EVERYDAY MATH - The Handy Math Answer Book

The Handy Math Answer Book, Second Edition (2012)

EVERYDAY MATH

NUMBERS AND MATH IN EVERYDAY LIFE

(Note: Medical information contained in this chapter should not be a replacement for seeing your doctor. Please consult your doctor for any exercise programs you want to employ or any medical problems.)

What are the various methods of keeping time?

There are three basic methods of keeping time in common use today. They include time based on 12-hour intervals (a.m. and p.m.); a 24-hour clock; and Coordinated Universal Time clock (called UTC or Zulu time). All of these have to do with simple mathematics—mainly addition and subtraction.

A 12-hour clock (using the 12-hour interval concept) is familiar to most of us. It represents half a day (24 hours divided by 2). The abbreviations “a.m.” and “p.m.”— terms that originated from the use of longitudinal meridians—are used to differentiate between the morning and afternoon hours. The term meridian (from the Latin meri, a variation of medius, or “middle,” and diem, or “day”) once meant noon. Thus, the time before noon was called “ante meridiem” and after noon was called “post meridiem.” They were eventually shortened to “a.m.” and “p.m.,” respectively. Whether the terms are capitalized or not is not an issue either, as both are used in texts. The 24-hour clock is, naturally, divided into 24 hour increments. It is commonly used by the U.S. military and other government agencies throughout many countries.

UTC (or Coordinated Universal Time; the letters are not a true abbreviation, but a variant of Universal Time) is equivalent to mean solar time at the prime meridian (0° longitude), formerly expressed as GMT, or Greenwich Mean Time. (The change was done to eliminate using the name of a specific location in an international standard.) Other names for UTC are World time, Zulu time, and Z time. UTC should not be confused with UT, or Universal Time, the basis for the coordinated broadcast of time signals, counted from 0000, or midnight.

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A brass strip marks 0° longitude (the prime meridian) in Greenwich, England.

What are the ways to convert between a.m and p.m. and 24-hour time?

When converting a.m./p.m. to 24-hour time for the hours between 12:00 a.m. and 12:59 a.m., just subtract 12 from the time. For instance, 12:45 a.m. minus 12 will equal 0045, and 12:59 a.m. minus 12 will equal 0059. Times between 1 a.m. and 12:59 p.m. are a straight conversion, such as 9:00 a.m. equals 0900 and 11:00 a.m. equals 1100. For the conversion between 1 p.m. and 11:59 p.m., add 12, such as 8:34 p.m. plus 12 equals 2034, and 11:59 p.m. plus 12 equals 2359 (also seen as 20:34 and 23:59).

To convert in the opposite direction from the above, add 12 for the hours between 0000 (midnight) and 0059, such as 0034 plus 12 equals 12:34 a.m., and 0059 plus 12 equals 12:59 a.m. To convert from 0100 to 11:59 and from 1200 to 12:29, there is a one-to-one conversion, the former to a.m. and the latter to p.m. For example, 0100 equals 1 a.m., 0345 equals 3:45 a.m., 1235 equals 12:35 p.m., and 1259 equals 12:59 p.m. To convert between 1300 and 2359, subtract 12, such as 1424 minus 12 equals 2:24 p.m., and 2359 minus 12 equals 11:59 p.m.

How are local a.m./p.m. and 24-hour times converted to Coordinated Universal Time (UTC)?

When it comes to converting to specific times, Coordinated Universal Time (UTC) is a “standard.” It is used for a variety of reasons, including certain scientific studies, such as weather forecasting and astronomical data. The following lists several examples of converting local standard time (a.m./p.m. and 24-hour) to UTC time, and local daylight savings time (a.m./p.m. and 24-hour) to UTC:

Converting 24-Hour Time to UTC

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Note: In the United States, for Eastern Standard Time (EST) add five hours to get UTC time, six hours to Central Standard Time, seven hours to Mountain Standard Time, and eight hours to Pacific Standard Time. For Daylight Savings Time, add four hours to Eastern Daylight Time to get UTC time, five hours to Central Daylight Time, six hours to Mountain Daylight Time, and seven hours to Pacific Daylight Time.

What is the Dewey Decimal System?

The Dewey Decimal System of Classification is a numerical method libraries use to classify nonfiction publications into groups based on subject. It was invented by American librarian Melville Louis Kossuth Dewey (1851–1931) as a system for small libraries. The subject of a book is classified by a three-digit numeral that represents ten classes of subjects (000–999). In order, these are Generalities, Philosophy and Psychology, Religion, Social Science, Language, Natural Science and Mathematics, Technology (Applied Sciences), Arts, Literature, and Geography and History. For example, The Handy Math Answer Book would be found in the Dewey Decimal System under the 500s for Natural Science and Mathematics.

A Dewey Decimal classification number is followed by the Cutter number, or Cutter. This method was invented by Charles Ammi Cutter (1837–1903) and is an alpha-numeric way to represent words or names by using one or more letters followed by one or more Arabic numerals used decimally. Both systems—the Dewey Decimal System and Cutter—are together called the “call numbers,” a way of locating every book in a library.

What is a ruler?

Usually made of wood, metal, or plastic, a ruler is a measuring stick. Most rulers have a straight edge used for drawing straight lines and measuring lengths. The simplest and most well-known ruler has small scales, measured in terms of inches (and/or centimeters).

In order to read a ruler, the user needs to know the main divisions. For example, when looking at a foot ruler, the longest increments are represented by inches and are usually numbered 1 through 12; the measurement starts on the left end of the ruler, which may or may not be marked with a “0.” The next divisions, from smallest to largest, are: The distances between the smallest increments represent a sixteenth of an inch, the distances between the next largest increments represent an eighth of an inch, the next represent a fourth of an inch; and, finally, a half of an inch. (For more about measurement, see “Mathematics throughout History.”)

What are some ways to measure when you don’t have a ruler?

There are some pretty standard items that you can use if you don’t have a ruler handy to measure something. Most of them are standard sizes, such as a sheet of paper measuring 8.5″ × 11″ so it will fit into most computer printers with ease. Here are a few of the best ways to do a quick measurement (remember, too—some of these are approximations):

· A sheet of standard letter paper is 8.5 by 11 inches long.

· U.S. paper currency measures 6 1/8 inches wide and 2 5/8 inches long.

· Most standard business cards measure 2 inches long by 3 ½ inches wide.

· Most postcards measure 3 inches by 5 inches.

· The diameter of a quarter is about one inch; the diameter of a penny is approximately 3/4 of an inch.

· A credit card is about 3 3/8 inches by 2 1/8 inches.

· A standard AA battery is about 2 inches long.

· The average adult toothbrush is about 7 inches long.

How can a person calculate the amount of carpeting needed to redecorate a room in a house?

Watching remodeling programs on television, visiting someone else’s house, or even getting cabin fever in the winter often triggers the need to redecorate from floor to ceiling. But just how much new carpeting does a house need? The main way to discover how to cover up the floors is by using simple geometry, the branch of mathematics dealing with area, distance, volume, and the properties of shapes and lines. (For more about geometry, see “Geometry and Trigonometry.”)

For example, if a person needs to buy new carpeting for a bedroom, just use the formula A = L × W, or the area equals the length times the width. For a bedroom that measures 10 feet by 12 feet, the area would equal 120 square feet. Thus, the room would need 120 square feet of carpeting.

If a living room has a circular alcove, there is another measurement one can use: A = pi (π) × r2, or the area (A) equals pi times the radius squared. If the room measures 12 feet in width at the long end of the alcove (or the diameter of the alcove), the radius of the alcove is 6 feet. Thus, A = 3.14159 × (6)2, or 113 square feet (rounded to the nearest square foot), would equal the area of an entire circle. Because an alcove is half of a circle, divide the 113 square feet in half, or about 56.5 square feet. Add this area to the rest of the room. For instance, if the rest of the room measures 10 by 12 feet, multiply 10 × 12 = 120, then add the alcove area of 56.6 to get the needed amount of carpeting: 176.5 square feet.

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Measuring a room for a circular rug involves some basic knowledge of mathematics.

Are ratio and proportion important in cooking?

Yes, ratio and proportion—two major mathematical concepts—are important in cooking (as is addition, subtraction, multiplication, and division). For example, when a recipe calls for 1 cup of flour and 2 eggs, the relationship between these two quantities is called a ratio. In this case, the relationship of cups of flour to eggs is 1 to 2, or ½, or 1:2. Any change in this ratio and the recipe’s result will not be the same—and the food might not be edible.

Another way of looking at cooking and mathematics is when changing the amount of ingredients in a recipe. For example, when a recipe calls for a certain amount of each ingredient, and the cook wants to make half a recipe, everything in the list is divided in half. Two cups of sugar becomes one cup, ½ teaspoon of vanilla becomes ¼ teaspoon, and so on. The same logic applies if the cook wants to double the recipe, but in this case everything is multiplied by two. Two cups of sugar becomes 4 cups; ½ teaspoon of vanilla becomes 1 teaspoon; and so on.

What does it mean when a test score is marked “on a curve”?

A test score marked “on the curve” means that the marks will roughly follow what is often called a Gaussian Probability Distribution—or, more commonly, a bell curve. This symmetrically shaped curve is based on the test scores of the exam. In a perfect world, one-sixth of the scores would be on either end of the curve, with more than two-thirds falling in the middle, creating a normal distribution.

But most test results are not ideal. Thus, when a plot of the number of students versus the marks received are viewed, the relative difficulty of the test is known. It is then up to the teacher to decide how to distribute the grades (this is usually done by comparing each student’s mark to the distribution curve). The teacher then decides where to cut off the passing and failing marks, which is often referred to as curving the grades. (For more about normal distribution, see “Applied Mathematics.”)

What’s the difference between wet and dry measurements when you’re cooking?

Many cooks know there is a difference in the kitchen when it comes to dry and wet measurements. In particular, such differences are seen when one measures a cup of something as an ingredient in a recipe. For example, when measuring dry ingredients, cooks use a “dry measuring cup,” in which the ingredient is easily scooped into the cup, filled to the brim, and leveled with a knife or other flat-edged utensil. Most of these cups are made so one cup of an ingredient will fill the cup.

A “wet measuring cup” usually has increments written on the side and a spout for easier pouring. A measured cup of liquid is below the outer rim of the cup, making it easier to see the meniscus (the concave upper surface of a still liquid caused by surface tension) that one lines up with the desired increment on the side of the cup. It would be rather messy to use a liquid in a “dry measuring cup,” whereas, it’s not as bad using a wet measuring cup with dry ingredients. Some cooking purists use a wet measuring cup for dry ingredients, then weigh the amount used. After making the desired recipe, the cook can judge just how much “dry” ingredients to use with a “wet” measuring cup.

For most of us who are not as picky in the kitchen, the dry or wet measuring cup is just fine for either type of foodstuff. The main reason for remembering that there is a difference may be for those who want to one day start their own cooking school; and it’s also for chefs who make up recipes for publication or special uses. This is because measuring accurately is the most important cooking skill that can mean a good tasting meal versus a bad one. In particular, test kitchens need to know the differences between dry and wet measurements to test recipes. They vary the amounts of the ingredients—called “tolerance testing”—to make the best-tasting recipes, all helped by the knowledge of the basic rules of measuring.

What do the numbers on gas and electric meters measure?

Gas and electric meters measure the daily gas or power usage, usually of a house or other building. The standard electric meters are clocklike devices that record the amount of usage. As a house or business draws on the electric current, a set of small gears inside the meter move. The number of revolutions are recorded by dials in the meter, with the speed of revolutions determined by the amount of power consumed. Newer digital models record power usage digitally.

A gas meter measures the amount of gas used by a home or business. In this case, the meter measures the force of the moving gas in the pipeline. The dials on the meter turn faster as the flow of gas increases, or slower as the gas flow decreases. To understand how the gas and electric companies charge a customer takes only an understanding of simple mathematics. In both cases, the difference between one month’s reading and the next month’s reading is the amount charged. For example, if the electric current reading is 3240 and the previous month’s reading is 3201, the amount of electricity used is 39 kilowatts for one month. Then the company multiplies the number of kilowatts by the amount per kilowatt hour to arrive at the charge to the customer.

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Electric meters like this one measure daily power usage.

What are some numbers associated with tire pressure?

Tire pressure is measured using a tire pressure gauge, with the most common measuring device being about the size of a pen. First, a bit about pressure: The atmosphere at the surface of our planet measures about 14.7 pounds (6.67 kilograms) per square inch, or a 1 inch (2.54 centimeters) square column of air weighs 14.7 pounds. This changes depending on the altitude. For example, at an altitude of 10,000 feet (3,048 meters), the air pressure decreases to 10.2 pounds (4.63 kilograms) per square inch.

But a car, truck, SUV, or bike tire needs more pressure in order to inflate. By increasing the number of atoms inside the tire, there are more collisions between the atoms and more pressure exerted on the sides of the tire. In other words, a pump stuffs more air into a constant volume (the confines of the tire), so the air pressure within the tire rises. A car tire’s pressure is typically at about 30 pounds (13.61 kilograms) per square inch; a bicycle tire’s pressure can be around 90 pounds (40.82 kilograms) per square inch.

How is political polling done?

Although polls seem to be magical, they are merely a matter of taking information and applying some simple statistics. Polling is a technique that uncovers the attitudes or opinions of a segment of the population, and is based on certain questions about politics, the economy, and even social conditions.

Polling as it is conducted today started in 1936, when George Gallup used statistical modeling to take the first political poll: the presidential election of Franklin D. Roosevelt over Alf Landon. FDR was successful, as we all know, and it was “predicted” by Gallup based on his model. After the further successes of his polling methods, he was quoted as saying that sampling public opinion was like sampling soup: “one spoonful can reflect the taste of the whole pot, if the pot is well stirred.”

How often are political polls wrong?

Just because statistics are used does not make polling infallible or even reliable. For example, some questions may be misleading. The media is notorious for asking such polling questions as, Do you agree that keeping the environment clean is important? as if the majority of people would say no. Such questions often make the results of polling somewhat questionable, especially if the original questions asked are not presented with the results. Errors and questionable outcomes are caused by people lying, bad interview techniques, and even the sample of people interviewed. All too often, the results of polling create another consequence: swaying public opinion and creating a “jump on the bandwagon” effect. (For more information about statistics, see “Applied Mathematics.”)

Finally, the average sample in a poll is around 1,500 people. This is said to reflect the entire population within around three percentage points (called the margin of error). In fact, this is also the reason why so many people don’t pay too much attention to the media’s political polls—too few “data points”—and why many politicians conduct their own polls before, during, and after the elections. Of course, others cite their own statistics, including one that says, since 1980, every presidential election has been accurately predicted by national polls.

His words continue to be followed today in justifying political polling: sampling a small number of people to reflect a larger population. The sample population can be chosen randomly, stratified, or by other methods; people can be polled via a telephone interview, questionnaire in the mail, or personal interview, such as an exit poll during an election (polled as a person leaves a voting place). Statistics such as averaging and resulting percents are then used, mainly by the media, to determine the overall “pulse” of the public. Many commercial poll takers not only claim their results help in market research and advertising, but also get the peoples’ concerns out in the open.

What is the purpose of the numbers found in a mailing address?

The Zone Improvement Plan (or ZIP) Code is a grouping of numbers assigned by the U.S. Postal Service to designate a local area or entity in order to speedily deliver and distribute mail. ZIP Codes most often refer to a street section, a collection of streets, a structure or building, or a group of post office boxes, but the numbers do not rigidly conform to boundaries of cities, counties, states, and other places.

Depending on the area, a ZIP Code includes 5, 7, 9, or 11 digits. In the most common codes—five-digit ZIP Codes—the first digits divide the country into ten large groups of states numbered from 0 in the Northeast to 9 in the far West. Each state is divided into geographic areas identified by the second and third digits of the ZIP Code. For example, New York and Pennsylvania have ZIP Codes starting with numbers between 090 and 199; Indiana, Kentucky, Michigan, and Ohio begin with numbers between 400 and 499. The fourth and fifth digits of a ZIP Code identify the local delivery area.

Why do we have certain telephone (and cellphone) numbers?

Telephone numbers have a long and numerical history. Starting around 1879 and before the 1920s, most telephone conversations started with cranking a crank, which would “connect” to an operator; from there, the caller would most often either give a person’s name or company and/or address—mainly to other local telephone owners. At this time, too, each customer who had a phone was usually assigned a four digit code, which would be used by the operator to make the call.

After the late 1920s, this progressed to calling the operator and using an “exchange name” to get the party one was calling; in addition, telephones with automatic dialers were becoming vogue in homes, making it easier to dial a number yourself.

But in order to accommodate the many people using the telephone, and to stop the confusion caused by many people living in larger cities having the same names, telephone companies turned to an alternate system of calling another party on the phone. Whether using an operator (usually if a person was not home) or a home dial phone, telephone company AT&T used a combination system of numbers to reach a party, whereas Bell System used a name and number system: the first two letters and number was the switching office or exchange, while the last four numbers represented the customer’s number. For example, the movie Butterfield-8 was actually named after an early exchange name. And, for example, for the Bell System name and number listing, an older number the authors are familiar with began with the colorful “PIoneer 8-XXXX.” No one really knows why the “letters and numbers” began, but it may have been that telephone companies believed that people would remember names better than the correct order of seven numbers.

By 1958 people began asking for their own telephone numbers, phasing out the exchange number/letter system. The first city to use all number calling was Wichita Falls, Texas. It took just over a dozen more years for the letter and numbers to be replaced all over the country. With this code, the first three digits are called the prefix (or exchange number); the last four digits represent the part of the town in which you live. Thus began the familiar area code numbers (to send the phone number to the right state) and seven-digit numbering system with which we’re all familiar.

MATH AND THE OUTDOORS

How is the amount of rainfall measured?

The amount of rainfall—liquid precipitation that falls to the surface—is measured by a rain gauge. The most commonly used free-standing rain gauge is a cylinder with increments (most often in inches) inscribed on the outside of the tube. It is put in an area that is not obstructed by buildings, trees, or other tall structures than can impede the collection of rainfall.

The rain gauge can also measure snow, but added steps are needed to calculate the measurements. In this case, a rain gauge measures the liquid equivalent of snow. This is why meteorologists will often say that a snowstorm that produces 10 inches (25.4 centimeters) of snow will have a liquid equivalent of 1 inch (2.54 centimeters) of rain, or a ratio of 10:1. But this generalization can be tricky. If the weather system is super-cold, such as an Arctic air mass over Canada and the northern United States, the below-freezing temperatures might create more than 10 inches of snow. Meteorologists often call this the “fluff factor,” because the snow seems “fluffier” due to the fact that there is more air between the snow crystals at much colder temperatures. In fact, in very cold air the snow-to-liquid equivalent ratio can be 15, 20, or even 30 to 1. (For more about weather and math, see “Math in the Natural Sciences.”)

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A weather station on top of the Alps, including a rain gauge and anemometer for measuring wind speed.

How is wind speed measured?

Wind speed is most often measured using an anemometer. The simplest ones are made with cup-shaped devices that spin and catch the winds. The number of spins translates into an approximate value for the wind speed. The more accurate electronic anemometer has a free-moving turbine suspended in the middle of the instrument that, when held correctly, measures the wind speed. The speed of the turbine sensed by an infrared light relays the signal to an electrical circuit. From there, the wind speed is digitally displayed.

How do you tell the BTUs of a certain type of wood for your woodburner?

There are numerous numbers involved in having a simple woodstove or fireplace. One of the most important is BTU, or the acronym for British Thermal Unit. In a physical sense, 1 BTU is equal to a unit of energy equal to the work done by a 1,000 watts working for 1 hour. When it comes to your fireplace or woodstove, the wood you put into the system has a certain amount of BTUs to keep you warm; some are higher, some lower, depending on the species of tree.

What do the numbers on an aneroid barometer signify?

The non-liquid aneroid barometer measures air pressure. As the atmospheric pressure changes because of storm systems (or lack of storms), the instrument records the changes. (Because atmospheric pressure changes with distance above or below sea level, a barometer can also be used to measure altitude.) The aneroid barometer contains a small capsule that acts like a bellows, but with the air removed. When the air pressure increases, the sides of the capsule are pushed in and the connected needle rises (moves clockwise). When the air pressure decreases or falls, the capsule’s sides puff out and the needle moves in a counterclockwise direction.

The numbers on a common barometer range from about 26 to 31, with divisions of 10 or more in between each number. A needle (actually a hand similar to a clock hand) points to the numbers on the barometer and moves in response to the changing air pressure. These numbers are based on the principle that atmospheric pressure supports 30 inches (76.2 centimeters) of liquid mercury in a tube with one end sealed; and this information is based on the mercury barometer, the first type ever made.

How is an aneroid barometer read? In general, a falling hand on a barometer indicates a low pressure system is on the way with poor weather (usually a storm with snow or rain); a steady barometer means there will be no changes with the ongoing pressure system; and a rising barometer means high pressure and fair weather; an even higher reading, around 31, means an extremely dry atmosphere. The timing of the barometric change is also telling: A change of a degree either way in a few hours means that the weather will change quickly; a slow change of 0.3 or so a day indicates weather arriving in 12 to 24 hours. A quick rise in the barometer also often indicates high winds and unsettled weather.

All barometers work because of our weather systems: Changes in air pressure are caused by differences in air temperature. And this, in turn, creates the wind and weather patterns that carry the high and low pressure systems around the Earth’s lower atmosphere.

It’s not cut and dry, so to speak. Each tree has not only a certain weight (some are denser than others), but BTUs per cord (a cord is equal to about 128 cubic feet of stacked wood, or about 4 by 4 by 8 feet; it’s interesting to note that only about 70 to 100 cubic feet of a cord is solid wood). A cord is different than a “face cord,” or a pile 4 by 8 feet long and only one “log” (or “stick”) deep.

The following lists some weights/cord and BTUs for some popular “burning woods” (note that the softer woods, such as pines, don’t offer as much in terms of heat energy BTUs; in addition, they do not last as long burning-wise, or burn as hot, as woods with higher BTUs):

· White ash—3,689 pounds per cord, offers 23.4 BTUs

· Cherry—3,120 pounds per cord, offers 20 BTUs

· Norway Pine—2,669 pounds per cord, offers 17.1 BTUs

· Spruce—2,100 pounds per cord, offers 14.5 BTUs

· White oak—4,012 pounds per cord, offers 25.7 BTUs

· White cedar—1,913 pounds per cord, offers 12.2 BTUs

· Red maple—2,924 pounds per cord, offers 18.7 BTUs

· Sugar maple—3,757 pounds per cord, offers 24 BTUs

What do plant hardiness zone numbers mean?

Plant hardiness zones are another way numbers are used. In this case, they indicate the average annual minimum temperatures for landmasses around the world. For example, one common Plant Hardiness Zone Map is broken down into 20 different zones based on the average annual minimum temperatures. In zone 5a, for example, the average annual minimum temperature ranges from -20 to -15 degrees Fahrenheit (-26.2 to -28.8 degrees Celsius; for example, Des Moines, Iowa). In zone 11, such temperatures are above 40 degrees Fahrenheit (4.5 degrees Celsius; for example, Honolulu, Hawaii). And in zone 1, such temperatures are below -50 degrees Fahrenheit (-45.6 degrees Celsius; for example, Fairbanks, Alaska). There are other maps, too, that break down the zones into even more detail.

What numbers are important when people fertilize their plants?

Not only do you have to pay attention to the zone where you plant, but how and when to use the best and complete fertilizers. In many cases, your average soil will not have enough nutrients to support certain plants, especially if you decide to cultivate a vegetable garden.

All plants need oxygen, carbon, and hydrogen, which are collected by the plant from the air, sunlight, and water, respectively. But they also need a range of other chemical elements, such as nitrogen (N), phosphorus (P), and potassium (K). These are usually present in what are called the complete fertilizers (which means it contains all three elements). Most general fertilizers—liquid or dry—that you find in your local nursery, farm-based, or even hardware store have these three elements in varying amounts.

For example, if a bag of fertilizer reads “10-5-6” that means the ratio of these three elements is 10:5:6, or that the fertilizer contains mostly nitrogen and lesser (and almost equal) amounts of phosphorus and potassium. This does not mean the bag contains all three of these elements only; it means that the entire bag of fertilizer contains 10 percent total nitrogen, 5 percent phosphate, and 6 percent potassium (often called potash). If a bag reads 5-5-5, then all three elements are found in equal parts within the fertilizer.

When buying such fertilizer for your vegetable (or even flower or herb) gardens, you should pick the best fertilizer based on what your plant needs, such as nitrogen for better leaf growth, or phosphate for activating root growth. In most cases, the fertilizer is either added to the soil before you plant in the spring, or added as a side-dressing as the plant grows. Each bag usually tells you how much you should add to the garden per square foot. Liquid fertilizers can also be used as something called foliar feed, which is sprayed on the plants’ foliage and absorbed quickly by the leaves. In the majority of cases, nitrogen is needed as a foliar feed, thus the container will have the first number (representing nitrogen, or N) as the largest number.

MATH, NUMBERS, AND THE BODY

Is a human’s normal body temperature really 98.6 degrees Fahrenheit?

Although everyone seems to be taught that the normal human body temperature is 98.6 degrees Fahrenheit, in reality, “normal” has a range. In fact, a person’s actual measured temperature is rarely exactly 98.6 degrees Fahrenheit. One reason for this has to do with how the old standard was calculated: using an oral mercury thermometer and basing the results on a small human sampling.

Gone are the days of putting a thermometer under one’s tongue for five to ten minutes. Today’s thermometers are more sophisticated, accurate, and faster. Thus researchers believe that, based on better data—and more people tested—normal body temperatures measured orally range from 97.5 to 98.8, with about 1 in 20 people having a bit higher or lower normal temperatures. These numbers change throughout the day, too, varying from 1 to 2 degrees (on the average, reaching a low at about 2 a.m. to 4 a.m., and a high twelve hours later). An even more accurate representation of a body’s temperature is to measure the core temperature, or the actual temperature inside the body, usually by using a rectal or inner-ear (tympanic) thermometer.

What do blood pressure numbers mean?

Blood pressure is a measure of how much the blood presses against the walls of the arteries. This creates two forces: The first comes from when the heart pumps blood into the arteries; the second is the force of the arteries to resist the blood flow. When a person “takes a blood pressure,” he or she is taking a ratio: The higher number (called the systolic or top number) represents the pressure when the heart contracts to pump blood to the body; the lower number (called the diastolic or bottom number) represents the pressure when the heart relaxes between beats. For example, for a blood pressure of 120/76 (said, “120 over 76”), the systolic reading is 120 and the diastolic reading is 76.

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When a nurse or doctor takes your blood pressure, the two numbers they read to you indicate systolic and diastolic pressures in your arteries.

In actuality, the numbers represent how high the blood’s pressure would force a column of mercury to rise in a tube. For example, a systolic reading of 120 means the mercury would rise 120 millimeters (usually labeled mm Hg, with Hg the symbol for mercury) in a tube. Based on the most recent information (and it keeps changing), a blood pressure below 120/80 is considered optimal for adults; 120 to 139 over 80 to 89 is considered “prehypertension.” Anything over 140/90 is considered hypertension, which includes three stages, with the highest hypertension reading, stage 3, being anything above 179/109.

How does one calculate resting heart rate?

Calculating resting heart rate (RHR) involves easy math. It is the number of heart beats per minute when the body is resting, with the beats per minute representing the number of times the heart contracts. To measure the RHR, just count the number of beats per minute via the pulse—usually taken either on the inside of the wrist or along either side of the neck—for 15 seconds. Then multiply this number by four (15 × 4 = 60 seconds) to get the heart rate per minute. Or count for 10 seconds, then multiply this number times six (6 × 10 = 60 seconds) to get the heart rate per minute. For example, if you count 12 beats in 10 seconds, multiply 12 by 6 to get a resting heart rate of 72.

How does one calculate heart rate during exercise?

One reason to calculate heart rate during exercise is to know if a person is getting beneficial exercise to keep the heart and body healthy. In order to determine the safe and effective range of exercise to get cardiovascular benefits, two measurements are often taken. The first is the maximal heart rate, a number related to a person’s age (the heart beats slower with age). To estimate the maximal heart rate, subtract a person’s age from the number 220. For example, if someone is 40 years old, his or her maximal heart rate is 180.

The next measurement is the target heart-rate zone. This number uses the maximal heart rate and represents the number of times per minute at which a heart should be beating during aerobic exercise. For most healthy people, the range is 50 percent at the lower limit to 80 (some charts say 75) percent at the upper limit of their maximal heart rate. When a person’s heart rate reaches a value within this zone during exercise, it means he or she has achieved a level of activity that contributes to his or her cardiovascular fitness. For instance, from the above example of a maximal heart rate of 180, the beats per minute for the lower range would be 180 multiplied by 50 percent (0.50), or 90 beats per minute; the upper range would be 180 multiplied by 80 percent (0.80), or 144 beats per minute.

If you work out and maintain a lower-than-50- or higher-than-80-percent limit, there are few beneficial effects from the exercise. In terms of the lower limit, the heart is not working hard enough for any cardiovascular benefit; in terms of the upper limit (besides the strain and injuries that can result), the heart is working too fast for any benefit and the body can’t replenish oxygen that quickly.

What do cholesterol numbers mean?

Cholesterol numbers indicate the amount of cholesterol in the bloodstream (cholesterol is a waxy, fat-like substance manufactured in the liver and found in all tissues). For humans, a total cholesterol number above 200 means there is an increase in the risk of heart disease (between 200 and 239 is considered borderline high cholesterol); for anything below 200, there is less of a risk for heart disease.

But total cholesterol is not the only number to know. There is also Low Density Lipoprotein (LDL), or “bad” cholesterol. LDL is the main source of the buildup and blockage in the arteries (risk levels are above 130, measured in milligrams per decaliter). There is also High Density Lipoprotein (HDL), the “good” cholesterol that helps keep the plaque from building up (risk levels are below 40, measured in milligrams per decaliter).

When there is too much cholesterol in the blood, it can build up on the walls of the arteries. Over time, the buildup (often called plaque) causes “hardening of the arteries,” or a narrowing of the arteries that restricts or stops blood flow to the heart. Blood carries oxygen to the heart, and if there is less blood (and thus, less oxygen) reaching the heart, chest pain can result. If there is a near-complete or complete blockage that cuts off the blood supply to a portion of the heart, a heart attack (myocardial infarction) is often the result. This is why most doctors recommend watching your numbers for total cholesterol, HDL, and LDL for signs of any change.

Why is measuring medications so difficult?

There are many ways that medications are measured, which often leads to confusion when someone asks what kind, amount, and dosage of medications you consume (and why doctors recommend you to carry a list of your medications, along with the amount of each pill and dosage).

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How much is a “teaspoon” of medicine? Because people often measure medicines in different ways at home, it is common that people do not administer correct dosages for themselves or their children.

Pills come in a wide variety of types, of course, ranging from tiny and round to what was once referred to as very large “horse pills.” Most pills have the needed ingredient—usually 5 to 10 percent of the tablet—along with other substances that either make them palatable or easier to swallow, and allow the pill to dissolve in your system. Dosage is the most confusing topic when taking a medication. Many times, a label will read, “one teaspoon before bedtime.” But the problem comes in when you realize that not everyone has the same teaspoon, or measures the same way. In fact, in one study, people were asked to pick out a spoon size that would measure 5 milliliters, the proper measurement meant by “a teaspoon” when using medications. Most people chose spoons (they ranged from 4.4 to 10 milliliters) that were higher in dosage, sometimes choosing a spoon that would give them double the dose. This is why doctors recommend a medicine spoon or dosing syringe.

The following is a list of numbers that you may see in your medicine cabinet. Many medicines (seen with pills mostly) are measured using metric weights:

· 1 kilogram (kg, Kg) = 1,000 grams (1000 g)

· 1 gram = 1,000 milligrams (1000 mg)

· 1 milligram = 1,000 micrograms (1000 mcg)

· 1 microgram = 0.001 milligrams (0.001 mg)

· 1 milligram = 0.001 gram (0.001 g)

· 1 microgram = 0.000001 gram (0.000001 g)

And in terms of measuring out dosages, the following lists some of the more common measurements:

· 1 cubic centimeter (cc) = 1 milliliter (ml)

· 1 teaspoon = 5 cc = 5 ml

· 1 tablespoon = 15 cc = 15 ml = 3 teaspoons

· 1 ounce (oz) = 30 cc = 30 ml = 2 tablespoons = 6 teaspoons

How do the body’s natural rhythms and mathematics seem to fit together?

The body has several natural cycles it experiences, including one called circadian rhythm (CR). In general, CR refers to the many body clocks that regulate our daily rhythms, natural cycles that control our appetite, energy, body temperature, sleep, and moods. The reason for such rhythms is mainly because your body’s cells don’t have enough energy to perform their many tasks at a constant rate and at the same time; thus, biological processes are rhythmic, giving cells time to alternately rest and be active. In its simplest form, this is why animals (and plants) have a dormant (sleep) cycle.

One of the more interesting interpretations of rhythms—albeit somewhat controversial, as most people believe it’s more “trendy” than scientific—are your bio-rhythms, or how your life is affected by biological cycles that cause oscillations (think of an up and down sine wave). It’s divided into cycles that show how you are doing intellectually (a 33-day cycle; peak days you’re at your best memory-, logically-, and alertness-wise), physically (a 23-day cycle; peak days you’re at your best strength-wise and well-being-wise), and emotionally (a 28-day cycle; peak days you’re at your best creativity-, sensitivity-, perception-, and mood-wise).

Each biorhythm varies throughout the days, affecting you depending on where on the sine wave you fall that day. To determine each part of the cycle, use the following: For physical = sin(2πt/23); emotional = sin(2πt/28); intellectual = sin(2πt/33); and an extra cycle, the intuitive = sin(2πt/ 38), with t being the number of days since you were born. Plot them each day to see how you’re doing through the week or year. Or just do it the easy way and go to a “biorhythm calculator” on the Internet.

What is a dog’s and cat’s age equivalent compared to a human?

All animals “age” at different rates, and our domestic dogs and cats are no exception. Unlike humans, these pets seem to age more rapidly in terms of human years, with both animals being of an age equivalent to about 15 human years by the time they are one year old. There are, however, some differences for dogs. Some people have determined an age range depending on the size of the animal. For example, a dog over 90 pounds may only be equivalent to 12 human years by the time it is one year old; a dog 51 to 90 pounds is about 14 human years old when it is one year old; and dogs under 50 pounds are the equivalent of 15 human years by their first year. These age differences are reflected for the entire life of the dog. Below is a simplified table illustrating dog and cat years and their human age equivalents.

Human Years versus Cat/Dog Equivalents

Human Age
Equivalent

Cat Age

Dog Age

10 years

5 months

13 years

8 months

14 years

10 months

15 years

1 year

1 year

24 years

2 years

2 years

28 years

3 years

32 years

3 years

4 years

36 years

5 years

5 years

40 years

6 years

6 years

44 years

7 years

7 years

48 years

8 years

8 years

52 years

9 years

9 years

56 years

10 years

10 years

60 years

11 years

11 years

64 years

12 years

12 years

68 years

13 years

13 years

72 years

14 years

14 years

76 years

15 years

15 years

80 years

16 years

16 years

84 years

17 years

17 years

88 years

18 years

18 years

89 years

19 years

92 years

19 years

93 years

20 years

96 years

20 years

21 years

99 years

22 years

103 years

23 years

106 years

24 years

MATH AND THE CONSUMER’S MONEY

How do I make change?

“Making change” is an important monetary endeavor no matter where a person travels or lives. It is defined as the change (in coins and/or bills) a customer receives after making a purchase, especially when the customer gives the merchant more money than the amount due. For example, say a person visiting a farmer’s market in the United States buys a half pound of garlic for $2.60 and gives the merchant a $20.00 bill. To make change, the merchant would first subtract $20.00 - $2.60 = $17.40; or the customer is to receive $17.40 in change.

The resulting change would start with the highest denomination and work down: a $10 bill, a $5.00 bill, two $1.00 bills, a quarter ($0.25), a dime ($0.10), and a nickel ($0.05). The merchant would then count the change out to the customer, stating each amount in the “opposite” fashion: “That’s a total of $2.60 out of $20.00; your change is two sixty-five [$2.65 as the nickel is given], two seventy-five [$2.75 as the dime is given], three dollars [as the quarter is given], four dollars [as the first dollar bill is given], five dollars [as the second dollar bill is given], ten dollars [as the five-dollar bill is given], and twenty [as the ten-dollar bill is given].”

What is a store discount?

Many stores—from apparel shops to bookstores—offer a discount on the retail price or, less often, on the sale price. For example, say a sweater that originally costs $50.00 is discounted by 25 percent during a sale at a department store. The customer will actually pay 75 percent (or 75/100 or 0.75) of $50.00. To determine the sale price, multiply $50.00 by 25 percent (50 × 0.25 = 12.50); then subtract the discount from the original price to find the sale price, or $50.00 - $12.50 = $37.50.

How does a person calculate sales tax on purchases?

Most states and cities charge sales tax on retail purchases made by customers—from clothes to certain foods. This sales tax is based on a percent of the purchase price, with the percentage of tax called the tax rate. In turn, each state’s tax rates vary greatly, often with the counties charging additional taxes: For example, in Vermont, the state and local sales tax rate stands at 5 percent; Florida has a state sales tax rate of 6 percent. And in various counties of New York, the sales tax rates start as low as 7.25 percent to as high as 8.75 percent.

How is the sales tax calculated at the checkout? Take, for instance, a purchase of a $100.00 ring in New York City, a place with a sales tax rate of 8.265 percent. Multiply $100 times 8.265 percent, or $100 × 0.08265 = $8.265, which rounds up to $8.27 for sales tax. The merchant would then charge the customer a total of $100.00 + $8.27 = $108.27. If the same $100.00 ring was purchased in Florida, the sales tax would be $100.00 × 6 percent, or $100 × 0.06 = $6.00, with a total purchase price of $106.00. Of course, not every purchase will be as straightforward as this, since some cities can also add local taxes and/or surcharges and fees to the bill.

How do you determine the interest rate on savings bonds?

U.S. savings bonds have been around for decades. During World War II, they were used to not only invest in the country, but to help pay for the war effort. Since then, buying bonds is still thought of as a patriotic way of “lending” money to the government to finance the country’s borrowing needs (for more information about bonds, visit the official savings bond site at Treasury Direct, www.treasurydirect.gov).

How is the price of gold determined?

The price of gold on the market has had a long and lustrous past. It’s been used as money for centuries because of its rarity and, no doubt, luster, heft, and malleability. Gold pricing is thought to have begun in London in the late 17th century; by 1717, then Master of the Mint, scientist Sir Isaac Newton (the same Newton of “laws of gravity” fame) declared that the gold price would be 4.25 pounds per troy ounce, an amount that was held mostly the same for close to two hundreds years.

Although the first gold standard was established by the Bank of England in 1696, it took until around the late 1800s for most other European countries to adopt a gold standard. Many of these countries halted the standards during the financially troubled times of World War I; then began again in earnest after World War II. The U.S. dollar represented a gold rate of $35 per troy ounce after the war; by 1971, the U.S. stopped the convertibility of the dollar to gold (yes, at one time, although difficult to do, you could “trade” a dollar bill for gold). One can see evidence of this on the older “silver certificate” one dollar bills printed before 1971. Other countries also dropped the currency-into-gold standards, with Switzerland being the last country to do so in 2000.

Today, gold is still sold most often as gold coins, with each coin having a certain amount of the metal, usually one troy ounce. The daily price of gold is determined by the trading on several exchanges, with the price of gold based on the London gold fix. The gold fix, or price, is set twice a day: at 10 a.m. and again at 3 p.m. in London, the latter time originally chosen to coincide with the opening of U.S. markets. The gold fix is determined by several banks (or bullion trading firms), a system that has not changed in decades, but the member banks have changed. Gold is also traded based on a “spot price”—the price paid for immediate delivery (versus “futures prices” in which delivery and payment is made at a future date)—and is derived from global over-the-counter gold trading markets.

There are several features to know when buying bonds and all of them have to do with a certain degree of mathematics. Bonds are backed by the government, and if you buy one for $500, your cashed-in value will never fall below the $500 you initially paid. That’s why they are considered safe, conservative investments. They have competitive interest rates, even between the different types of saving bonds. Although there are several rules concerning cashing certain bonds (including not cashing one in for over a year in some cases), no state or local taxes are charged, and the federal tax is deferred until the bond is cashed in. Depending on the bond, it can be for face value or half the face value. For example, people who bought Series I bonds bought them for $100 with a face value of $100; and people who bought Series EE bonds bought them for $50 with a face value of $100.

The interest rates also represent mathematics. Some bonds have a fixed rate— especially older bonds—never changing in interest over the life of the bond. But there are several types of bonds that are affected by changes in the economy, including some that have fixed rates. And although it seems as if “fixed” means never moving, in this case, it is not. The government can still change the rates in certain situations, such as in “deflationary situations.” Thus, every May and November, based on the changes in the Consumer Price Index for all Urban Consumer (CPI-U) and combined with the fixed rate, the earning rate of this type of bond is determined. This means that if you have one of these types of bonds, and it started with a rate of 4 percent, under certain economic conditions it can change to, for example, 2 percent (at this writing, I-bonds were in this category).

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Gold, surprisingly, is not as rare an element as many people believe, and, despite some industrial uses such as in circuitry, is not as useful as other, stronger metals. What gives it its worth is that peopler perceive it to have great value, as well as the fact that it is a very pretty substance.

What are some interesting gold and numbers facts?

Gold and numbers go together for many reasons. The following lists some interesting gold and number facts from the World Gold Council:

· The 40,000 miners who joined the California Gold Rush in 1849 were called “49-ers”; only a very few ever “struck it rich.”

· One ounce of gold can be stretched to a length of 50 miles, with the resulting wire just five microns wide.

· One ounce of pure gold could be hammered into a single sheet covering 9 square meters (96 square feet).

· Gold melts at 1,064 degrees Celsius (1,947 degrees Fahrenheit) and only boils at 2,808 degrees Celsius (5,086 degrees Fahrenheit).

· Over 90 percent of the world’s gold has been mined since the California Gold Rush.

· The number of grams in a troy ounce of gold: 31.103.

· Fort Knox holds 4,600 tonnes of gold (10,141,264.06 pounds); the U.S. Federal Reserve holds 6,200 (13,668,660.25 pounds).

· There was a 394 percent increase in the price of gold from December 2000 to October 2010.

· There are 750 parts per thousand of pure gold in 18-carat gold.

· The largest ever true gold nugget weighed 2,316 troy ounces (158.8 pounds) when found at Moliagul in Australia in 1969; it is called the “Welcome Stranger.”

· Even at only 10 parts of gold per quadrillion, the world’s oceans are estimated to hold up to 15,000 tonnes (33,069,339.33 pounds) of gold.

How does a person calculate the amount of a tip?

Atip, or gratuity, is the money given to a person who performs a service for a customer, such as a waiter or waitress at a restaurant. Depending on the service, in the United States a 10 percent to 20 percent tip is usually left, with the most common being 15 percent, although many people have stories about the 0 percent tip they left after a bad experience. The tip is based on the total bill—the meal and the tax—although some people base the gratuity on just the meal. For example, if a meal at a restaurant costs a total of $10.00 (meal and taxes), a 15 percent tip would be $10 × 0.15 = $1.50. The tip is usually left at the table (or given to the waiter or waitress), or taken out by the establishment and added to a “tip pot” shared by all the wait staff.

There are some mathematical tricks to remember when leaving a tip at a restaurant, to a hairdresser, doorman, or in other appropriate circumstances. A good way to estimate a tip is to round the total bill to the most significant place value. For example, an $18.50 meal would round to $20. Next, move the decimal point of the rounded amount one place to the left ($20 to $2.00), or 10 percent of the total cost. Then divide this amount in half to determine 5 percent (or $2.00/2 = $1.00). Add the two resulting amounts to estimate 15 percent of the total—in this instance, $2.00 + $1.00 = $3.00 tip. (In reality, 15 percent of $18.50 is $2.78, which is close enough to $3.00.)

But remember, not ever country tips the same. Tipping is a way of life in Egypt, but taxi drivers don’t accept tips. French restaurants must add the tip to the bill by law, usually at 15 percent. Tipping in Australia is almost nonexistent; no one tips in mainland China (mainly because the government tacks on enough charges to visitors); there is no tipping in New Zealand, either, as the price usually includes services; and don’t even think about tipping in Japan (it’s almost an insult)!

How is unit price used to determine total price?

The unit price is simply the cost for each item (or unit). The term is often used to compare the cost of the same quantity of items that come in different sizes, or it is used to determine total costs for services. For example, if a person was having a birthday party at a local restaurant for 100 guests, and each meal (unit) cost $7.50, the total cost of the celebration would be $7.50 × 100 = $750.00. Add a tip to the total (see above), and it’s easy to see why most people celebrate at home or only invite a few friends.

How do some businesses price their products to sell?

Did you ever wonder why some businesses price their products with a 99-cent cost at the end of the price? In this case, it appears that how we view numbers in association with money—or interpret our change—can affect the way we buy, especially when it comes to our impulse purchases. According to researchers, consumers are actually more likely to buy a product with the “.99” included in the cost.

To see what numbers most affected the consumer, researchers set up three catalogues, one with traditional prices, one with prices ending in .00, and one with prices ending in .99. They found that the 99-cent catalogues did much better than the .00 one, or the traditional catalogue, even though it only meant a one-penny difference. They believe the reason for reaching for the 99 cent article versus the $1.00 one is the 99-cent ending makes the consumer feel the price is less.

The researchers called the perception of 99-cents the “left and right digit effects.” The study showed that the biggest impact in pricing items at 99 cents comes when it changes the left-most digit in the price; for example, $29.99 instead of $30.00 would be thought of as a “better value” than $2.49 versus $2.50. Other research found that 99-cent pricing in the opposite directions also had an affect on the consumer. For example, with the prices $29.99 versus $29, one out of four consumers would not even notice the .99 after the $29.99—or the right hand digits are ignored, yielding one penny short of an extra dollar for the business.

To further prove that money perceptions of consumers are often skewed, in one college experiment students who compared the prices of $99.99 with $150.00, then compared the prices of $100.00 with $150.00 concluded that the gap between the $99.99 and $150.00 to be significantly larger—even though there was only a cent difference between the two.

What is the lesson? Pay attention to what you’re really paying for, and don’t ignore the 99 cents at the end.

What is balancing a checkbook?

Balancing a checkbook is often a challenge. For some people, forgetting to enter checks written against or deposits made into the account creates the biggest balancing problems . For others, it is not depositing enough money to cover written checks. There really is no “art” to keeping a checkbook. It is just a matter of checks and balances—or debits and credits—and a little bit of simple mathematics.

To keep a healthy checkbook, there are several things a person can do. For example, keep a running balance of distributed checks in a check ledger. Whenever you write a check, write the amount in the ledger booklet most banks give with the checks. In the proper column, list the check number, who the check is made out to (and any other important information), the amount in the negative (-, or debit) column, and subtract the check amount from the last balance.

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Balancing a checkbook involves reconciling one’s bank statement with the record that you yourself have kept track of in expenses and deposits. By keeping tabs on deposits and withdrawals, you will hopefully not become overdrawn on your account.

Along with making out checks (taking out money), keep a record of deposits made in the checkbook register. Deposits are usually written in the positive column (+, or credit). Don’t let the money get low—if the account balance goes into negative numbers, the account does not have enough money to cover the checks. If more money is not put into the checking account at this point, the checks will “bounce,” or not clear with sufficient funds. (This is notgood!) Most banks charge substantial fees to the account owner for bounced checks, not the person to whom the check is made out.

Whenever you receive your bank statement, check to see if the balance agrees with your checkbook. This is called “balancing the checkbook.” As you compare the checks that have cleared with the listing in the register, check each off with an “X” or check mark. Also subtract any bank charges, such as ATM (Automated Teller Machine) fees. If all of the checks have cleared, and all charges have been accounted for, the balance of the checkbook and statement should agree (unless the bank gives interest on checking accounts; if so, add the interest to the checkbook under the “+” or “credit” column). If not all the items have cleared, check the bank statement and note the ones not marked; total all these outstanding transactions. Subtract the total of the outstanding transactions from the end balance on the bank statement; then add any deposits that are not on the bank statement to this new balance. The numbers should match the balance in the check register. If they don’t, go back over the addition and subtraction in the checkbook register to catch any inaccuracies, which is often the reason why a checkbook doesn’t balance.

How do modern cash registers automatically know how much an item costs?

Modern cash registers are actually computers that are able to read a code consisting of a series of vertical bars varying in width. These are called bar codes (or barcode) and represent numbers and other symbols. The bar code is scanned by a laser beam that is sensitive to the reflections from the line along with space thickness and variation. The reader translates the reflected light into digital data that is transferred to a computer for immediate action or storage—usually both—resulting in the addition of items purchased and an immediate inventory for the store. (For more information about computers, see “Math in Computing.”)

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Many store items these days are labeled with bar codes, which use lines of varying widths to indicate numbers that can be read by laser scanners.

But bar codes aren’t only for stores. They are also used to check out books from the library, identify hospital patients, and track manufacturing and shipping movements. There are even very small bar codes used in scientific research, for example, to tag and keep track of honey bees.

What is the Annual Percentage Rate, or APR?

The Annual Percentage Rate (APR) is an expression of the yearly interest rate that will be paid on a loan (which includes credit cards). It differs from the advertised interest rate, as it includes one-time fees in an attempt to calculate the “total cost” of borrowing the money. Therefore, it is wise to always look for a low APR. Because lenders are required to disclose the APR before a loan or credit application is finalized, it makes it easier to compare lenders. For example, if a person borrows $100 for one year at 5 percent simple interest (they will owe $105 at the end of the year), and the lender charges a $5.00 fee, the total cost to borrow the money will actually be $10, meaning the APR is 10 percent.

What are some terms to know when getting a credit card?

There are several terms—many that include a bit of mathematics—one should know when getting a credit card. An important one is the annual fee, which is often charged by the credit card company and is a flat, yearly charge similar to a membership fee. A finance charge is the dollar amount you pay to use credit; it should be listed on your credit card statement. It usually includes the interest on the borrowed money and other charges associated with transactions, such as cash advance fees or exchange rate calculation fees when paying for an item in a foreign country.

As stated in the question above, the annual percentage rate (APR) is a measure of the cost (or relative cost) of credit on a yearly basis. With credit cards, APR commonly includes interest and other charges, such as a yearly rate. Credit cards often offer two types of interest rates, too. In the variable-rate plan, as the name implies, the interest is variable; it is usually tied to other interest rates, such as the Treasury Bill or prime rates. A fixed-rate plan is a rate not tied to changes in other interest rates; it remains steady, unless the credit company raises or lowers rates for everyone, which they can periodically do.

What are the ways credit card companies calculate finance charges?

When the credit card issuer calculates the finance charge on a card, it applies a periodic rate to a balance. In order to calculate that balance, the company uses various methods. The most common is the average daily balance method, in which the balance is calculated by taking the amount of debt in the account each day during a specific period and averaging it. The previous balance method uses the outstanding balance at the end of the period to compute the finance charges. The adjusted balance method derives the balance by subtracting any payments made during the cycle from the previous balance, with new purchases not being counted.

What is a mortgage?

A mortgage is a method of using property as security for the repayment of a loan. It is based on a 14th-century coinage of a Latin word meaning “dead pledge.” The interpretation was that the property was “dead” to the borrower if he defaulted on the debt, and the pledge was “dead” to the lender after the loan was repaid. For centuries, it’s been figured out using mathematics.

What mathematical concept is used to calculate mortgage payments?

In most cases, a mortgage is based on amortization, which is the gradual elimination of a liability (a financial obligation or debt, such as a mortgage) in regular, fixed, systematic payments (such as monthly) over a specific period of time. These payments must be enough to cover both the principal borrowed and the interest. Although it is usually written in a complex set of mathematical calculations, simply put amortization means a part of the payment goes toward the interest cost and the remainder goes toward the principal (or the amount borrowed). The interest is then recomputed on the amount owed, and therefore it gets smaller and smaller as the ending balance of the loan becomes less and less. That is why the homeowner pays a great deal toward interest and not the principal for the first several years of a home mortgage.

How are “points” determined in a real estate transaction?

When buying a home through a real estate group or bank, “points” may be paid by the borrower at the time a loan is made. This is usually to get a lower interest rate, because the lender often offers certain rate/point combinations that may help the homeowner save money. Actually, points usually refer to the commission charged by the mortgage broker or the loan fee charged by the lender when the loan is made. Points can be also negative, in which case they are called “rebates” from the lender to the borrower and are often used by borrowers to defray other settlement costs.

In general, each point is 1 percent of the loan amount. For instance, three points would be equal to 3 percent of the total loan amount. There is no set number of points offered by a lender, as it is not controlled by any laws. For example, on a $100,000 loan, one point is equal to $1,000; 10 points is equal to $10,000. A homeowner looking for a loan should try to find a mortgage broker or lender that charges fewer points. Some financial institutions might even be willing to negotiate for lower points. But beware of lenders offering no or zero points, because they usually charge much higher interest rates than those offering loans with points.

Although it is a matter of mathematics and a person’s budget, there are some very general guidelines when choosing a mortgage. Low-rate, high-point loans are usually used by borrowers who can meet the down-payment cash requirement and either want to stay in a house a long time or want to reduce their monthly mortgage payments. High-rate, low-point combinations are for borrowers who don’t expect to be in their houses very long, or who are short on cash.

For example, if a mortgage is taken out for $100,000 at 6.5 percent for 30 years, the fixed monthly principal and interest payment is $632.07. For the first month, the homeowner pays interest on the $100,000 (or $541.67), with the remainder of the payment ($90.40) going toward principal. In other words, the debt on the principal is reduced by $90.40. By the next month, the homeowner owes interest on a lesser amount of money—on $99,909.60 (or $100,000 - $90.40), not the $100,000, with $541.18 going toward interest and $90.89 going toward principal. As payments are made month after month, the interest decreases and the principal reduction increases. By the 360th payment (or 30 years later), the payment contributes $3.41 to interest and $628.66 to principal.

MATH AND TRAVELING

How are positions on Earth determined?

Positions on Earth are determined using two numbers that represent latitude and longitude. These numbers are actually two angles, measured in degrees (°), minutes of arc ('), and seconds of arc (''). On a globe of the Earth, latitude lines circle parallel to the equator, and differ in length depending on their location. The longest line is at the equator (latitude 0 degrees); the shortest lines—actually pinpoints—are at the poles (90 degrees north at the North Pole; 90 degrees [or -90 degrees] south at the South Pole). In the Northern Hemisphere, latitude degrees increase as you move north away from the equator; in the Southern Hemisphere, latitude degrees increase as you move south away from the equator.

Longitude lines, or meridians (once called “meridian lines” and eventually shortened to “meridians”), are those that extend from pole to pole, slicing the Earth like segments of an orange, with each meridian crossing the equator. In the Western Hemisphere, longitude increases as you move west from Greenwich, England (0 to 180 degrees). In the Eastern Hemisphere, longitude also increases as you move east from Greenwich, England (again, 0 to 180 degrees). All points on the same line of longitude experience true noon (and any other hour) at the same time. But note: Longitude lines are not to be confused with time zones, most of which follow a more erratic demarcation. (For more information on time zones, see below; for more information about latitude and longitude with regard to polar coordinates, see “Geometry and Trigonometry.”)

How does a GPS work?

What is often called a GPS, or Global Positioning System, is actually a receiver that locks on to the signals from satellites in a constellation of 27 Earth-orbiting crafts (only 24 are used, with three as backup). The GPS-receiving device allows people on Earth to determine their positions on the globe—and that’s all about mathematics.

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Solar-powered satellites (with backup batteries in case of a solar eclipse) travel about 7,000 miles per hour and circle the globe twice in just less than 24 hours. As they orbit about 12,000 miles above the Earth, they send signals to the ground that can be received by a GPS. When the GPS locates four or more of the satellites, the data can be used to pinpoint the user’s location through trilateration, or the use of geometric techniques that determine the location in three-dimensional space, which is actually the user’s location in terms of latitude, longitude, and altitude.

Why is Greenwich, England, called the Prime Meridian?

The reason for Greenwich, England, being the Prime Meridian is historical. An imaginary line passes through the old Royal Astronomical Observatory, which was chosen by astronomers of the day as zero longitude. The observatory is now a public museum located at the eastern edge of London. It is a great spot for tourists, who can find there a long strip of brass that stretches across the yard marking the “prime meridian.” Here it is possible to straddle the line with one foot in the Earth’s Eastern Hemisphere and the other in the Western Hemisphere.

Simply put, the receiver compares the time a signal was transmitted by a satellite and compares it to the time it was received, a way of telling the distance to the satellite. From there, it contacts the other satellites, gathering the same information, thus pinpointing the receiver’s location. The reason it works is that any time, anywhere on Earth, there are at least four accessible satellites in orbit, which is why you can determine your location even on such remote places as Mt. McKinley or even Mt. Everest. (In fact, in 1999, the U.S. Expedition Team used a GPS to measure the elevation of Mt. Everest at 29,035 feet [8,850 meters], although this result is still debated.)

How does one convert latitude and longitude to degrees from readings containing degrees, minutes, and seconds?

It takes simple mathematics to convert the degrees, minutes, and seconds of latitudes and longitudes into degrees only. It helps to know that there are 60 seconds in one minute, and 60 minutes in one degree. Therefore, to translate 65° 45' 36'' (also written as 65:45:36) south latitude into degrees you would do the following calculation: -65 degrees (south makes the number “negative”) + 45 minutes × (1° /60') + 36 seconds × (1’/60'') × (1°/60') = -65.76° latitude.

What are time zones?

Time zones are any of the 24 regions on the globe (loosely divided by longitude but more erratic in their demarcations), which is divided according to the number of hours in a day. Within each zone, all clocks are set to the same time. Time zones include the international date line, an imaginary line of longitude generally 180 degrees east or west of the prime meridian; the date becomes one day earlier to the east of this line.

Time zones are truly a product of mathematics. Initially, people used local solar time, resulting in slightly different times between towns. With technology—especially trains and telecommunications—more accurate timekeeping became a necessity. Thus, time zones helped solve many of the problems by setting the clocks of a region to the same mean solar time. The time zones made it easy for neighboring time zones to be labeled “one hour apart.” Not that the system is perfect. The hour separation is not universal, and because they often follow political boundaries, the shapes of the time zones can be extremely irregular.

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The world is divided up into time zones, sometimes rather irregularly; and not all time zones are separated by a full hour— some only have a half hour difference from adjacent zones.

How does one determine the distance and amount of time a trip will take?

There are several ways to determine the distance and amount of time a trip will take. For example, if the traveler knows that his or her car travels 20 miles for every gallon of gas in the tank, he or she could represent the total number of miles it is possible to drive based on how much gas is put into the tank. This can be expressed as 20g, in which g stands for the number of gallons in the tank. For example, if someone bought 10 gallons of gas to reach a destination (10 gallons × 20 miles/gallon), the traveler can go about 200 miles (the gallons in the top and bottom of the equation cancel each other out, leaving miles).

Another example involves cars and mileage. If the traveler drives down a highway at a steady rate of 65 miles per hour, the driver can determine how long it will take to get to a destination, especially if he or she knows how many miles it is to that destination. The equation is mileage divided by 65 miles per hour (m/65). For example, if the traveler is going 650 miles, the (650 miles)/(65 miles per hour) equals 10 hours (the miles in the top and bottom of the equation cancel each other out, leaving hours).

What is the scale of a map?

Most travel, street, or highway maps show a measurement scale, usually in terms of miles and kilometers. To determine a straight line (horizontal) distance on a map, take a piece of paper and mark the origin and destination as tick marks on the paper. Then measure the “distance” between the tick marks based on the map scale to find the distance in miles or kilometers.

Topological maps also have scales, but in this case the scale is a ratio representing the measure on a map to some number of the actual units of measure on the Earth’s surface. For example, a map with a scale of 1:25,000 means that one inch on the map is equal to 25,000 inches on the ground. Because both numbers have the same units, it can also be interpreted as any unit measure. For example, the same map could also be interpreted as 1 centimeter equals 25,000 centimeters on the ground, or 1 meter equals 25,000 meters, and so on. For those who prefer to measure in miles and kilometers, most topographic maps also offer a graphic scale in the legend. (For more about scales, see “Math in Engineering.”)

What is a currency exchange rate?

When traveling to another country, it is important to know the currency exchange rate, which is the value of a traveler’s home currency compared to the currency of the country being visited. Like all currency, the U.S. dollar fluctuates daily when compared to other countries’ currencies. For example, if you travel to Canada, and the U.S. dollar can buy 1.02 Canadian dollars, then the exchange rate is 1.02 to 1. If you go to New Zealand, and the U.S. dollar exchange rate is 0.5477, then 1 New Zealand dollar is worth 54.77 U.S. cents. The daily change in rates is based on many global economic factors, including the economic stability and health of the various countries, the price of commodities such as oil, gas, and gold, and the ups-and-downs of the various stock markets around the world. So before you travel, it’s best to check the Internet for the latest currency exchange calculator to compare your money to the currency rate in the place you intend to travel.

What are some non-Hindu-Arabic numerals encountered around the world?

Although Hindu-Arabic numerals are the dominant numerals used around the world, there are some places in which other number symbols are used. For example, there are Chinese, Japanese (kanji), Greek, Thai, and Hebrew numerals. The following lists some of the ones encountered by world travelers (for more information about numbers, see “Math Basics”).

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Besides our familiar Hindu-Arabic system, numbers are expressed using different characters in such languages as Greek, Hebrew, and Japanese. (For Chinese characters, see table in the chapter “Math Basics.”)