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# Math Fundamentals

150 150 Deborah

### Overview

Percents are based on the decimal system and have many useful applications in business, sales, and finance. Other applications of percents include percent increase and decrease.

### Estimating Percentages

Many of the most useful applications of percents involve estimating and calculating percentages for sales tax and tips at a restaurant. Suppose the sales tax is 8.7% in a particular town. That means that for any taxable item, the cost to buy it is the cost C plus .087C, calculated automatically at the cash register. It is always better to estimate high to make sure there is enough money for a purchase, and it is easier to use a value such as 10%. Then, the price of that item is the cost C plus .10C. If the cost of a CD is \$15.99, estimate \$16.00, and add 10% of \$16.00 or \$1.60, so the estimated cost will be a little less than \$16.00 + \$1.60 or \$17.60. If the cost is higher, the estimated tax will also be higher. Suppose a DVD is \$24.00. Its estimated price with tax will be \$24.00 + \$2.40 or about \$26.40.

### Tips

Estimating tips works on the same principle, except the percentage of tip to leave is up to the diner, and depends on the quality of service and the type of service. Suppose the waitperson brings the check and it is \$25.36, including tax. The quality of service is good, and the diners decide to leave a 20% tip. The cost of the meal will then be the price P plus 0.20C. It is easier to estimate the tip by adding 0.10C + 0.10C, or \$2.54 + 2.54, so the cost of the meal is \$25.36 + \$5.08 or \$30.44.

### Commissions and Fees

Commissions are typically a percentage of total sales and are either added to a base salary or paid as commission only. Suppose that Lori works in a salon, and her employer pays her \$9.50 an hour plus a 1.5% commission on all products she sells. Her gross wages after a 40-hour work week are \$9.50 times 40, or \$380.00. Suppose she sells \$125.00 in products during that week. Her commission will be \$125.00 times 0.015, or \$1.88. (When using percents in a calculation, change the percent to a decimal by multiplying by 1/100 and using decimal points.) \$380.00 +\$1.88 = \$381.88.

### Calculating Simple Interest

Simple interest follows a formula of I=Prt, where I is the amount of interest, P is the principal amount of money invested or loaned, r is the interest rate, given as a percentage, and t is the amount of time in years. It is only paid on the principal. In order to find the amount of simple interest paid on a loan of \$1000.00 at a rate of 15% over 2 years, first substitute the known values in the formula such that I = \$1000.00 (0.15)(2). Again, the 15% is changed to a decimal 0.15. Solving the equation, I = \$300.00.

### Percent Increase and Decrease

Percent change refers to the amount of increase or decrease given as a percentage. Percent increase is often referred to as a markup, and percent decrease is a discount. Suppose the regular price of an item is \$12.00 and it is being sold at 30% off. In this case, the sale price is equal to the regular price P minus the discount of 30% P, or the sale price is equal to \$12.00 – (0.30) P or \$12.00 – 3.60 = \$8.40.

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150 150 Deborah

### Overview

Percents are special types of ratios that compare the number to 100. Similar to any ratio, a percent can be expressed as a fraction. Since percents compare to 100, they can also be expressed in decimal form.

### Fractions to Percents

Percent or the percent sign (%) means “per 100,” so 5% means the same as 5/100, 31% means 31/100 and so on. Many common fractions can be expressed as percents by converting the fraction in terms of 1/100, so ½ is 50/100 or 50%, ¾ is 75/100 or 75%, 4/5 is 80/100 or 80%. Percentages, like fractions, can be greater than 1, so that something that is 200% is two times as much, just as 200 is twice as much as 100.

### Decimals to Percents

Any decimal can be expressed as a percentage, and any percentage as a decimal. (This can be a handy shortcut if the calculator does not have a percent key). For example, 20% is 0.20, and 7% is 0.07. Similarly, 0.003 is .03%, and 3.12 is 312%. A percentage such as 120% can be expressed in decimal form as 1.20.

### Percents as Proportions

Percents can be expressed by the formula part/whole =percent/100, so that they can be compared as 2 equivalent ratios, the definition of proportion. Crossproducts can be used to then solve the equation. Suppose the problem were to find 30 is 20% of what number. 30/x = 20/100 is the equation for the proportion, or 20x = 30·100, or 3000. Dividing both sides by 20 to isolate the variable, x = 3000/20, or 150. Similarly, find 50% of 50. In this case, 50/100 = x/50 gives the proportion, so 100x =2500, or x= 2500/100, or x = 25. Also, 30 is what percentage of 150? This is also a proportion, as 30/150 = x/100, or 150x =3000, or x=3000/150, x = 20%.

### Applications of Percents

Percents are used in many different ways. For example, if a ring is 24-karat gold, it is nearly 100% pure gold. It is too soft for most jewelry, so most gold is in the 10-karat (41.7% pure gold) to 18-karat (75% pure gold) range. Suppose a ring weighs about 10 g. If it were 10 karat, 4.17 g would be gold and the rest of the ring would be other metals, but if it were 18 karat, 7.5 g would be gold. Percents are often used in sale prices, as items are often on sale for 30% off or 40% off.

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150 150 Deborah

### Overview

Not all equations can be solved in one step. Many take several steps to solve, but the principles of solving one-step equations can be applied. Use the order of operations to simplify the equation; math properties to combine like terms; then apply inverses to isolate the variable on one side of the equation in order to solve it.

### PEMDAS to the Rescue

A messy equation can almost always be cleaned by applying the order of operations. Start simplifying the equation by solving within parentheses, clearing exponents, multiplication, division, addition and subtraction. Sometimes the order of operations will suggest shortcuts. Suppose a problem were (3x –x) +2 =8. Solving within the parentheses, 3x –x =2x + 2 = 8.

### Review Math Properties

The commutative properties of addition and multiplication; associative properties of addition and multiplication, and the distributive property of multiplication over addition or subtraction are good strategies when solving equations. Students are able to apply them to tame an equation.

### Combine Like Terms

Whenever possible, combine like terms. They become obvious when they are not hiding behind parentheses, standing next to one another in the equation. For example, an expression such as 5x +2 + 3x +4 +6y +3-y can be simplified to 5x + 3x + 6y –y +2 +4 +3 or 8x +5y +9. After like terms are combined, the variable can be isolated on one side of the equation and solved. If 2x +2 = 8, then 2x +2 -2 =8-2 or 2x =6. Both sides can then be divided by 2 so that x = 6/2 or 3.

### Multi-Step Equations Represent the Real World

Many problems in the everyday world are modelled by equations that represent more than one step. Suppose that the regular price of a gallon of gasoline is \$3.00 a gallon, but a discount card gives 10 cents less per gallon. If a car has an 11 gallon tank, and it is empty, how much will it cost to fill the tank? The first step in solving the problem is to find the cost of the gasoline with the discount card. \$3.00 – 0.10 = 2.90 per gallon. The second step is to multiply \$2.90 ·11, the number of gallons, or \$31.90. In this problem, the x was already isolated on one side of the equation.

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150 150 Deborah

### Overview

Some algebra equations can be solved with one step, using operations such as addition, subtraction, multiplication, and division. In order to solve them, use inverse operations so that the variable is on one side and real numbers are on the other side. Then, solve the problem with arithmetic.

### Balanced Equations

An equation is solved when both the left side of the equation and the right side of the equation mean the same thing. Take a math fact such as 2 + 3 = 5. It is true for real numbers because 2 + 3 equals 5, and 5 equals 5. Suppose the equation were 3 +7 =x. In order for the equation to be balanced on both sides, x = 10, because 3 +7 equals 10 and 10 equals 10.

### Inverse Operations

Addition and subtraction are inverse operations; and multiplication and division are inverse operations. That is because adding a number and its inverse equal the additive identity, or zero; and multiplying a number by its inverse equal the multiplicative identity or 1. In math symbol language a – a =0, and a/a, as long as a doesn’t equal zero, is 1. Adding or subtracting zero doesn’t change anything, and multiplying or dividing by 1 doesn’t change anything. The equation stays balanced by using the inverse on both sides.

### Solving Problems by Adding or Subtracting

An equation such as 12.5 +7.2 = x is easy to solve because the variable x is already on one side and the real numbers (also called constants) are on the other. By adding up the constants, 12.5 + 7.2 = 19.7, so x equals 19.7. Suppose the equation were 2 + x = 5. Keep the equation balanced by using the same inverse on both sides. Since it is an addition problem, the inverse of addition is subtraction. In this example, 2 -2 +x = 5 -2, or 0 + x = 5-2, or x =3. Similarly, y -3 = 10 remains balanced when y – 3 +3=10 +3, or y +0 = 10 +3, or y = 13. The equation is a subtraction problem, and the inverse of subtraction is addition.

### Solving Problems by Multiplying or Dividing

In an equation such as 4·11 = z, z equals 44. The constants are already on the left side and the variable is already on the right side of the equation. Suppose the equation were 9x =72. In that case, it is a multiplication problem, and the inverse of multiplication is division. 9 ·1/9·x = 72 ·1/9, because 9 ·1/9 ·x = 1x, or x, and 72/9 = 8, so x = 8. Similarly, x/3 = 12 becomes 3 ·1/3 ·x = 12·3 or x = 36.

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150 150 Deborah

### Overview

Students can use the commutative, associative, and distributive properties of addition and multiplication to simplify algebraic expressions. Problems are more easily solved if compatible terms can be combined first. Some problems can then be solved mentally.

### Compatible Terms

Real numbers, whether whole or mixed, are constants that can be added, subtracted, multiplied or divided. The only exception is dividing by zero, which is undefined. Compatible terms can also be simplified, as long as each term has the same variable or variables at the same degree. The terms 2x + 4x can be combined to equal 6x, and the terms 3y2– 8y2 equal -5y2. However, 2x + 3z cannot be added, and 3x3 + 2x -4x is in its simplest form as 3x3 –2x.

### Commutative Property of Addition and Multiplication

According to the commutative property of addition or multiplication, the order of adding or multiplying numbers does not matter, as long as all of them are added or multiplied. For example, 30 + 40 = 70, and 40 +30 = 70. The law in math symbol language is a + b = b + a for addition, and ab=ba for multiplication. Similarly, 2x + 3y +3x – 2y is the same as 2x +3x +3y -2y. The commutative property is a useful tool in mental math. Suppose the column of figures is 51 + 25 + 25 +49 + 50. It can be rearranged as 51 +49 +25 + 25 +50 to equal 200.

### Associative Property of Addition and Multiplication

According to the associative property of addition or multiplication, the way numbers are grouped doesn’t matter, as long as all the numbers are added or multiplied. In math symbol language, (a + b) + c = a + (b +c).   The associative property is also a tool that can be used in mental math. When the mental math problem 51 +49 +25 + 25 +50 was rearranged, it was also grouped, as (51 + 49) + (25 +25 + 50). Solving within parentheses, 100 +100 =200. Both the commutative and associative problems were used together to solve the problem.   An expression such as 4x + 3y – .5y +7x can be rearranged as 4x +7x +3y -.5y and then regrouped as (4x +7x) + (3y – .5y) to equal 11x + 2.5y.

### Distributive Property

The long formal name of the Distributive Property is the Distributive Property of Multiplication over Addition, or the Distributive Property of Multiplication over Subtraction. In math symbol language, it means that (a +b)c = ac + bc, or (a-b)c = ac – bc. This property is very useful, because it allows monomials to be combined that use the same variables. For example, 3y + 2y = 5y because (3 +2)y equals 5y. Similarly, 17a -11a =6a because (17-11)a = 6a.

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150 150 Deborah

### Overview

Operations with real numbers are the most familiar types in mathematics. Those numerical expressions with constants and operations include such actions as addition, subtraction, multiplication, and division.   Many algebraic expressions are usually operations with real numbers.

### Real Numbers

The set of real numbers include the entire set of rational numbers, such as positive and negative integers, whole numbers, and natural numbers. Positive and negative fractions are rational numbers, because the numerator and denominator of the fraction represents a ratio. Whole numbers are rational numbers, because they are ratios in simplified form. For example, the fraction 4/1 is usually represented as 4. Irrational numbers, such as pi π, the constant e, √7, or √2 are also real numbers, but imaginary numbers and the infinities are not.

Addition of real numbers can be represented on the number line by moving right on the number line. For example, 4 +3 =7 can be modeled on the number line by starting at 4 spaces from 0 and then moving right 3 more spaces. Fractions can be added by finding their common denominator, then adding the numerators. For example, ½ +1/3 = 3/6 +2/6 = 5/6.

### Subtraction of Real Numbers

Real numbers can be subtracted on the number line by moving to the left. Subtraction is the same as adding the additive inverse, so that 7-5 means the same thing as adding 7 + (-5). The additive inverse of a number is the opposite of that number, so that the additive inverse of 3 is -3, the additive inverse of 1/5 is -1/5, and the additive inverse of -12 is 12. Fractions can also be subtracted by adding the additive inverse, as long as the common denominators are found.

### Multiplication and Division of Real Numbers

In the most common model of multiplication and division, multiplication is repeated addition and division is repeated subtraction. When multiplying two positive or two negative numbers, the product is positive, but if one number is positive and the other is negative, the product will be negative. Division by zero is undefined, the product of any number and 0 is zero, and the quotient of 0 and any nonzero number is 0.

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150 150 Deborah

### Overview

The language of algebra is written in symbols, numerals, variables, constants, and expressions. In order to be successful in math, students should be able to translate its language into words and words into math language. Numerals and many symbols should already be familiar, as well as many operations with real numbers.

### Common Symbols and Numerals

Several mathematical symbols should already be familiar, such as + for addition, – for subtraction, x or ·for multiplication, and ÷ for division. Similarly, most students are familiar with = for equals, < for less than, and > for greater than. The integers 0, 1, 2, 3 … are common, as are fractions such as ½, 1/3, ¼, 2/3, and so on. By the time students begin studying algebra, they are familiar with working problems that involve basic operations, as well as some that aren’t so basic.

### Variables, Constants, and Expressions

Variables are letters or symbols used to represent any value that can change, depending on the needs in the expression. Constants do not change. They may be numerals or particular values. For example, the letter pi π is a constant that always means the same thing, 3.1416, give or take a few decimal places. Numerical expressions contain only constants and symbols for operations, such as 2 +2 = 4. Algebraic expressions are similar, but they also contain variables, constants and symbols for operations. While 2 + 2 =4 is a numerical expression, x +y = 4 is a simple algebraic expression. It contains two variables, x and y; one constant, 4; and two symbols for operations, + and =.

### Translating from Symbols to Words

In order to translate from symbols to words, understand what the symbols mean and analyze them. For example, the expression “x +2” can mean the sum of x +2 or x increased by 2. The expression “5 –a” can mean the difference of 5 and a, or a less than 5. The expression “8z” can mean 8 times z or the product of 8 and z. The expression “y/5” can mean y divided by 5 or the quotient of y and 5.

### Translating from Words to Symbols

In order to translate words into mathematical expressions, such as for a word problem or real-life application of math, analyze the statement in words and the relationships between them. Drawing a picture can be a useful strategy to translate a problem into an algebraic expression. Certain words and phrases mean particular operations. Are things to be put together? Addition might be the operation. Is one thing more or less than another? Subtraction might be needed. Are equal numbers of things to be grouped? Multiplication might be the key. If equal groups need to be separated, they are divided.

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150 150 Deborah

### Overview

The binary (Base 2) system is based upon powers of 2. It has many applications to situations that have only two alternatives, such as off and on, 0 and 1, yes and no. Electronics, computers, and computer-based hardware such as cell phones and satellites use the binary system.

### Switching

The binary system was adapted in the 17th century by a mathematician named Leibniz from a system used extensively in China. It was quickly adopted by engineers, because 0 and 1 could represent a switch turned on or off. That simple on-off code could be used by sequences of electric lights hooked to vacuum tubes in early computers. As technology developed, vacuum tubes were replaced by transistors, as well as tinier and tinier microchips and circuits. The most powerful computers in the 1950’s and 1960’s filled entire rooms. Personal desktops and laptops today are just as powerful, taking a fraction of the space.

### How Computers Calculate

Computers calculate very rapidly, and can use many shortcuts to common operations. The traditional method for adding binary numbers can be cut down by many operations by a shortcut called the “long carry method.” A system of advanced algebra based on Boolean logic and logical operations cuts the number of steps still further.

### Bits

Bits, or binary digits, can be represented by different voltages, different polarities of a magnetic disk, or other alternatives. Bitwise operations and patterns provide even faster applications. Think of a bit as one letter in a word, 8 bits as a byte (B), 2 or 3 paragraphs of text as a kilobyte (KB), four 200-page books as a megabyte, and so on.

### Smartphones and Other Applications

Smaller and smaller chips have been used to integrate computer processes with mobile phone technology, producing smartphones. As new operating systems are developed, new generations of mobile devices are running smaller, faster, and ever smarter. Newer forms of computing architecture such as quantum computers and neural networks are in development.

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150 150 Deborah

### Overview

Mathematical patterns underlie everything in the natural world, including those human extensions of the natural world that surround us. Some of those mathematical patterns are expressed in the arts, including visual arts such as painting and sculpture, and the physics and mathematics of music. Other mathematical patterns are expressed in performance arts, such as the choreography of dance.

### Patterns in Art

Renaissance artists represented mathematical patterns and concepts in visual form, using mathematical concepts such as symmetry and perspective. A two-dimensional canvas is used to portray three-dimensional surfaces. Painters used tricks of perspective, optical illusions of geometry such as the vanishing point and the horizon line. They were especially skilled at making objects appear to converge at a single point, just as people might see parallel train tracks in the distance. Those converging points fall along the horizon. Also, objects appear larger the closer they are and smaller the farther away they are, even though the real size of the objects has not changed. Symmetry was also important to create pleasing patterns of objects or shapes in a painting or artwork made of mosaic, fiber, or glass.

### The Golden Rectangle, Golden Ratio, and Standards of Beauty

From the time of the ancient Greeks, through the Renaissance. The Golden Ratio, also known as the divine proportion, was represented by the Greek letter phi φ. It is an irrational number, about 8/5. Sculptors used it to determine the most pleasing lines in their works. The Golden Rectangle is related to the Golden Ratio. In the Golden Rectangle, the long side of the triangle is φ times as long as the short side. Leonardo da Vinci used the Golden Rectangle in his paintings and sculpture, putting important figures inside one. Painters from Seurat to Mondrian, as well as painters and artists of the present day, use the relationships of the Golden Ratio and the Golden Rectangle in many of their works.

### Patterns in Music

Pythagoras wrote extensively about the geometric foundations of music, the relationships of pitch, modes, and octaves. For example, if a string of a certain length is plucked, dividing it into shorter lengths will produce a higher pitch, as anyone who has ever played a stringed instrument knows. Similarly, a Pan pipe consists of four or five wooden recorders of different lengths that are bound together. Octaves are pleasant because of the relationships between their pitches and frequencies. The Greeks used a 7-note diatonic scale, in various relationships between tones, called modes. Modes predated the 12-tone scale that was well-tempered by Johann Sebastian Bach. Time signatures are standardized, and the length of time each note is played in a time signature can be represented by fractions and patterns.

### Patterns in Kinetic Arts

Dance and other kinetic arts have mathematical relationships of their own, modeled by dynamic equations and the behavior of solid objects in space. Think of the parabolic leaps made by dancers as they seem to fly across the stage and pause in midair, or the relationships between dancers as they cross the stage, from ballroom dancing to hip-hop.

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150 150 Deborah

### Overview

Mathematical patterns underlie human activity, as far back as we can discover ancient artifacts. Architectural wonders such as the Parthenon, the Pyramids, and other ruins show that civilizations were able to apply geometric and mathematical relationships. Neolithic sites as far away as Newgrange, Stonehenge, and many others around the world, show an ancient grasp of complex mathematical calculations. Artifacts from the smallest decorated cup to the largest mosaics and tessellations show an appreciation of symmetry and balance.

### Mathematical Patterns in Architecture

The ancient Greeks built the Parthenon and other complex building as monuments. Not surprisingly, the Parthenon, their most important temple on a high hill overlooking their capital city, is based on their divine proportion phi φ, also called the Golden Ratio. The Golden Ratio is also evident in the Great Pyramids of Egypt. Mayan temples in Central America show complex mathematical relationships that reflect their intense interests in mathematics and astronomy.

### The Mystery of Stonehenge

Stonehenge is a ring of standing stones as part of the Amesbury complex. It is not known how the Neolithic builders used it, but the rings have relationships to each other, as well to the position of the sun during the summer and winter solstices and the spring and fall equinoxes. Other standing stones exist in many other parts of the world, such as Newgrange in Ireland, Carnac in France, Cueva de Menga in Spain, and other locations throughout Europe.

### Patterns and Symmetry in Mosaics and Tessellations

Many civilizations around the world have developed intricate patterns of mosaics, as well as repeating tile patterns called tessellations. Tiles have been found in many different areas around Mesopotamia, including some intricate mosaic flooring in ancient temples, homes, and other public buildings. Mosaics were used in ancient Greece and Rome, as well as throughout the Byzantine Empire. Tessellations, or repeat patterns, are found all over the world, with some of the most beautiful and intricate patterns in old Seville, Spain, through Moorish architecture.

### Patterns and Symmetry in Textiles and Fiber Arts

The earliest examples of weaving show a knowledge of symmetry and design. For example, symmetrical embroidery stitch patterns are echoed in traditional embroidery from East Asia, throughout India, throughout Egypt and the Middle East, to be duplicated in forms throughout Europe. The Hardanger embroidery techniques found in the Scandinavian countries are based upon multiples of threads and complex geometric relationships.

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