Census - opening a can of worms?

    Counting people is a hassle. There are plenty of people who don't want to be counted, some who want to be counted more than once, others who don't want YOU to be counted, and so on. But I'm not worried about the national census. I'm worried about counting my worms.

    We have a worm farm. We are keeping them like pets - not really doing anything productive with the worms, but they get to eat all our scraps and lots of our newspaper.

    We recently bought a Worm Factory, which is a plastic container for the worms to live in. It's a bit like a beehive.

    I'd like to know how many worms we have. But it's not that easy to count, as you can imagine!

    They hide when you open up the lid, they intertwine, wriggle, and do lots of other wormy things.

    Looking for ideas on the Internet, I saw several people suggest killing the worms, so they would be easier to count. Another said suck them up through a straw taped onto the vacuum hose.

    Sorry, not going to happen.  I'm not going to kill all my worms just to count them.

    I learned about a company - Union Biometrica - that makes high-tech machines for laboratories, to count cells, seeds, mosquito larvae, etc. That was out of my budget, and too fancy for my purposes.

    Westgate Laboratories does "worm counts" but I learned these are really a worm egg counts. They take a small sample of dung (horse poo) and examine it under a microscope to find out how many worm eggs are present.  From that count they extrapolate an estimate of the worms in your horse. Yukk!

    I found a story of a group of kids doing a worm census out in the forest. They put ground mustard in water, poured the liquid on the ground, and the worms came up, spitting and gasping. They get counted while on the surface. I can see where that's a possible way to get them up out of the soil, but I don't want to anger my worms - they can disappear overnight if you get them upset.

    Some other kids did their own census at home, using the mustard technique. They were frustrated by too few worms - only 3 in an afternoon. No problem counting them!

    The worm farms that supply mail-order worms to home worm factory buyers tell me there are about 600-1200 worms per pound.

    Another source says 1000 worms per square foot (or layer) of the bin.

    The Worm Factory people say maximum capacity is 3000 worms per tray.

    Well, which is it??? I give up.

    We started with a  handful 6 months ago. Now we have thousands. Since the Factory can accept more layers, I guess we should have plenty of room...

    Tuesday, October 13, 2009

    Have trouble making a decision?

    If you can't decide what to order on a menu, don't look in the Duluth Trading Company catalog! They sell a seemingly-endless variety of clothing and related products for hard-working Men and Women.

    As I was shopping, I saw descriptions of clothing that used terms that were unfamiliar to me, all seemingly related to weights of fabric.

    The one fabric I do understand is their Fire Hose canvas. They promote it well and I own a couple  pairs of Fire Hose jeans.

    I thought I need to make a list!

    Here are the varieties of fabrics and weights in the printed Oct '09 catalog and online:
    • 14.3 oz. "hulking" cotton/polyester
    • 14.0 oz cotton denim
    • 13.7 oz. cotton terry 
    • 13.4 oz. Fire Hose
    • 12.7 oz. Fire Hose
    • 12 oz. Fire Hose
    • 12 oz. cotton denim
    • 10.9 oz. Fire Hose
    • 10.7 oz. Fire Hose
    • 10.5 oz. Fire Hose
    • 10.0 oz. Fire Hose
    • 10.0 oz. cotton/polyester blend
    • 9.0 oz. cotton chamois
    • 8.4 oz. cotton twill
    • 8.0 oz. pre-washed cotton canvas
    • 8.0 oz. cotton twill
    • 7.6 oz Fire Hose
    • 7.5 oz. cotton
    • 7.5 oz. cotton/spandex pique
    • 7.0 oz. cotton pique knit
    • 6.5 oz. waxed cotton canvas
    • 6.5 oz. polyester
    • 6.5 oz. cotton knit
    • 6.4 oz. cotton denim
    • 6.2 oz. nylon/cotton
    • 6.0 oz. pre-washed cotton twill
    • 5.9 oz. polyester
    • 5.9 oz. cotton jersey
    • 5.7 oz. CoolMax
    • 5.6 oz. cotton twill stripe
    • 5.6 oz. nylon
    • 5.5 oz. cotton/hemp blend
    • 5.5 oz. flannel 
    • 5.3 oz. cotton interlock
    • 5.3 oz. polyester
    • 5.0 oz. polyester/nylon microfibre
    • 5.0 oz. ripstop polyester
    • 3.5 oz. cotton/nylon
    • 2.5 oz. rayon/polyester
    Then there were some items whose weight was indicated in grams
    • 265 gram polyester fleece
    • 160 gram nylon-lined insulation
    • 140 gram synthetic insulation
    • 100 gram insulation
    • 80 gram Thinsulate
    • 40 gram Thinsulate
    Based on a few clues in some of the Duluth descriptions, I decided the weight figures should indicate the weight of a square yard of fabric. I hope that's right.

    Other non-clothing synthetic fabric items contained additional unknown units:
    • 2000-denier nylon/polyester
    • 1680-denier ballistic nylon
    • 1200-denier polyester
    • 1000-denier polyester
    • 600-denier polyester
    • 500-denier nylon
    • 300-denier polyester
    • 150D x 600D nylon
    • 5000-lb. tensile strength 1.2 mm polyester
    • 16 mm polyurethane
    Denier is a fabric industry term that describes the heaviness of fibers in a fabric. It's calculated by weighing the mass of 9000 meters of thread or yarn. A measure of 1 means weight of 1 gram for 9000 meters of thread. The larger the number, the thicker the fiber.

    NOTE: this is not the term used for someone who denies (the Holocaust, for example).

    Many man-made synthetic fabrics are also rated by tear strength (ability to resist the start of a tear) and tenacity (ability to resist additional tearing once a rip has started). Tenacity is indicated in grams per Denier and ranges from 2.0 grams (weak) to 9.5 grams (very strong). Other qualities include hydrostatic resistance to water being forced through the fabric, and color-fastness in bright light.

    And just a few more, for you miscellany specialists! I haven't the time or space to define them all. And I've intentionally omitted the trademarked fancy fabric names (sunblocking, stink-stopping, tree-hugging, etc.)
    • 550-fill-power goose down
    • sharkskin
    • waxed cotton
    • anti-microbial wicking
    • 100s fleece 
    • 21 oz. #4 cotton canvas
    • 20 oz. cotton twill
    • #10 cotton duck canvas
    Could there be a few too many choices? A little too much information (TMI)?

    In general I am in favor of choices. But not choices that mean nothing to me - the consumer. I did not know what these terms meant in general, or what they meant to me. As a wanna-be hard-working man, I could stand a bit more education on the subject.

    I went to the Fashion Incubator blog, for some background on fabric weight and conversions. I hate to admit it, but there I also found TMI. But I had an enjoyable visit. That site is right up my alley when it comes to using math.

    Monday, October 12, 2009

    You can never be too tall or too thin or too rich

    But a wall can be too tall and a coat of paint can be too thin.

    I need to paint a tall, strangely-shaped wall in our house. The painters left me a little bit from the last job, so I can have the color matched down at the paint store. The question is - how much do I need? The color is sort of a medium blue. I want to cover it with white. I know I will need more than one coat of paint.

    The wall is a complex shape. Here's a drawing I made of it.

    We can calculate the area if we think of it as a rectangle at the bottom with a right triangle on top.

    The rectangle is 172 inches long by 93 inches high. That's 16,000 square inches.

    The right triangle is about 150 inches on one side, and 48 inches on the shortest side, so that's 7200/2 or 3600 square inches.

    I round the totals to 20,000 square inches. Divide by 144 to get 139 square feet.

    A quart of paint will cover 400 square feet. I could apply at least 2 coats of paint if I buy a quart.

    So I bought the quart of paint and put the whole can of paint on the wall (it did 2 coats plus a bit more). As I sat there looking at my wall, I wondered, How thick is the paint?

    My wife said, Excuse me? How thick is the paint?

    Yes, I said. Putting a quart of paint on the wall should be easy enough to figure - how thick is the coat of paint?

    Sounds like a Blog Problem, she said, turning back to her mystery novel...

    So here we go. First we need a few rules:
    • Let's make an assumption that 10% of the paint gets wasted by the brush, spilled, etc. 
    • We'll ignore "soaking in" and assume the paint dries on the surface.
    • It will be easier to do this in metric units. Let's convert to metric before we begin.
    A square foot is 930 square centimeters. Multiply 139 square feet x 930 to get about 130,000 sq cm.

    A quart is 946 cubic centimeters. If we waste about 10% we'll have 850 cc of paint.

    Our question can now be stated: How thick a layer will 850 cubic cm of paint create if poured out on 130,000 sq cm of wall?

    If the wall was a floor we would just pour the paint out, let it spread, and let it dry. But here's another way to think of it.  Let's re-shape our 850 cc cube to 20 cm x 20 cm (400 sq cm) x 2.125 cm thick (20x20x2.125=850).

    Imagine the paint as a block of cheese. We slice the block, and simply spread the 20x20 cm slices around on our 130,000 square cm piece of bread. How many slices do we need?

    If we divide 130,000 (wall) by 400 (slice) we get 325. We need 325 slices of the paint to cover the surface of the wall. So we need to slice the 2.125 cm thick block into 325 slices. Each slice of cheese/paint is 1/325th of 2.125 cm. That's .0065 cm, or if converted to mm it's .065 mm.

    Seems thin, doesn't it? Let's recheck our calculations.

    .0065 x 130,000 = 845  ≈ 850 It checks!

    Still a bit uncertain, I found a company that makes ultra-sonic paint testing gauges and discovered  most painters put one coat of primer and two coats of finish paint on interior walls. Here's how the measurement readings look using their ultra-sonic paint thickness tester:

    These readings are shown in mils but can be converted to mm.
    The three coats of paint vary from 1.4-4.1 mils of thickness. If we take my paint thickness of .065 and multiply by 39 to convert to mils, we get 2.5. That's close enough for me.

    These cool gauges are $500-2500 and another trip to the paint store would have taken at least as long as these calculations. I saved both money and time.

    See how useful math is?! I exclaimed (standing tall) with pride.   Zzzzzzzzzzzzzzz ... she replied.