Additional Math Pages & Resources

Wednesday, December 21, 2011

Multiplication Mysteries

Yesterday I shared the discovery of a new material called Magic Gold. It's an alloy of gold and ceramic. In some respects, the fusion of these materials gives us more than the sum of the individual parts. Magic Gold is still gold, but lighter than gold, stronger than gold, and tougher than gold. It's a form of magical multiplication (I had to get math in here somehow, this is a math blog!)

Alchemy is a philosophical and magical tradition that seeks a philosopher's stone, an object that transforms base metals into gold or silver, and creates a drink that confers immortality - the elixir of life or fountain of youth.

Here's a poem from centuries ago:

Augmentation is of the Elixer indeed,
In goodness and quantity both for red and white
Multiplication is therefore as they do write,
That thing that doth augment medicines in each degree,
In colour, in odor, in virtue and also in quantity.

And why may you multiply this medicine infinitely,
Forsooth the cause is this,
For it is as fire, which kindled will never die,
Dwelling with you, as fire does in houses,
Of which one spark may make more fire this way,
As musk in pigments and other spices more,
In virtue multiplied, and our medicine right so ...

Keep in your fire therefore both morning and evening,
So that you do not need to run from house to house,
Among thy neighbors to seek or borrow your fire,
The more you keep, the more good shall you win,
Multiplying it always more and more within your glass,
By feeding with Mercury unto your lives end,
So shall you have more than you need to spend.

Did you notice the distinctly modern tone in this ancient poetry? Multiply what you have, magically, so you will have more than you need to spend for the rest of your life? Sadly, despite the promises of our 401(k) brochures, and the ready availability of alchemic symbols on the Web, magical multiplication eludes our grasp today.
As human beings, we are pretty good at division - diluting and sweetening (this blog's topics of the past few days). We can water anything down and make it go farther for less cost. It's a lot more difficult to multiply and get something out of nothing, because in non-magical math, zero times any quantity is still equal to zero.

In his Philosopher's Stone, Van Morrison articulates this eternal search, lamenting his unending task of writing new songs:

Out on the highways and the by-ways all alone
I'm still searching for, searching for my home
Up in the morning, out on the road
My head is aching and my hands are cold

I'm looking for the silver lining in the clouds
I'm searching for the philosophers stone

It's a hard road, Its a hard road daddy-o
When my job is turning lead into gold
I'm on the road again and I'm searching for
The philosophers stone

Even my best friends they don't know
That my job is turning lead into gold.

Up in the morning, up in the morning
When the streets are white with snow
Up in the morning, out on the job
Searching for the philosophers stone.

Monday, December 19, 2011

Dilution and Density

In the most recent series of math blogs I have been talking about drinks - water, juices, tea, etc.

What are drinks?

Drinks are liquids that we consume to sustain or entertain ourselves. They are carefully-contrived blends of various substances (carbonation, fruit or vegetable juice, water, sugar, etc.) that are wet, drinkable and nutritious (or intoxicating), etc.

As I discussed earlier, whenever companies create blends or combinations, government gets interested. Blending may be used to disguise fraudulent or unhealthful products. Thus we regulate what vendors can claim or sell. We often need some elementary school math skills to understand these regulations and the products being controlled.

Today I would like to turn your attention to a different type of "blend". I invite you to compare cranberry juice and gold.

Pure cranberry juice is extremely tart, sour, bitter and acidic. It's nearly impossible to drink without dilution and sweetening. We need to mix the juice with sweeter fruit juices or add water and sugar. Most cranberry juice products end up being about 25-30% cranberry juice.

 Now let's make a big mental jump from red to gold.

Pure gold (24 carat) is very valuable, very heavy and very soft. It's nearly impossible to use without dilution with silver, copper, nickel, etc. We need to mix the gold with other stronger substances, which make gold much stronger, a bit cheaper, and change its color. Many gold products are 18k gold, which is a blend (alloy) of 75% gold and 25% other materials.

ALLOY: A metallic substance made by mixing and fusing two or more metals, or a metal and a nonmetal, to obtain desirable qualities such as hardness, lightness, and strength.

OK, with all that background established, this week a brand-new gold alloy has appeared. It's called MAGIC GOLD. It was developed by Hublot, a Swiss watch company. I have visited the company, and met with its dynamic president, Mr. J-C Biver.

To clarify the magic, I will use a trick question "Which is heavier, a pound of feathers or a pound of lead?"  Of course the answer is "A pound of one is the same weight as a pound of the other."

This reveals the MAGIC at the heart of MAGIC GOLD. It weighs less than any other alloy of gold, yet it is still 75% gold. Why? How?

Here is the pound of lead part of the question:

The Swiss Federal Customs Administration (FCA) manages any claims made about gold, and Hublot quotes them when saying:
  1. MAGIC GOLD contains 750 grams of pure gold per 1000 grams of alloy (proportions by mass), thus FCA gave us permission to use a hallmark of 750/1000 (18 carat)
  2. MAGIC GOLD is a composite (alloy) made of pure gold [density 19.32 g/cm3] and boron carbide ceramic [density 2.5 g/cm3]
  3. The density of this new material is found with this equation:
((% gold x mass of gold) + (% ceramic x mass of ceramic) = (mass of the alloy) 
((0.3 x 19.32) + (0.7 x 2.5)) = (5.796 + 1.75) = 7.546 g  so density = 7.546 g/cm3
Now here is the pound of feathers.
  1. MAGIC GOLD is by volume 30% pure gold and 70% boron carbide (proportions by volume)
  2. A watch case in 18k gold that weighs 100g contains 75g of pure gold (75%)
  3. The same watch case in MAGIC GOLD weighs 48.68g (about half of 100g) and contains 37.38g of pure gold (about half of 75g)
  4. So, for the same volume of material, the watch case contains half as much gold by mass
  5. Total weight of the case is also half, but the proportion of pure gold remains at 75%
Where is the MAGIC here? The MAGIC is that Hublot can make watch cases and cuff links, rings, etc. using half of the gold but adding lots of research, ceramics, temperature and pressure. The resulting substance is stronger, more scratch-resistant than the old stuff and it's still 18k gold.

Sweet, eh?  Just like cranberry juice punch!

Friday, December 16, 2011

Pour boiling water ...

In the past few blogs, I have outlined the things that we can do to make water more palatable by chilling, carbonating, mineralizing and flavoring it. I described drinks that involve various blends of fruit juices, but skipped sodas and sport drinks.

Today we will consider hot beverages - without which most people would stay in bed, or fall asleep at their desks.

Do we need any math skills to understand these drinks?

With a cup of hot coffee or tea, the groggy sleeper ingests some warmth, some caffeine, some flavor and some aroma - and perhaps a few calories. There are plenty of other hot drinks in the world - infusions and heated spirits and hot chocolate, but coffee and tea are the two favorites.

Neither of these beverages has much nutritional value. We get no calories, no fat, no protein, but coffee is apparently a good source of Pantothenic Acid, and a very good source of Riboflavin, while tea is a very good source of Manganese. Both contain caffeine, with coffee supplying 2-3 times caffeine as much as tea.

Neither of these beverages can be easily made at home, from scratch. You can't grow the plants or prepare the raw materials yourself unless you live in certain tropical regions. [click images to enlarge]

But looking at the production process or chemical content doesn't tell us why people drink either one. There's plenty of intangible value to be obtained from some human activities, even when that value can't be quantified by the scientists.

NOTE - For detailed nutritional statistics on brewed, un-blended beverages made with tap water, check here for coffee, and here for tea.

I don't drink coffee myself, so I can't speak to its intangible virtues. But I can certainly see people all around me who enjoy it. Historians tell us this has been the case for about 5 centuries. How much do these coffee lovers drink?
  • I found estimates suggesting half the US population of 310 million drinks coffee, and the average drinker consumes 3.2 cups per day, so ( 310 ÷ 2 ) x 3.2 =  496 million cups
  • Other sources told me 400 million cups a day
And all this because someone thought of picking, drying, roasting and grinding these pretty berries, before forcing hot water through them!

The tea drinkers of the world, and I am one, find pleasure in drinking tea. We have been doing so for about 30 centuries! We have been taking a short break with it, blending it, sweetening it, diluting it with milk and cream, washing down cookies and cakes with it, and so on. How much do we drink?
  • I found that tea drinkers comprise 66% of the 65 million people in the UK and each consumes about 3 cups per person daily. That means ( 65 x .66 ) x 3 =  128 million cups
  • The Tea Council says the number is 165 million cups a day
Just for fun, a comparison: the UK has about 1/5 the population of the US but drinks about 1/3 of the tea compared to the coffee we drink.

It's amazing that all this activity centers around pouring heated water over some leaves, isn't it? Ok, it's a bit more involved than that, but still, it's hot water and plant leaves. And some math.

Non-water, Non-Alcoholic, Non-sense Drinks

For the last few days I've written about water and how we prefer to consume our water and other drinks (meaning a beverage which is NOT water and not primarily a food). We like them a bit cooler or a bit warmer than our body temperature, depending on the drink.

It turns out that a little elementary mathematics comes in handy in dealing with drinks, as we must consider heat, cold, volume, percentages of dilution, etc. Here are a few reasons why some math is necessary when shopping (or indeed when trying to sell juice). Imagine if these were applied to the proverbial Lemonade Stand!
[Code of Federal Regulations]
[Title 21, Volume 2][Revised as of April 1, 2002]
From the U.S. Government Printing Office via GPO Access [CITE: 21CFR102.33]

Subpart B--Requirements for Specific Nonstandardized Foods
Sec. 102.33  Beverages that contain fruit or vegetable juice.
(edited slightly for ease of reading)


For a carbonated or non-carbonated beverage that contains more than 0% and less than 100% fruit or vegetable juice, and, if the usual name uses the term juice, it shall include a qualifying term such as beverage, cocktail, or drink to advise the consumer that the product is less than 100% juice (diluted grape juice beverage or grape juice drink).

If the product is a diluted multiple-juice beverage or blend of single-strength juices, then the names of those juices must be shown in descending order of volume unless the name specifically shows that the juice with the represented flavor is used only as a flavoring (raspberry-flavored apple and pear juice drink).  The presence of added natural flavors is not required to be declared in the name of the beverage.

If a diluted multiple-juice beverage or blend of juices contains a juice that is named or implied on the label (represented juice), and also contains a juice other than the named or implied juice (non-represented juice), then the product name shall indicate that the represented juice is not the only juice present (Apple blend; apple juice in a blend of two other fruit juices).

In a diluted multiple-juice beverage or blend of single-strength juices where one or more, but not all, of the juices are named on the label, and where the named juice is not the predominant juice, the product name shall indicate that the named juice is present as a flavor or flavoring (Raspcranberry; raspberry and cranberry flavored juice drink); or it shall include the amount of the named juice, declared in a 5% range (Raspcranberry; raspberry and cranberry juice beverage, 10-15% cranberry juice and 3-8% raspberry juice).

The common or usual name of a juice that has been chemically modified shall include a description of the exact nature of the modification (acid-reduced cranberry juice, or deflavored, decolored grape juice).

If the product is a beverage that contains a juice whose color, taste, or other properties have been modified to the extent that the original juice is no longer recognizable, or if its nutrient profile has been diminished to a level below the normal nutrient range for the juice, then the fruits or vegetables from which the modified juice was derived may not be depicted in pictures on the label.
      (1) If one or more juices in a juice beverage is made from concentrate, the name of the juice must include a term such as from concentrate or reconstituted, in a type font no less than one-half the height of the letters in the name of the juice.

      (2)  If the juice is 100% single-species juice consisting of juice directly pressed from a fruit or vegetable whose Brix level is adjusted (raised) by the addition of juice concentrate from the same fruit or vegetable, it need not include a statement that it is from concentrate. However, if water is added to this 100% juice to adjust (lower) the Brix level, the product shall be labeled from concentrate or reconstituted.

Now that you know all this, have fun inspecting the labels of these juice-like drink beverages we found in the local market. The first is what I would call a brightly-colored, flavored soft drink:

After tasting this, I don't know what I would call it, but it doesn't taste like either Pomegranate or Blueberry juice. It's sweetened with sugar and with concentrated pear and apple juices.

 This is some tart red stuff that looks like juice, tastes like juice, but can't be called juice.

This is the most complicated label and very tasty stuff it is too, but what would you call it? Do we benefit from these complex creations? It's not a hot summer day but a cool winter day - yet it still tastes very nice, in my opinion.

Ah, orange juice. Not from concentrate. Turns out it was the cheapest of all these choices, too!

Thursday, December 15, 2011

Cold, Clear, Sparkling Water

What makes water appealing to drink? Can elementary math help? Let's see.

Which type of water is most appealing to you (select 1 of 3 answers below):
  1. A glass of luke-warm, 85° F rusty tap water, with a bit of chlorine odor?
  2. A bucket of greenish water with a few particles drifting (or swimming) around the bottom?
  3. A small trickle burping up out of a rusty fountain with greasy kid-prints on the push button?

NONE!  I'd rather go without!, you say? I completely agree with you.

I wondered why we tend to like water that is a bit colder or hotter than we are ... lukewarm or tepid aren't very nice, either for bathing, swimming, or drinking. Too cold is difficult - chattering or cracking our teeth - and too hot burns our tongues or scalds us.

I read that drinking cold water burns calories, as the body has to work to heat the water up. That sounds a bit fishy to me, but you can see here how people try to explain it with math. Other researchers suggest that beverages 10 degrees or less below room temperature are more platable, so we drink more. Of course they are not talking about plain water, but:

"Fluid replacement beverages that are sweetened (artificially or with sugars), flavored and cooled to between 59 and 70° F, should stimulate fluid intake."

Some studies show cooling a beverage will increase consumption between 40 and 80%.

Of course there are calories in sugar-sweetened water (and you get cavities from it, say dentists) or chemicals in artificially-sweetened waters.

Others say that drinking warm water relaxes the mouth and throat, making it a healthier choice. (Yawn).

As long as we are talking additives, here are some numbers that define salinity for you (salt in the water). The unit we are using is parts per million:
  • drinking water - 100 ppm
  • limit drinking water - 1000 ppm
  • limit for agriculture irrigation - 2000 ppm
  • sea water - 30,000 - 50,000 ppm
  • brine / slush > 50,000 ppm


Take out dirt, chemicals and colors. Make it clean and clear. There are many filter systems available - they fit in the garage, or under the sink, or on the counter, or in the refrigerator.

Sometimes water tastes "flat." Air dissolved in water occupies up to four times the space that carbon dioxide consumes, so get the air out of it first by boiling it.

Warm or room-temperature doesn’t taste very good. It doesn't carbonate well either - CO2 dissolves better as water temperature decreases - the closer to freezing the better. 

Mineral salts enhance flavor, which is why mineral waters work well with food, and they are more appealing internally as the correct salinity of water allows it to be integrated more easily into the body.

Carbonation means bubbles of CO2 - adding a tingle and sparkle in your mouth to fire up your sensory and olfactory systems. The bubbles help lift the aromas to your nose and up the back of your throat.

There are lots of things you can add to water, but more than a very small amount means it becomes a "flavored drink" or simply dilutes the other liquid (not always a bad thing).

I'm thirsty - it's off to the Excel Math water cooler for me!

Wednesday, December 14, 2011

How to get into hot water

A common idiom in English is "you're in hot water" which means you're in trouble. Why? Is the phrase derived from the old image of an explorer being cooked in a pot by a bunch of cannibals?

I think hot water is universally considered to be a very good thing, as any one who's bathed in luke warm or cold water can attest. Today in the math blog we'll investigate this subject - a moderate amount of elementary math is enough for the main issues related to hot water.

US Average Soil Temperatures
Most of our water comes from pipes buried well below ground level, where it stays a relatively constant 40-70° F or so, depending on location and weather. Since our body temperature is around 100° F, we prefer to wash (if we can) in water that is a bit warmer than the water coming out of the pipes - say about 104–120° F.

A water heater in our homes should be able to raise water temperature about 40-60° F and supply us with at least 5-6 US gallons per minute (gpm) or more. Why? Because sink faucets and appliances can draw up to 3 gpm (each).

You may know that plumbing codes restrict shower water flow to 2.5 gpm. But remember, this is for ONE shower head. If TWO or more showers might be going at the same time, you need a much greater supply of hot water, or there will be angry shouting in your house!

A BTU is the energy required to raise one pound of water by one degree F and a gallon of water weighs 8.3 pounds. Assuming 100% efficiency, raising 40 gallons of 50° F water up sixty degrees to 110° F requires a fair amount of energy, as you can see here:

(40 gal × 8.3 lbs) × (110 − 50) degrees
332 lbs x 60 degrees
19920 BTU
19920 / 100,000 
(natural gas is sold by the "therm" which is 100,000 BTU)
.20 therms

NOTE: For more on Natural Gas, check my first blog on the subjectHere's the second one, too.

We have a number of choices for heating our water. None are without compromises:
  • A 40,000 BTU/hr gas heater takes about a half hour to heat 40 gallons, at 100% efficiency. At $1 per therm, the cost of the gas would be about 20 cents. The tank will give us at least a 16-minute shower (40 gallons ÷ 2.5 gpm = 16 min) or longer since we usually mix some cold into the hot water flowing from the tap.
  • A typical electric water heater with a 4500 watt heating element takes a bit more than an hour to heat the same tank of water for roughly $ .50. 
  • A tankless water heater with 40,000 BTU/hr can only supply about 1.6 gpm. To supply 2 showers (5 gpm and 60° F temperature rise), tankless heaters need capacity of about 150,000 BTU per hour, which means larger gas lines or electrical wiring.   
  • Combination boilers in the UK and Europe heat water for both the shower and heating the house. They are compact and efficient and can generate 100,000 BTU or more. Since many are located inside the house, some fancy plumbing or venting may be required.
  • Solar heating is "free" but not fast. It requires a solar array and piping. The details vary so greatly with location, climate, and sun exposure that I can't do any reasonable math on solar water heating. While the heat is free, solar requires large holding tanks, because it works best when we are least likely to be taking hot showers (in the middle of a sunny day).
Lukewarm water is no treat. Neither is a cold shower.

Hot water is a luxury we all appreciate, enjoy and (most of us) happily pay for!

Tuesday, December 13, 2011

Best Price or Lowest Payment, Part II

This blog deals with issues in adult life that we are capable of handling with elementary math skills, as taught in our Excel Math curriculum.

Yesterday I touched on the difference between getting the lowest initial price (good if you pay cash or are buying for the short term) or the lowest monthly payment (fine if you are financing for the long term). My comments here apply most directly to houses and sometimes to automobiles.

How can paying a higher price save you money in the long run? It can ONLY IF you are able to get better financing and thus a lower total cost, OR if money is declining in value (inflation). It might be worth paying a bit more IF you can refinance later for less, or IF you don't plan to stay very long in your house.

Let's look at an example. We borrow $100,000 at 5% or 6% interest. Here's our cost:
  •  $100,000 at 5% interest over 30 years; 360 monthly payments of $537 = $193,300
  •  $100,000 at 6% interest over 30 years; 360 monthly payments of $600 = $216,000
Now imagine that we have to pay a slightly higher price but we can choose from 3 different financing alternatives. Here's the first option:
  •  $110,000 at 6% interest over 30 years; 360 monthly payments of $660 = $237,600
A total cost of $237,600 over the 30 years. Hmmm. What happens if I get a better interest rate?
  •  $110,000 at 5.5% interest over 30 years; 360 monthly payments of $625 = $225,000 
This loan at one-half percent less interest ends up costing $12,600 less.
  •  $110,000 at 5% interest over 30 years; 360 monthly payments of $590 = $212,400
This deal at one percent less interest saves me $25,200 compared to the first deal. 

What happens if I fail to get a good loan rate, and I have to pay higher interest, say 7%?
  •   $110,000 at 7% interest over 30 years; 360 monthly payments of $732 = $263,520
A change of 1 or 2% on the interest makes a huge difference!
  • monthly payments jump from $590 to $732 for $142 per month increase
  • overall cost jumps from $212k to $264k for $52k increase
Shop hard for good loan rates, but don't overpay for the house or the loan "points".

In the good old days we used to buy at the very limit of what we could thought we could afford, counting on inflation and time to raise our salaries and raise the value of the house. We gambled.

We bet on the interest rates coming down to reduce the cost of re-financing that hopefully would coincide with our greater income.

We gambled against interest rates going up, taxes increasing, losing a job, cars breaking down and/or divorce. All things that occur frequently, if not with certainty. Any one of these factors could make your deal collapse.

Even in "the good old days" people gambled and lost. But not too many people, and not too often.

Why? Among other reasons, Banks and Savings & Loan Associations kept us from getting too close to the edge. They made us put 20% or more down in cash (so we had a lot to lose), they talked to us about taxes and maintenance and other unanticipated expenses. They checked our pay stubs. They looked us sternly up and down, made us shuffle and stammer and invent good reasons for why we should get the loan. They didn't let us buy cars and houses at the same time.

In the past 10-15 years, it got easier. Loan applications were "streamlined" and less attention was paid to the property, or the buyer, or the ultimate holder of the loan. Everyone in the line was handling the paperwork with "tongs" or "gloves" rather than putting their fingerprints on it.

The loan was like a hot potato. Everyone expected to be in the house, or the loan, or the paperwork for just a short time, get paid well for their services, and move on without getting burned.

I call this processing. We were all gambling - everyone from top to bottom - both the naive and the sophisticated players.

We forgot that houses are also homes, and that we live in them as well as invest in them. Bankers forgot they were not just loan originators, but also in the business of holding loans over time, collecting the payments so they could pay interest to the folks dependent upon that income - making their house payments with the income. In my mind, this is holding.

Monday, December 12, 2011

Best Price or Lowest Payment?

What's more important to you when making a major purchase like a house or car - the best initial price or the lowest monthly payment?

You might object, saying, "Aren't they pretty much the same thing?", and the answer is "No, not really."

The initial price is driven by the price of other competitive properties or vehicles. It is somewhat related to the supply of money and its cost (interest rates). The monthly payment is related to the price, of course, but it is also driven by the supply of money and its cost (interest rates). 

I have younger friends who when they hear what we paid for our house, say "You bought your house at just the right time!" As far as the initial price goes, yes we did.

But when I say we got a 30-year, fixed-rate loan at around 10% interest, they say, "Whoa! You bought your house at the wrong time!" As far as the monthly payment (based on loan rates), we bought at the wrong time (or so it seems today).

The red lines on this chart indicate when I got (or re-financed) my 3 real estate loans. At least I got better at this as time went along!

35 years of average home loan interest rates
It's interesting (for me, in hindsight) to see this but doesn't help you in deciding if now is the time to buy.

A Buyer most likely to be interested in getting the lowest initial price is a
  • short-term owner
  • putting down lots of cash
  • buying an asset in a stable or declining market
A Buyer most likely to be interested in the lowest monthly payment is a
  • long-term owner
  • putting down little cash
  • buying an asset in a stable or improving market
Today we are at a point in our local market where the prices of homes are reasonable, but interest rates are low. That makes payments low. But the housing market is still chaotic; prices are likely to continue to decline; therefore short-term or cash buyers are not so inclined to jump in.

For more information on this subject, and how it particularly applies to housing in the Southern California area, read this nicely-written article. If this is totally confusing to you, ask your kids to help. We teach interest rates and percentages in our Excel Math curriculum.

Friday, December 9, 2011

Self-imposed Deflation

It seems most of us prefer to buy food that's on sale.

I noticed a story on BBC News today (my wife and I used to live in the UK, so we are attuned to the issues there). UK supermarkets are as brutally competitive as the stores here in Southern California, and the BBC story compared 4 big chain stores.  Asda, Morrisons, Sainsbury's and Tesco each have their own strategy to catch your attention, build loyalty, and get you to open your wallet:
  • Tesco gives you vouchers if their food is more expensive than Asda, when you prove it using their Price Check online.
  • Sainsbury's automatically gives you a voucher at checkout if any brand-name item would have been cheaper at Tesco or Asda. They do the research.
  • Asda supplies a voucher if their prices are not 10% cheaper than Tesco, Asda or Morrisons when you use their Price Guarantee software online or via mobile phone app.
  • Morrisons is taking the high ground and trying to focus on offering good food from local suppliers.

Can they all (at the same time) be the cheapest place to do your shopping?

"Yes," says Peter Lunn, an economist in Dublin, Ireland, "because we are drawn to discounts."

When you come out of a supermarket, you will have a cart full of discounted items. Your combination of items at that store will probably cost less than the same combination in a another supermarket.

However, if you'd gone to the competitor's store for their discounts, the resulting unique combination of stuff which you would assemble there will probably be cheaper than at the first store.

When researchers study a "standard cartful" of groceries (and unlike you and me, ignore all the special offers) the total prices vary by only about 3%. Asda (owned by Walmart) was cheapest.

But each of us buys different things, and many grocery items are bought due to a sale, regardless of we intend to buy when we go to a store. We only enjoy buying things that seem reasonable or cheap to us. We avoid the expensive items, or get them elsewhere.

Imagine the frustration of managers at a supermarket that drops its prices, pulls in more customers, and yet still falls behind in total sales to the competition. That happened to Tesco this year. One analyst said, “With more products available for less, the amount of cash taken has understandably dropped, despite Tesco having successfully attracted more shoppers. They call this strategy self-imposed deflation."

Why do they do it? To attract and (hopefully) retain customers for the long term.

That's exactly why we at Excel Math keep our curriculum prices low - after trying some tricky marketing ourselves,

Discount pricing formula courtesy of Dave Coverly at Speed Bump Cartoon!

we've found that special sales prices don't always work!

Thursday, December 8, 2011

Everybody loves a discount, but this is ridiculous!

I got my auto insurance renewal this week. Like mobile phone contracts, or buying a cable entertainment package, understanding the details demands that we use mathematics. Will the simple operations we learned in elementary school be adequate ... or not?

At least I know how to construct a 3-column, 10-row table. That's useful:

 Insurance Category Car 1 Car 2
 Liability 112 157
 Medical 5 8
 Uninsured Motorist 24 26
 Comprehensive Damage 14 31
 Collision Damage $500 ded. 63 94
 Towing & Rental Car 4 4
 Collision Ded. Waiver 11 11
 Miles Driven Annually750012000

Despite my boring blue cars (Honda, Volvo) I still have a total premium of 564 dollars and no cents.

After looking at the total price, I dove into the 3 pages of fine print regarding discounts on my policy premium. Here are the possible discounts off the "list price" for my policy:
  • 20% for a Good Driver discount (no more than one moving violation "point", no more than one accident) Applied
  • 25% for being At-Fault Accident Free for five years (5% for each year, up to 25%) Applied
  • 22.5% for Cross Sold Credit (this does not mean I am angry, but I have Homeowner's insurance with this same company) Applied
  • 12% for Continual Renewal for six years without lapsing (2% for each year, up to 12%) Applied
  • 30% for Physical Damage Deductible of $500 rather than $100 Applied
  • 15% for Multi-car Discount when two or more cars owned by the same person are insured Applied
  • 5% off Medical premium for Passive Restraints (airbags) Applied
  • 15% off Comprehensive for Passive Anti-theft Devices (factory car alarm/security system) Applied
  • 10% off for Educator Discount (my wife is employed by a school district) Applied
  • 5% off for Mature Driver Improvement Certificate (I'm insulted by the very thought of this!) Not applicable
  • 1.5% discount for Good Student Grade Point Average (B or above) Not applicable
  • Discount for Distant Student living more than 100 miles from home (without a car) Not applicable
  • Discount for low annual mileage driven Applied
Now I suspect you are beginning to think the same way that I was thinking. Are all these discounts additive?


If you add them all up, you get:   20 + 25 + 22.5 + 12 + 30 + 15 + 5 + 15 + 10 = 154.5%

Take 154.5% times $564 and that means they owe me $871. Right?

Alas, no. That's not how it works. Each of the discounts can apply to only one type of coverage, and the discounts have already been applied to the prices shown above. Elementary math is really no use at all in deciphering exactly how the discounts decreased my price.


As Bob in our office is fond of saying, "It is difficult to say at this time."  Comparison shopping with other companies is your best bet, or you might ask your insurance agent (if you have one) to go through the details with you.


I was thinking about getting a more interesting car for driving in my spare time. What do you think of the engine in this red Pantera? Do you suppose my insurance company will give me a discount, or no discount?

Wednesday, December 7, 2011

The Cold Hard Facts

Yesterday I talked about the saturation of the US market, by television and then by personal computer. The statistics were fairly simple, appropriate for we who have only studied math in elementary or secondary school. This isn't a blog with "rocket science" analysis, just a place to investigate interesting subjects and a bit of the math behind them.

Today we take a look at the refrigerator - an appliance so common in America today that it's nearly impossible to imagine living without one. Can math help us understand the refrigerator?

My wife's great-grandfather Morris was a mining engineer in the northern Mexico desert 100 years ago. He loved to surprise visitors from the United States by offering them ice cream in the middle of summer. How  did he keep it frozen? In an icebox - a large cave filled (during the winter) with blocks of ice and straw. Go here to see his book.

My wife's great-uncle John sold refrigerators in the 1930's. He was a top executive with Frigidaire, making his personal fortune selling "boxes full of cold air." These appliances were greatly appreciated in Los Angeles, a fast-growing, hot and dusty corner of the US. Refrigerators were much better than ice boxes. Here's Frigidaire's one-millionth unit.

When I was growing up, my uncle Dick was president of LA Cold Storage, a company with the largest refrigerated warehouse on the West Coast. Today they chill 5.7 million cubic feet! I guess refrigeration runs in our family, although personally I am not that fond of the cold - I really suffered when visiting this Swedish Ice Hotel!

Visiting the JukkasjÀrvi ice hotel
Sorry for the detour - now back to our topic of the day.

In the United States, England and France, ownership of refrigerators after WWII rose from under 10% (late Forties, early Fifties) to over 85% (mid-Seventies). My guess it it's now over 100%, with some households having more than one.

Since US households all have refrigerators already, manufacturers try to encourage us to replace our old ones BEFORE they wear out, thus reducing electricity consumption. This chart shows how energy is being saved (red line) as we replace (blue line) our smaller (17 cubic foot) refrigerators with cheaper (green line), larger (21 cubic foot) refrigerators. The data came from an energy blog/newsletter. [click the chart to enlarge it]

To see how refrigerators are adopted by households, we need to look at other countries coming up the curve of household automation. Here's a chart from a study of appliances in developing countries. I found the study very interesting even though it had a fair amount of complex statistical analysis.

The data implies this sequence of major purchases after electricity reaches a household: television, followed by refrigerator, washing machine and air conditioner. This information supports what we Americans know from experience:

There's a close link between watching television and needing to drink a cold beverage, while the clothes are in the washer. Hence the development of a recliner chair with built-in drinks cooler.

La-Z-Boy "Chill" recliner with cooler

Tuesday, December 6, 2011

How Much Time Do You Spend Staring at a Screen?

This blog describes and demonstrates how adults can use the math they learned as kids. I write it because my "day job" is creating Excel Math curriculum materials.

Today I want to talk about television and computers.

During the decade I was born (the 1950's), the population of the United States grew from 150-180 million. The number of households increased from 44-53 million. There were about 3.4 people in a household back then. Only about 1 out of 11 households owned one or more televisions. Our family didn't have one for much of that decade. We were late adopters. Consequently, I don't know much about the famous TV shows of the Fifties.

In 2010 the US population was 309 million people, divided into 116 million households. That's an average of 2.66 people per household, with an average of 2.8 televisions per household!

USA Households Total
 Households w/Television
 Households w/Computers
 195044    4  0
 196053  46 0
 197063  60 0
 198081  80 5
 199093  88 14
 2000105 102 55
 2010116 116 94

For people born in the 1980's, there was a similar revolution. The decade started with only a very small percentage of people with a personal computer. Not wanting to be "late" again, I got my first, a Radio Shack TRS-80 Model 100, in 1979.

Now more than 80% of all US households have at least one personal computer, and virtually all of them have internet access.

(Most households that have a current computer also have a few obsolete computers too. How many do you have sitting in the garage or basement?)

In my 2-person household we have one television. We have 7 computers - an aluminum MacBook, a G5 tower, a silver iMac, 3 iPhones and one Windows Laptop.

In my 1-person office, I have no televisions. Today I have 4 computers - a titanium MacBook, a G4 tower, a white iMac and a brand-new silver iMac. Plus my phone and an iPod.

But hey, I'm in the publishing business. This is what I make math books with. What's your excuse justification for your computer(s) and television(s)?

Monday, December 5, 2011

Price of a Brownie vs Value of an Avocado

Oscar Wilde said a cynic knows "the price of everything and the value of nothing."

Today I am going to examine a similar proposition - the price of a brownie versus the value of an avocado. This is the sort of useful analysis that an adult can engage in, using the math skills learned long ago in elementary school.

A couple hours ago, Dave and I went up to the food court to get 5 of their 3-tacos specials for folks in our office. When we received the food, the lady at the counter said:

"We ran out of avocado on the last 2 tacos, so I gave you a brownie instead."

It was very nice of her, but raised the question of equivalence in my mind. What I mean is:

How is the value of a brownie related to the value of avocado in a taco?

The brownie was $1.75 as a dessert item, and the avocado was rolled into the price of the taco special. But let's estimate it as a $1.00 upgrade to the normal taco lunch. So we got $1.75 worth of brownie to compensate us for a shortage of $1.00 worth of avocado.

Now let's take a look at the numbers - can we determine who's getting the best end of this deal?

 150 grams / 6 ounces Avocado BrownieRatio
 Calories 150 675 1:4.5
 Carbohydrates 13g   72g 1:6
 Cholesterol   0g 108g infinity
 Protein   3g     6g 1:2
 Total Fat 22g   42g 1:2

If you are hungry for a dessert, the brownie definitely wins in this comparison test. It's got more of almost everything, including subjective things like intensity of flavor. Why not? It contains lots of chocolate, sugar and fats. Everything except "health."

On the other hand, an avocado is filled with exotic things, like:
  • phytosterols, including beta-sitosterol, stigmasterol, and campesterol
  • carotenoid antioxidants, including lutein, neoxanthin, neochrome, chrysanthemaxanthin, beta-cryptoxanthin, zeaxanthin, violaxanthin , beta-carotene and alpha-carotene
  • non-carotenoid antioxidants, including epicatechin and epigallocatechin 3-0-gallate, vitamins C and E, and manganese, selenium, and zinc
  • omega-3 fatty acids (alpha-linolenic acid) and oleic acid
  • polyhydroxylated fatty alcohols (PSA)
  • vitamin K, vitamin B5, vitamin B6 and potassium
  • dietary fiber

Which would you chose: Brownie or Avocado?

Which would you say has the most value?

Did you base your decision on reason (and math) or emotion?

Friday, December 2, 2011

Counting Up to 180, Part III

Today I am showing the final list of countries that have visited our Excel Math Blog.
On these pages we talk about the math(s) we learned in elementary school and show how those skills can be utilized by grown up people.

This list is in descending order, with the most visitors at the top. If you have come to the Excel Math blog from one of these countries, were you looking for assistance in helping your kids learn math, or were you looking for Microsoft Excel software?

Country   Unique Visitors Last New Visitor

November 29, 2011
Palestinian Territory11

November 28, 2011

November 12, 2011

October 13, 2011
Saint Vincent and the Grenadines10

November 28, 2011

September 14, 2011
Netherlands Antilles9

November 24, 2011

November 20, 2011

September 22, 2011

November 29, 2011

November 21, 2011

November 8, 2011

November 7, 2011

October 31, 2011

November 30, 2011
US Virgin Islands7

November 29, 2011

November 17, 2011
Saint Kitts and Nevis7

November 10, 2011

October 20, 2011
Isle of Man7

October 11, 2011

September 13, 2011

September 9, 2011

November 17, 2011

November 7, 2011
Antigua and Barbuda6

November 6, 2011

October 24, 2011
El Salvador6

October 18, 2011
Saint Lucia5

November 22, 2011

November 18, 2011

November 18, 2011

September 28, 2011

August 30, 2011
Northern Mariana Islands5

May 23, 2011

May 18, 2011

November 20, 2011

October 26, 2011

June 7, 2011

November 21, 2011

November 17, 2011
Turks and Caicos Islands3

November 1, 2011

October 26, 2011

October 20, 2011
Papua New Guinea3

October 14, 2011

October 5, 2011

September 26, 2011

July 7, 2011

November 16, 2011

November 4, 2011

November 3, 2011

October 6, 2011
Faroe Islands2

September 20, 2011

August 17, 2011

July 16, 2011

April 15, 2011
Cote D'Ivoire2

March 23, 2011

March 12, 2011
New Caledonia2

February 25, 2011

January 20, 2011
British Virgin Islands2

December 2, 2010

November 28, 2011

November 27, 2011

November 21, 2011

November 16, 2011
Cayman Islands1

October 10, 2011
San Marino1

October 8, 2011

September 30, 2011

August 21, 2011

August 12, 2011

July 12, 2011
Aland Islands1

July 2, 2011

May 5, 2011

March 21, 2011

February 28, 2011

February 3, 2011
American Samoa1

October 24, 2010

October 21, 2010

October 15, 2010

June 19, 2010
Burkina Faso1

February 26, 2010

February 3, 2010

Unknown - European Union66

November 30, 2011

Unknown - Satellite Provider24

November 30, 2011

Unknown - Anonymous22

October 11, 2011

Unknown - Asia/Pacific                  12

November 22, 2011