Additional Math Pages & Resources

Friday, February 26, 2010

AMAZON, The River

Water seems to be on my mind lately, so today we'll see if we can use some metric math about rivers. Not just any rivers.

The biggest river in the world. And a little local river.

Before the "A" word was associated with an Internet retailer, or fierce women warriors, the Amazon was flowing across South America to the Atlantic Ocean.

It drains an area of 7 million square kilometers.  It has more than 1,100 tributaries flowing into it. During flood season it's 100-200 kilometers wide! There are ZERO bridges across the Amazon. That is partly because of its size, but also because it flows through the jungle and there aren't roads that need to cross it.

Here's a Wikipedia map showing the main tributaries of the Amazon River. Approximately 15-20% of all the fresh water in the world flows through the Amazon and into the Atlantic!

An interesting unit of river size is called Strahler number. This scheme is named after Arthur Strahler, a professor at Columbia University. About 60 years ago he came up with a strategy to describe how many branches and tributaries make up any hierachical system (which includes rivers).

The Amazon is a Strahler Order 12 river, the most complex drainage network in the world.

In contrast, my local waterway, the San Diego River, is 83 km long, with 625 km of tributaries and drains 1140 sq kilometers.  At its widest point it's only about 200 meters wide.

Most of the time it looks like this.

The San Diego River is an Order 5 river. Even though it's only a fraction of the size of the Amazon, it's a nice place to spend some time.

Click here to see all about it.   Or watch a sunset there.

Thursday, February 25, 2010

Patching Potholes, Part 2: Plan

Yesterday we discussed patching problematic potholes. I did some research and I have a plan to propose today. A business plan. That means we need some math. Just elementary school math, nothing complicated.

My plan involves a JetPatcher machine - a faster way to patch potholes. Go on and take a look at their site; I'll wait here, checking out this used truck I found.

Maybe I could buy one and rent myself out to the city and make a ton of money... Shall we do a preliminary business plan here?

My used JetPatcher will cost $70,000 dollars compared to $200,000 for a new one. Let's say I keep it one year and sell it for $50,000 at the end. My equipment cost is $20,000.

I'm going to estimate (guess) at the other expenses.

We assume $20,000 for the machine.
We estimate $50 per day in fuel, insurance, repairs, etc. times 250 working days (5 x 50 weeks)
(We might also have to work a few weekends if there are too many rainy days.)

Fixed Expense Total = $20,000 + ($50 x 250) = $32,500

We estimate $5.00 per patch in materials (sand, asphalt and oil).
We estimate 20 minutes per repair including driving from one to another = 3 potholes per hour
We estimate paying $15 per hour per person for 2 people = $30/hr or $10 pph [per pot hole]

Variable Expense Total = $5.00 materials + $10.00 labor = $15.00 pph

(5 days x 50 weeks) x (3 per hour x 8 hrs/day) = (250 x 24) = 6000 potholes per year maximum

What if I could charge the city $20.00 per pothole?  If I could do this, we have:

STEP 1. $20.00 - $15.00 = $5.00 pph that I can put towards my fixed expenses.

(REALITY CHECK $32,500 ÷ $5 = 6500 potholes to break even = 26 per day to break even.)

STEP 2. $20.00 per pothole x 6000 repairs = $120,000 income - $32,500 fixed expense = $87,500

STEP 3. ($87,500 - (6000 x $15.00)) = ($87,500 - $90,000) = $-2,500. That's a small loss.

We're not going to make any money at this $20.00 price  :-(

What if I could sell the city on a $25 per pothole price? We'd work fast and make the streets look good. If I could get this price, now we have:

STEP 1.  $25.00 -$15.00 = $10.00 pph towards the fixed expenses.

(REALITY CHECK $32,500 ÷ $10.00 = 3250 potholes to break even = only 13 per day to break even.)

STEP 2. $25 per pothole x 6000 repairs = $150,000 income - $32,500 fixed expense = $117,500

STEP 3. ($117,500 - (6000 x 15.00)) = ($117,500 - $90,000) = +$27,500 That's a profit.

We're making a good profit at this $25.00 price  :-) so now we will have to pay a few taxes  :-(

PS - A newspaper article said Tijuana's pothole patching pace is 200 per day! Maybe I could hire THEM to fix our potholes! Or is something wrong with our math? Or are their potholes smaller, closer together and easier to fix?

Wednesday, February 24, 2010

Patching Potholes, Part 1: Problem

A month or so ago we talked about roads.  Today we talk about a lack of roads - or small gaps where roads ought to be.

We could call them missing-pieces-of-roads. But pothole is the word we normally use.

Potholes are problems. Nobody likes them. They appear everywhere, even in NASCAR's Daytona 500. The race was stopped twice last week because of potholes!

Do potholes require elementary math? YES

Two years ago my city said it was repairing 30,000 potholes a year on its 2735 miles of road.

How many potholes per mile is that? Just about 11.

Now they say the numbers are much higher. In 2009 they filled 53,046 potholes on 2800 miles of road.

How many potholes per mile is that? About 18.9. 

At a cost ranging from $15-30 pph (per pothole), that adds up. Let's do some math.

How much do the potholes cost us to fix?  $22.50 x 53,046 = $1,193,535,  or $1.2 million.

Although the city says they have 64 people dedicated to fixing potholes, they can't keep up.

How many potholes per person did they fix last year? 53,046 ÷ 64 = 829 per person

That sounds like a lot, doesn't it?

How many potholes per day is that?  It's 829 ÷ ((52 weeks x 5) - (10 days vacation + 10 holidays)) =

829 ÷ (260-20) = 829 ÷ 240 = 3.45 potholes per day per person, or 7 per day per 2-person team.

The streets division has created a web application so you can locate a pothole on a map and send its coordinates right to the patching crews. That's faster than relying on the crews to find it visually.

I must say the repairs don't seem to be evenly distributed, but then I live in an older part of town. The city says it divides the repair funds evenly among the 8 major districts of the city. If it's true that newer roads are in better condition, then newer neighborhoods can spend more to fix fewer potholes.

It's not the bad weather ruining our roads, because we have great weather. So what is it? Do we drive too hard? Do our cars and trucks weigh too much? Is the hot sun ruining the asphalt? At least we have fewer than Tijuana, which estimates its pothole population at 800,000!!

Tuesday, February 23, 2010

Ship Shape Limbo

Limbo is a game where people bend over backwards to shimmy under a moveable stick. See the photo to get an idea about the game.

You win if you make it underneath without touching the stick. You lose if your chin or hands touch the stick, if you knock it off the holder, or if you fall down or touch the ground - it's much harder than it looks!

Where does the math come in? Well, a similar activity is played out every time a ship passes under a bridge. In most cases, there's plenty of room above the ship. But not always. And the pilot MUST be good at math.

The distance a ship sticks up out of the water is called the air draft. The clear air below a bridge is called the air gap.

The air gap minus the air draft has to be a positive number, or the ship will hit the bridge. Here's a drawing from the NOAA website showing the Air Gap under the Bayonne Bridge in New York.

This bridge has an air gap of 154.1 feet. If the ship's air draft is say 153 feet, it should clear, right? NOT RIGHT!  Here is the measurement a few hours later...
Things can change quickly in the ship-limbo game!
  • Ships float higher in salt water, and lower when in fresh water
  • Increasing a ship's load makes it float lower in the water, until it sinks or scrapes the bottom
  • Tides cause the water levels to change constantly
  • Winds cause the water to be higher or lower on one side of the canal
  • Waves and wakes cause ships to move vertically 
  • It can be difficult to know the exact air draft of a ship
  • If ships aren't loaded correctly, the front or back might be slightly higher than the middle
  • Traffic on the bridge causes the bridge to sag slightly
  • Winds cause bridges to move from side to side (up to 27 feet for the Golden Gate bridge!)
Our government is very concerned that ships don't hit bridges. Here's the Air Gap graph for today. Notice that it changes by more than 4 feet in less than 4 hours and might now be under 150 feet.

This is such important information that the numbers are calculated every 6 minutes and posted on the web. Here's a text summary for right now:
Observations for New York/New Jersey PORTS 2010-02-23 15:01:02 EST
High water conditions exist for Kings Point, The Battery, and Sandy Hook.
Water levels are rising above predictions by up to 2.1 feet at Kings Point, Sandy Hook, and Battery.
Currents at The Narrows are flooding at 1.1 knots.
Reported winds are generally from the northeast, between 5 and 16 knots with gusts to 19 knots.
Reported air temperatures are in the mid to high 30s °F, and reported water temperatures range from the high 30s to the low 40s °F.
Barometric pressure is falling, with readings between 1007.4 and 1008.8 mb.
Air gap is 150.5 feet and decreasing at Bayonne Bridge Air Gap, and 227 feet and decreasing at Verrazano-Narrows Air Gap
    Miscalculating the clearance can ruin a bridge, sink a ship, and/or result in deaths. Here's a picture of a freighter that failed to clear a bridge. Four lives were lost and a ship and bridge ruined.

    A bridge over the Chesapeake was knocked down by the freighter you see here.

    I found a story about the Narrows Bridge in Vancouver that was struck by a run-away barge. The barge didn't ruin the bridge at first, it was simply stuck in place. But as the tide lifted the barge, it also lifted the bridge span off its supports, and the span fell into the water - thus ruining the bridge and the waterway access at the same time!

    Monday, February 22, 2010

    Do you prefer Capesize or Capsize?

    I saw an article today that used the word Capesize. It looked like capsize, which is not a good thing when we are talking about ships!

    The author means ships so large they don't fit any locks - to traverse oceans they must go around the Cape of Good Hope (South Africa), or Cape Horn (South America). Most of these are freighters or oil tankers, though some new cruise ships are very large too. Here's the Wikipedia photo of the largest cruise ship, the Oasis of the Seas.

    Ships can be described by the size of the locks or canals that they can fit through, or the ports they fit. For example, HandyMax means the biggest cargo ship that fits almost any port.

    Besides being the right length, width, draft (amount below water) and air draft (height above water), ships must be trimmed so they sit completely level in the water. Their captains must adjust for the fact that most locks are filled with some fresh water which is less buoyant than salt water. And if they are going under a bridge, steaming full-speed-ahead actually lowers the ship by a few percent!

     Class Oasis PanamaxPanamax II Suezmax Capesize
    Length 1187 ft (360m) 965 ft (294m) 1,200 ft (366m)1,400 ft (427m) 1400+ ft
    Width 208 ft (64m) 107 ft (32.6m) 160 ft (49m)164 ft (55m) 200+ ft
    Draft 30 ft (9m)39.5 ft (12m) 50 ft (15m)62 ft (18.5m) 50+ ft
    Air Draft 213 ft (65m) 190 ft (58m) 201 ft (61.3m)223 ft (68m) 200+ ft
    Displacement 100,000 tons 80,000 tons 120,000 tons?150,000 tons ≥200,000 tons

    The largest (longest) ship ever built was known as Knock Nevis. She was 1504 ft (485m) long,  226 ft (69m) wide, and had a draft of 81 ft (25m). That made her too big even to go through the English Channel! Knock Nevis was built in 1979 and after living a short, rough life, was beached last month. She's being dismantled and scrapped right now. 

    Friday, February 19, 2010

    Why elementary math?

    I work for Excel Math. We create math curriculum. Writing this blog is part of my job. In it we explore how people (grown-ups or youths) can use math they learned in elementary school

    Wikipedia states: Mathematics is the study of quantity, structure, space, and change.

    A Maths Dictionary for Kids says: Mathematics is the study of numbers, quantities, shapes and space, using mathematical processes, rules and symbols.

    I guess I put it a bit differently.

    Excel Math Blog #2 said, Math is a language of counting, measurement, shapes and calculation. A language with precise definitions and specialized terms.

    I think Math involves first learning, then using a variety of skills and procedures for thinking about and manipulating numbers and shapes.

    Math is composed of a wide range of things that are lumped together in school for convenience (or because Reading, History or PE teachers won't teach them)!

    In Excel Math these math areas include (arranged roughly in order from first-taught to last-taught):
    1. Place Value and Counting
      • Number sequences, missing numbers and ordering numbers
      • Number lines
      • Mathematical symbols
      • Number patterns
    2. Addition of Whole Numbers
    3. Subtraction of Whole Numbers
    4. Multiplication of Whole Numbers
    5. Division of Whole Numbers
    6. Fractions
      • Fractional parts of a set
      • Converting and simplifying
    7. Money
      • Units of currency
      • Prices and unit pricing
      • Making change
    8. Time
      • Clock
      • Calendar
    9. Odd and Even Numbers
    10. Geometry
      • Two-dimensional space and geometric figures
      • Three-dimensional space and figures
      • Flips, Slides, Turns, Scaling
      • Area, volume, perimeter, surface area
      • Angles and degrees
      • Calculating pi
    11. Problem Solving
      • Word problems
      • Reasoning
      • Giving directions
      • Map reading
    12. Measurements
      • Units and tools of measure
      • Conversions and equivalents
      • Comparisons
      • Abbreviations
    13. Estimating
    14. Probability
    15. Graphing, Plotting and Charting
    16. Pre-Algebra
      • positive and negative numbers
      • exponents
      • square roots
      • factors and permutations
      • Venn diagrams
    17. Decimal Numbers
    18. Percents
    19. Statistics
    Sheesh! That's a long list. We have a huge spreadsheet (prints 4 feet by 12 feet) to track what's taught when and where.

    That's what kids are doing throughout grades K-6, BEFORE they get Algebra, Trigonometry, etc.- learning CONCEPTS, TECHNIQUES, WAYS of THINKING, ACTIVITIES.

    For most of us, math is not a life-long process of "proofs" about conceptual possibilities that might be true or false.

    Instead, MATH is skills we use every day. Most do not involve intensive calculation, so fingers, calculators and spreadsheets are useful at times but insignificant in the larger picture.

    Math requires precise thinking and a healthy skepticism about the numbers thrown at us by advertisers, government, bankers and friends. It's the ability to find our own answers to questions. Like,

    Why is my budget off? Is is the numbers, is it the spreadsheet, or is my analysis faulty?

    Excel Math assumes that people will need to learn to think mathematically, perform calculations (using any tools at hand) and trust their own judgement. These skills are a necessary math foundation for any self-confident citizen.

    I search for interesting topics and do math on them in this blog - to entertain myself as well as do useful things like figure out a utility bill, calculate fuel economy, etc.

    Thanks for coming along for the ride.

    Thursday, February 18, 2010

    Eggzactly, Part 2

    Yesterday I talked about units of measure for eggs. Specifically, sizing which is based on the weight of 12 eggs. An individual egg can be larger or smaller as long as the average in the carton is equal to USDA standards.

    What I haven't covered yet are quality ratings. They are:
    • Dirty
    • Cracked
    • B
    • A
    • AA 
    How do eggs get dirty or cracked? Well, chickens walk around, get dirt or poop on their feet, and that gets on the eggs. Hens may lay them while standing up so the eggs drop. If it's a hard surface, the eggs can crack. Washing doesn't necessarily remove all the flaws.

    Dirty and Cracked mean what they say, and eggs graded this way (after being washed!) are only for sale to factories or commercial egg processors. The assumption is these commercial egg users will process them quickly and in a controlled and sanitary manner. Besides that, the eggs are most likely to be used without anyone seeing the shell anyway (liquid, powdered, separated, etc.).

    I suspect most of us wouldn't want to buy anything less than A or AA anyway. When did you last see a B egg?

    But how are those ratings determined? It is based on the albumen (white) being measured using the Haugh Scale. That's our indication of freshness. The process was developed by Raymond Haugh, in 1937.

    An egg is stored overnight at a specific temperature (40° F or 4° C), weighed in a room between 45-65° F, then broken out onto a clean flat surface. The depth of the white is measured with a precision micrometer, halfway between the yolk and the outer edge of the white. If you want to do the calculations, here's the formula, though I have to warn you it is beyond elementary school mathematics.

    HU = 100 log(h-.01 x 5.6745 (30w.37 - 100) + 1.9)

    or the simplified version 

    HU = 100 log(h - 1.7w.37 +7.57)

    The variables in the formula are h = height of white in mm, and w = weight of the egg in grams.

    I read a fascinating (yawn) paper that recommended further simplification - just measure the height of the albumen and leave it at that. That approach had an almost perfect correlation to Haugh scores, and takes a fraction of the time to achieve. Sounds good to me. Sometimes less math is better.

    Here is the criteria for grading:
    • B eggs have an HU ≤ 60
    • A eggs have an HU = 60-72
    • AA eggs have an HU ≥ 72
    This simply means the freshest eggs have the tallest albumens. Over time, the whites collapse and are lower (closer to the table top).

    We had to have an omelette for dinner last night because of all the eggs I broke onto plates, taking photos and comparing...

    PS - I also found a new site today - It discussed units for measuring almost anything!


    Wednesday, February 17, 2010

    Eggzactly what am I buying?

    Math is focused on quantity. How much? How large? How heavy? Specific units of measure enable us to define how much we are getting when we buy something. Then we can calculate cost per unit.

    We buy fuel in gallons or liters. We buy paper by the ream (500 sheets) or the case. We buy produce in ounces, pounds, kilos or individual items. But when we buy eggs ... well what are we getting?

    I suppose a carton of one dozen, large, grade A eggs is the normal unit of purchase. But it gets complicated from there!

    The US Department of Agriculture monitors what happens in the poultry business in our country. Some of the standards they've set are voluntary. Egg producers have to pay extra to have the USDA seal on their packages. Other producers can save some money by complying only with state regulations, not Federal guidelines. The USDA has established standards for eggs that include:

    • Grade: from AA, A, B then to (industrial use only) cracked or dirty
    • Size: from PeeWee, Small, Medium, Large, Extra-Large to Jumbo
    • Source: Caged, Free-Range
    • Feed: Vegetarian, Organic, Omega-3, etc.
    I was most interested in learning about the sizes of eggs. Why? Well because we bought some from an egg ranch last week and the eggs looked enormous. See for yourself:

    You can see four grade AA Large eggs from the store at the left side of the carton. The four Egg Ranch Jumbo eggs at the right. Here's another look. The size difference is very apparent:

    Now you might be wondering HOW BIG are those JUMBO eggs?

    It turns out that the size is not really graded by the USDA regulations. It's the weight of a dozen eggs combined that determine how they are "sized". The scale looks like this:

    Jumbo = 30 oz         (30/12=2.5)
    Ex-Large = 27 oz     (27/12=2.25)
    Large = 24 oz          (24/12=2.0)
    Medium = 21 oz       (21/12=1.75)
    Small = 18 oz           (18/12=1.5)
    PeeWee = 15 oz       (15/12=1.25)

    (I did an average in the parentheses, but it's not required that every egg match that size)

    I read that there is a high likelihood of getting double-yolks in Jumbo eggs. In fact, the BBC news had an article talking about probabilities, saying it was millions to one against getting multiple double-yolk eggs in a dozen. That was just doing the simple mathematics.

    The Daily Mail reports that the chances of finding a double-yolked egg is one tenth of 1%. So imagine how Fiona Exon felt when she started to make scrambled eggs for Sunday's breakfast. The first one had two yolks. The second had two yolks. And as the Mail says, she called her husband when the third also turned out to have a double yolk. When all six in the box were the same, she admits to being astonished. Experts say the chances of it happening are one trillion to one.

    A reader went one step further and called a local egg ranch. The manager said, Oh yes, when we grade the eggs we put all the double-yolk ones into cartons for ourselves. Once in awhile we have too many and we sell those packages.

    There's no probability in sorting eggs - they don't come out of the chickens and drop into boxes by random chance - every egg is graded, sized and sorted. The speculations of the amateur mathematicians were completely  incorrect because they knew nothing about the egg business.

    Tuesday, February 16, 2010

    Marking the days

    Calendars are part of math.
    They help us count and organize what happens on certain days.

    In today's blog we ask:
    How can we express calendar events with a watch or clock?
    (Day of the week, date in the month, month of the year and phase of the moon)

    It seems there are two very popular displays: hands on dials or words in windows.

    Here's an old and very expensive Patek watch that uses the words in windows approach.
    There's a number for the date, then words for day and month (and moon phase at the top).

     This IWC watch shows the calendar using yellow hands on dials
    It has day of the week, month, and date on 3 sub-dials.
    The moon is at top center, and the year in a small window at bottom left.

    Here's another IWC watch, but it uses numbers in windows for month and day.
    A tiny L indicator in the bottom window tells us if it's a leap year. 
    For Americans, this layout isn't ideal, as we tend to put the month first (08 / 25 / 2010) 
    rather than having the day of the month first like most of the world ( 25 / 08 / 2010 ).
    An advantage is the watchmaker doesn't need to print dials in different languages.

    This gold Girard-Perregaux watch clearly expresses the day, month, date and moon phase.
    Windows and words display the day and month, while a hand on dial is used for the date.


    Here's another watch with windows and words for the day and month. 
    In this case the date is around the outer edge of the dial so the numbers can be larger.

    This one uses the opposite approach, a window for the date (at top),
    and at the bottom, dials display the day on the left, and month at the right.
    The center dial indicates a second time zone on a 24-hour scale.
      Here's an unusual piece just shown this year, 
    with fully-spelled-out month and day, on rolling word in window indicators.

    The following clock uses neither words in windows or hands on dials
    Little black and red tabs move in the background, highlighting a specific day and date.
    That way all the motion is created with a rotary clock motor, no flipping, rolling words here!
    This is a creative design and used by some watches too.  We'll call it back-lit indication, okay?
    This Seiko clock cost about $75 but has been discontinued.

    Now that we have discovered this 3rd method of indication,
    we're in the position to say Just because you can do it doesn't make it better!
    The photos here show two new and different JLC calendar watches.
    They show the same calendar features, but which is easier to read?

    Hands on dials are better than back-lit indicators, in my estimation!
    Granted, with the watch on the right you get to see the movement too, but it's very hard to read.

    In case this blog has you wanting a clock calendar on your desk,
    here is is a new, easy-to-read, hands on dials,

    Chelsea clock you can buy today for $350.

    Or do you prefer the words in windows calendar display?

    It isn't a new idea. Here's an Ithaca
    clock calendar mechanism that was patented 150 years ago!

     This is how it looks in a complete Ithaca clock.
    We see words in windows for day and month.
    The date is shown with hand on dial around the the outside of the dial.
    It's almost exactly like the older JLC watch shown earlier.
    The time is on a separate upper section.
    This clock is 145 years old and in beautiful shape.

     Finally, here's a Dussalt clock with the months around the outside of the dial.
    This is a real antique work of art. 

    I'll keep looking, and I will let you know if I find any other method besides
    words in windows, hands on dials, and back-lit indicators.

    Friday, February 12, 2010

    Water water everywhere, Part 3

    Water is something we each use in small quantities. But because there are lots of us and we use little bits of water all the time, we have to store it in huge quantities.

    When people in the USA talk about water, we use ounces, cups and gallons as our units of measure. In most other places we'd use liters. But what good are cups and liters when it's raining and you want to figure out how much water is going into the reservoirs? Do you say

    We just had 52 gazillion cups of water added to our lake? or 800 trillion windshield-wiper sweeps of rainfall?

    No, water keepers use larger units. One common water storage unit around here is the acre-foot. That's the amount of water needed to cover an acre (43,560 sq ft) one foot deep in water.

    For people who like Olde English units, that's 66 feet (one chain) by 660 feet (one furlong) by a foot deep.

    In metric, it's an acre-foot = 1,233.5 m(cubic meters)

    That's a big number. You might ask How many gallons?

    One Acre Foot = 325,851 gallons
    One Acre Foot = 43,560 cubic feet or 435.6 HCF

    The US water folks used to say an average family uses one acre-foot of water in a year. However in the Western US, where conservation is much more important (mostly desert!) we tend to use about a quarter of that, or one-fourth of an acre-foot.

    Since our family's average consumption per month is 9 HCF, what percentage of an acre-foot do we consume in a year?

    9 x 12 = 108 ÷ 435.6 = .248 acre-foot per year. That's about the average.

    A few months back, I did a blog about our yard, called A Square Root.  We calculated that our property covers about 1/3 of an acre. So here's my question:

    How long would it take to get a foot of rainfall on our property, and how long would it take us to use it up?

    The San Diego County Water Authority says average rainfall in our area = 9.9 inches. Let's just say 10 inches.

    If we caught all the rain that fell in 1.2 years, we had a place to save it, and none evaporated or got to water the plants, we would have a foot of rain times 1/3 of an acre, or .33 acre-foot.

    Our indoors consumption is .248 x 1.2 years = .297 or about .3 acre-foot. In theory that means we might be able to be self-sufficient when it comes to water, but I wouldn't want to try it!

    Our plants would certainly resent it. Even though we mostly have California natives so we don't irrigate, they need every drop that falls. Thank goodness we have the water authority to catch, store, clean and distribute our drinking water.

    (Although it was a bit of a mess when they recently decided to replace the old pipes under our street!)

    Thursday, February 11, 2010

    Water water everywhere, Part 2

    More water today.
    I forgot to include figuring out the bill in yesterday's post on water. This is one good reason for studying math in school - utility bills are always complicated - it's not like buying a gallon of milk!
    Water isn't free but in the United States it isn't very expensive either.  Most cities monitor the amount consumed, but a few have fixed prices per month and don't charge based on how much you use.
    Here are the prices for the water that appear on my water bill:

    $35.42 Basic fee for having water service at a single-family house
    $ 3.29 per Hundred Cubic Feet of water, from 0-14 14 HCF
    $ 3.57 per HCF for the next 14 HCF used (from 14-28 HCF)
    $ 4.01 per HCF for all water used (over 28 HCF)

    My average bill for the past few years is under 10 HCF. Last month it showed 29 HCF.

    That's enough of the facts, now to the math questions!

    How much did it cost for the water?

    $35.42 + (14 x 3.29) + (14 x 3.57) + (1 x 4.01) = $35.42 + 46.06 + 49.98 + 4.01 = $135.47

     How much is that per gallon?

     29 x 748 = 21,692 gallons so $135.47 ÷ 21692 = $.0062 or about 2/3 of 1¢ per gallon.

    We normally consume 9 HCF per 60 day billing period. What is the price of that water?

    $35.42 + (9 x 3.29) = $35.42 + 29.61 = $65.03

    How much is that per gallon?

    9 x 748 = 6732 gallons so $65.03 ÷ 6732 = $.0096 or about 1¢ per gallon.

    The cost per gallon is higher because the basic fee is a much greater proportion of the bill and is spread out over fewer gallons.

    What would the cost per gallon be without the basic fee?

    $29.61 ÷ 6732 =  $.0043 or less than half the price that we pay including the fee.

    There are other charges on the bill too, for sewer service and storm drains, but they don't impact the price of the water we consume.

    Wednesday, February 10, 2010

    Water water everywhere, Part 1

    I live right next to the ocean. But in a desert. So we have to import our water. The city provides us with water and sends us a bill every 2 months.

    Back in November I discovered a leak in one toilet, and a broken pipe in the back garden. I didn't know what the impact of the leaks would be, but we fixed them as soon as we noticed. Then we had 3 sets of visitors over the Christmas holiday time.

    When the water bill arrived the amount we were faced with was huge! MUCH greater than we had ever used before.

    The meter had been read one week before, and the reading on the bill was 1613.00

    I decided to check the meter myself. We found the box in the sidewalk, opened it and took a photo. It says 1614.03 (+.68). If their figures were correct, we had used 1 HCF in the week since the meter was read.

    I called the water department and asked some questions. They said we could check to see if there were any leaks, by turning off all water sources, checking the meter, waiting an hour, then checking again. If the meter needle moved, there might be a leak somewhere underground or in a pipe under the house. So we turned off everything and waited.

    After a few minutes my wife started washing the dinner dishes. WAIT! I shouted. We're checking the water! Sorry, she said, as we waited some more, tapping our toes impatiently. Finally I took another photo.

    The meter now said 1614.03 (+.75). That small movement of the needle must have been my wife filling the kitchen sink. Seems reasonable. So we waited a bit more. No change in the readings, so probably no leaks.

    But I was not completely convinced. We were very careful with our use of water for another week, then I checked the meter again.

    Now we could see a real change. The meter measured 52 cubic feet, or just about half of a "hundred cubic foot" unit used by our water department.

    How could I tell if this is normal water consumption? I checked my previous bills, prior to all the fuss. We have averaged 9-10 HCF for an average billing cycle of two months (60-62 days).

    Let's make a number sentence problem from this.

    Is 9 HCF over 62 days less than, equal to or greater than .52 HCF over 7 days?

    900 ÷ 62 days = 14.5 cubic feet per day
    52 ÷ 7  days = 7.43 per day

    It's almost exactly half our average from previous months! (Was there a leak then too?)

    I think we need a better sample. A bigger one. So let's recheck the figures using the water company reading from two weeks ago. It was then 1613 (roughly) so we now have this calculation:

    1614.55-1613.00 = 1.55 HCF  or 155 cubic feet

    155 ÷ 14 days = 11.1 cubic feet per day

    That's under our normal usage. But reasonable.  Remember it has been raining around here. It was not dry as a bone this January and February. As my poor cat found out ...

    Are you wondering how much water is in that strange HCF unit? It's 748 gallons. Now we can ask: How many gallons have we used in 2 weeks?

    If 100 cubic feet = 748 gallons, then 1 cubic foot = 7.48 or about 7.5 gallons.

    Since we used 11.1 cubic feet per day that's 11.1 x 7.5 = 83.25 gallons of water per day.

    That seems like a lot! But indoor water use guidelines for US consumers suggest a maximum of 65 gallons per day per person. With 3 people in our house these past few weeks, our average usage is below those guidelines.

    83.25 ÷ 3 = 27.75 gallons per person

    Tuesday, February 9, 2010

    Hitting a (mathematically) moving target

    Many numbers are like a hummingbird's wings - they are constantly moving. The speed of the wind is a good example (the Blowin' in the Wind blog). Others are fuel economy of your car at any single moment, or your weight on a given day.

    We plead, "Honest, Doctor, I normally weigh less."

    We can measure things, but we don't know if a single sample reflects the normal, or the unusual.

    Moving numbers prompt a math-oriented person to use averaging. This is taught in 4th grade Excel Math.

    Average is one of many math terms used to describe choosing a single number out of a set of numbers:
    • arithmetic mean (average) is the result you get when you add the sample numbers in a set, then divide that sum by the number of samples
    • median is the center sample number, when a set is put in order from least to greatest; if there is no single center number, then the two numbers on either side of the center are added and the sum divided by 2 
    • mode is the sample value that appears most frequently in a set of numbers; there is no mode if no value repeats
      We choose different techniques for differing circumstances. For example, we might want to minimize the effect of outliers (numbers at the outer ranges of a set).

      Set One: (15, 16, 18, 12, 11, 4, 10, 27, 15, 14, 23)

      Here is an unsorted set of 11 numbers. Outliers are in red
      The mean is 15 (165 ÷ 11) we add the numbers and divide the sum by 11
      The median is 15 (4, 10, 11, 12, 14, 15, 15, 16, 18, 23, 27) we have sorted to find the median (middle)
      The mode is 15 we notice that this value appears twice out of 11 samples

      Set Two: (15, 16, 18, 12, 11, 2, 10, 44, 15, 14, 23)

      Now I have changed 2 numbers - making the outliers further away from the center value.
      The mean now is 16.36 (180 ÷ 11) we add and divide
      The median is still 15 (2, 10, 11, 12, 14, 15, 15, 16, 18, 23, 44) we sort and find the middle
      The mode is still 15 we notice it appears twice out of 11 samples

      Set Three: (15, 16, 18, 12, 11, 10, 10, 10, 15, 14, 23)

      I changed the previous outliers. Both became 10. Now the 10s and the 23 are the outliers.
      The mean now is 14 (154 ÷ 11) we add and divide
      The median is now 14 (10, 10, 10, 11, 12, 14, 15, 15, 16, 18, 23) we sort and find the middle
      The mode is now 10 the 10 appears three times, so it displaces the two 15s

      Which of the three averaging methods resulted in a value that changed the least? 
      In these examples, it was the median.

      The questions we consider when averaging are:
      • how often do we take samples; how many do we need?
      • how frequently do we average; how fast are we with our math?
      • how do we account for outliers; do we include or ignore them?
      • which method do we use; what kind of result is most useful?
      There are no right answers to these questions.  But the answers DO make a difference.

      Can mathematicians complicate these simple concepts in pursuit of more accuracy? YES they can!

      Monday, February 8, 2010

      We sell you the same for less and we make more. How?

      Pfizer (Big Drug company) offers a preferred-customer discount program to heart patients using long-term drugs, who are outside the USA and buy drugs with their own money (no government, no insurance). Here's the deal:
      • customers pay Pfizer rather than the pharmacy, through a direct buyer card
      • customers continue to pick up their drugs from the pharmacy
      • Pfizer still pays the pharmacy a commission for handling the customer and drug
      Pfizer gets customer names, and keeps track of their drug use (pharmacists don't monitor patients well). Pfizer continually encourages patients to take their medicine, and to reorder. They give a discount to customers as an incentive to participate.

      Does this make financial sense? How can a company sell something for less money and still earn more on the business?  Let's see if math can help us figure out this question.

      First, some research:

      A study from 15 years ago reported 15-46% of heart patients discontinue their drugs within a year. The rate was somewhat dependent upon cost and the side effects that were experienced.

      Another study found that 54% of heart patients discontinued for at least 90 days in the first year. Of those who stopped, 48% restarted within one year, and 60% within 2 years. Some restarted treatment after conversations with their doctors, a negative cholesterol test, or another heart incident.

      A third study showed that just 66% of heart patients given aspirin, beta blockers and statins continued taking all of them for more than one month after leaving the hospital. So 34% of patients stopped taking one or more of the drugs after only 30 days.

      A fourth study reported 77% of patients who were given statin drug prescriptions WHILE IN the hospital were taking them a year later. Only 25% of patients who were given statin prescriptions AFTER leaving the hospital were still taking the drugs a year later (75% discontinued).

      Many (most?) people stop taking their medicine soon after it is prescribed. People who stop taking medication may hasten their "mortality risk" (newly-approved drugs appear to work for only about half the people who take them) but are certainly former customers of the drug companies.

      Therefore, Pfizer started this program to encourage people to stick to taking drugs. They cited a 162% increase in customer retention through this discount purchasing program.

      So can we demonstrate drug costs and savings with math? It's hard to determine! Congress can't do it.

      Let's assume:
      • Average price per month is $75
      • Average pharmacy share is $15
      • Duration of dose is 12 months
      One year dose  12 x 75 = $900

      Pharmacy gets 15 x 12 = $180
      Big Drug gets  60 x 12 = $720

      This income stops if the patient stops buying the drugs!

      Assume we have 1000 patients for 2 years.
      Pharmacy = ($180 x 1000) + ($180 x 1000) = (180,000 + 180,000) = $360,000
      Big Drug = ($720x 1000) + ($720 x 1000) = (720,000 + 720,000) = $1,440,000

      TOTAL $1.8 million.

      If 50% stop taking the drugs within one year, and never reorder, then

      Pharmacy = ($180 x 1000) + ($180 x 500) = (180,000 + 90,000) = $270,000
      Big Drug = ($720x 1000) + ($720 x 500) = (720,000 + 360,000) = $1,080,000

      TOTAL $1,350,000

      Pharmacy is down $90,000 and Big Drug loses $360,000 per 1000 patients. Each year.

      TOTAL LOST $ 450,000 or 25% of the total potential revenue

      Big Drug has to work out the discounts that will earn them MORE money than the current pricing is losing, and covers the cost of the follow-up program. 

      We can't do all the math on this topic, but Pfizer most certainly can, or they wouldn't be expanding this program! More than $3.5 billion dollars of "heart medicines" were sold by Pfizer last year.

      Will this extra effort to sell drugs keep us from dying? I think the math is fairly clear on that (human mortality = 100%) but perhaps a few of us could get a temporary reprieve ...

      Friday, February 5, 2010

      Math is an international language

      There's a small counter on the Excel Math blog page that counts our visitors and figures out what countries they are from. I was astonished yesterday to see that we've had visitors from 77 countries around the world. Thank you for taking a few moments from your busy lives to learn more about math on our web page!

      The following list includes the flags of all our blog's visitors. I also noted the 27 countries which I have been privileged to travel. Thanks for welcoming me and my family and friends. In most places we have been able to drive, camp, eat and enjoy conversations with local residents. I don't even want to calculate how much money I have spent on all those trips!

      For those of you whose countries I haven't yet seen, perhaps someday I can visit and learn how you use math. At least I will be able to work out how to buy a meal and a souvenir ...

      We've had special relationships with a few places: the UK where I lived and worked and have an adopted second family; Canada where I worked off and on for 4 years; France where I encountered Citroens, great food, fine wine and fabulous friends; and Switzerland, which enticed me into Swiss watches.

      OK, no more talking - here is the list in order of most frequent visitors to the blog:

      United States ~ Resident and Citizen

      United Kingdom ~ formerly lived there; many friends

      Canada ~ formerly lived there; had a work permit, friends




      Sweden ~ visited 5 times

      Italy ~ visited 2 times

      Germany ~ visited 10 times, friends

      Hong Kong ~ visited 2 times


      Spain ~ visited 2 times

      Netherlands ~ visited 6 times


      Poland ~ visited 1 time




      Denmark ~ visited 3 times, friends


      United Arab Emirates ~ friends


      Honduras ~ friends

      Greece ~ visited 1 month

      France ~ visited 20 times; many friends

      Croatia ~ visited 1 time

      Republic of Korea


      Finland ~ visited 3 times

      New Zealand


      Switzerland ~ visited 10 times, friends and watches







      Slovenia ~ visited 1 time


      Sri Lanka



      Norway ~ visited 2 times

      Bahamas ~ visited 1 time



      Belgium ~ visited 5 times

      Japan ~ visited 2 times

      Saudi Arabia

      South Africa

      Macedonia ~ visited 1 time



      Czech Republic ~ visited 1 time

      Dominican Republic

      Taiwan ~ friends

      St Vincent and Grenadines

      Puerto Rico



      Iceland ~ Sportacus and Lazy Town

      Jamaica ~ visited 1 time

      Panama ~ friends

      Jersey ~ visited 1 time, family





      Guernsey ~ visited 1 time




      Mongolia ~ friends


      Austria ~ visited 4 times; friends