BASIC MATHEMATICAL CONCEPTS

Transcription

1 CHAPTER 1 BASIC MATHEMATICAL CONCEPTS Introduction Science and Technology rely on accurate measurements and calculations. In order to acquire mastery in mathematical operations, it is important to have good understanding of basic properties of numbers and operations. In this chapter we will study basic mathematical concepts, symbols and operations. Objectives Upon completion of this chapter, you should be able to: Describe the concepts of numbers, numerals and numbering systems. Illustrate and differentiate the characteristics of whole numbers, integers, decimals and fractions. Describe and explain the properties and usage of number ZERO. Use math symbols to compare numbers and indicate arithmetic operations.. Explain the properties of basic mathematical operations. Describe the basic rules of the decimal system. Explain the significance of measurements, quantities and units. Translate simple word statements to mathematical expressions and viceversa. 1.1 Numbers and Numbering Systems Definitions Number: A number is an expression of quantity. We use numbers to represent quantities. Examples: the number of people in a classroom, the length of an object ( with given units ), the weight of an object. Numeral: A numeral is a symbol used to represent a number. The same number can be written in different ways. Example: 5 (Arabic numeral) or V (Roman numeral) Digit: A digit is any of the numerals 0 through 9 used to represent numbers. ( from Latin, digitus = finger) ELPT

2 Example: the number 34 is a two-digit number, the number 751 is a three-digit number. Numbering system: A set of rules which determines the symbols, base and weights and regulates arithmetic operations. The decimal system is the most widely used in everyday life; however, binary, octal and hexadecimal systems are used by computers and programmable controllers. Example: the number 26 in the decimal system (base 10) can be written as in binary (base 2), 32 in octal (base 8) and 1A in hexadecimal (base 16). Whole number: A whole number is a complete unit, containing no fractional parts. Whole numbers are considered positive. Example: 0, 1, 2, 3, etc. are whole numbers. Natural or counting numbers: The natural or counting numbers are the numbers used for counting. Example: 1, 2, 3, 4, etc. are natural or counting numbers. Integers: An integer is a whole number but it can be positive or negative. Example: 1, 3, -2, 4, -7 are integers. Fractions: A fraction is a portion of a whole amount. It describes into how many parts the whole is divided and how many of those parts are used. Example: the fraction 3/4, means the whole amount is divided into 4 parts and 3 are used. Mixed number: A mixed number contains a whole number and a fraction. Example: 1 ½ is a mixed number. Decimal numbers: A decimal number (sometimes called decimal fraction) represents the fractional part of a number. It is a fraction written in a decimal form. ELPT

3 Example: the fraction 3/4 can be written as 0.75 or.75. Rational numbers: A rational number is a number that can be expressed as a fraction or ratio of two integers. Example: ¾, 7/11 and 89/100 are rational numbers. TIP: All integers are rational numbers since they can be written as the ratio of the number and one. Example: 4 can be written as 4/1. Irrational numbers: An irrational number is a number that cannot be expressed as a fraction or ratio of two integers. Example: the number pi, written B = cannot be expressed as a fraction or ratio of two integers and it is an irrational number. Real numbers: Real numbers are the set of all numbers that we can possibly use in most common mathematical operations; the real numbering system consists of integers, rational and irrational numbers. Decimal point: The decimal point separates a number in two parts: the whole number appears to the left of the decimal point and the fraction appears to the right of the decimal point. Example: the number 2.75 is two whole units and 3/4 (.75) of the whole unit. TIP: A number written without a decimal point is a whole number; the decimal point, although not shown, is understood to be after the number. Thus, the number 12 is the same as 12.. Exercise Write T for true or F for false next to each expression. 1] All whole numbers are integers. 2] Negative integers are whole numbers. 3] All rational numbers are integers. 4] 3/4 and ½ are fractions. 5] 1/10, 4 and ½ are rational numbers. 6] All integers are rational numbers. 7] Rational numbers are real numbers. 8] 1\2, 4, and 1 are irrational numbers. 1 9] 3 2 is a mixed number. 10] Irrational numbers can be written as fractions. ELPT

4 The Number ZERO The number zero by itself indicates the absence of quantity. When used with other digits, it can indicate the size of a number or hold digits in their proper places. Example: 430 is not equal to 403, 0.05 is not equal to 0.5 It is important to understand the role that number zero plays in various numbers. A zero placed to the left of a whole number doesn t change the number; however, if it is placed to the right of a whole number, it changes the number. Example: 18 equals 018, but 18 is not the same as 180 A zero placed to the right of a decimal number doesn t change the number. Example: 0.23 = A zero placed between the decimal point and a number does change the number. Example: 0.23 is not the same as , and 4.5 is not the same as 40.5 A decimal fraction (without a whole number) can be written with or without the zero to the left of the decimal point. Example: 0.34 =.34 Exercise Rewrite each group of numbers in ascending order; i.e. 1 smallest number first, largest number last Example: Given the group 4.5,.5, 5, 1.5 write.5, 1.5, 4.5, 5 1] 6, 1.6, 0.7, 6.7 2] 12, 4, 9, 19 3] 100, 23, 78, 98 4] 760, 706, 697, 907 5] 1.5, 4.5, 1.45, ] 18, 1.8,.18,.018 7].01,.005,.5,.03 8] 408, 804, 480, , 12, 14, 10] 39, 139, 319, : Latin, id est meaning that is (to say). ELPT

5 Exercise Rewrite each group of numbers in descending order; i.e. largest number first, smallest number last. Example: Given the group 4.5,.5, 5, 1.5 write 5, 4.5, 1.5,.5 1] 98, 88, 108, 18 2] , 4, 12, 2 4 3] 6.1, 5.8, 5.9,.61 4] 10, 45, 59, 290 5] 1.7,.17, 17,.017 6] 120, 180, 12, 18 7] 601, 106, 160, 610 8] 2.9,.9,.09, 3 9] 17, 117, 170, ] 6,.6, 6.6, Basic Math Symbols Symbols are used in math, science and technology to denote equality, operations, indicate and compare numbers and quantities and to express formulas. Symbols of Equality, Inequality and Comparison The most common symbols are: = is equal to is not equal to > is greater than < is less than $ is greater than or equal to # is less than or equal to Examples: a = b means number (or quantity) a is equal to number (or quantity) b a > b means number (or quantity) a is greater than number (or quantity) b ELPT

6 x # y means number (or quantity) x is less or equal than number (or quantity) y a = 3 means number (or quantity) a is equal to number 3 or has a value of 3 7 = is a true statement = 6 is not a true statement 8 > 6 is a true statement 5 < 2 is not a true statement 0.2 =.2 is a true statement TIP: When using > and < to compare numbers, the number located on the wide end of the symbols > and < is the larger one. Example: 5 > 3, 5 is larger than 3; likewise, 4 < 9, 9 is larger than 4. Exercise Write T for true or F for false next to each expression. Example: Given 5 > 4 write T 1] 18 > 16 2] 48 < 84 3] 30 > 31 4] 9 < 8.9 5] 4.2 > ] 16 < 14 7] 120 < 115 8] 874 < 784 9] 13 < ] 0 < 1 Exercise Use =, > or < to compare each pair of numbers. Example: Given the pair 4 7 write 4 < 7 1] ] ] ] ] ] ] ] ] ] ELPT

7 Exercise Use the equal to ( = ) symbol or the not equal to ( ) symbol to compare each pair of numbers. Example: Given the pair write ] ] ] ] 7,002 7,020 5] ] ] ] ] ] Find a number to complete each number sequence. L O H ACTIVITIES N F 7 1] ] ] ] ] Operational Symbols The basic mathematical operations are addition, subtraction, multiplication and division; they are represented by the following symbols: Addition plus 4 Subtraction minus 2 Multiplication x or * 12 x 3 12 times 3 12 * 3 3 Division / or 15 / 3 15 divided by 3 or 3 divided into 15 or _ ELPT

8 Exercise Perform the following operations 1] = 2] = 3] = 4] 18 / 6 = 5] 4 x 6 = 6] 6 / 18 = 7] 9 3 = 8] = 9] 10 * 10 = 10] = Definitions Variable: A variable is a quantity that can have any value. Variables are usually represented by letters. For example: the letter v can be used to represent velocity; the letter P is used to represent power. Equation: An equation is a mathematical statement that indicates that two quantities are equal. For example: = 7 is an equation. Inequality: An inequality is a mathematical statement that indicates that two quantities are not equal. For example: 5 7 and 8 > 3 are inequalities. Formula: A formula is an equation involving two or more variables. For example: v = d / t is a formula to find velocity ( v ) when the distance ( d ) and the time ( t ) are known. Grouping Symbols Parenthesis ( ), brackets [ ] and braces { } are used to group numbers and variables. They indicate the priority of operations; the operations indicated inside these symbols are performed before any other operations. For more complex expressions parenthesis are used first, followed by brackets and braces. For example: in the expression 4 x ( ), is performed first; the result is multiplied by 4. 4 x ( ) = 4 x 7 = 28 ELPT

9 Order of Operations For expressions that contain a combination of parentheses, addition, subtraction, multiplication and division, the order of operations is as follows: 1] Parentheses 2] Powers and Square Roots ( to be studied later ) 3] Multiplication and division - whichever comes first from left to right 4] Addition and subtraction - whichever comes first from left to right. For example: the expression 2 ( ) is solved by multiplying 3 and 6 first, then 4 is added and the result is multiplied by 2. 2 ( ) = 2 ( ) = 2 ( 22 ) = 44 Perform the following operations Exercise ] 3 x ( 12-4 ) 2] ( ) / ( 4-1 ) 3] 14 - ( ) 4] ( 14-6 ) x ( ) 5] ( 10 x 5 ) / ( ) 6] 3 x ( 8-5 ) x ( 7-2 ) 7] ( ) x ( 5-0 ) 8] 16 / ( 9-5 ) 9] 1 / ( ) 10] ( 2-1 ) x ( ) / ( 7-5 ) 1.3 Basic Properties of Mathematical Operations Definitions Notation of Positive Numbers: If a number is written without a + or - sign preceding the number, the number is positive. ELPT

10 For example: 24 is positive and can be written as Opposite, negative or additive inverse of a number: The opposite, negative or additive inverse of a number is the number written with opposite sign. For example: the opposite of 24 is - 24 ; the opposite of - 3 is + 3 or simply 3. Likewise, the opposite of the variable a is - a. Reciprocal or multiplicative inverse of a number: The reciprocal or multiplicative inverse of a number is the number divided into 1 ( 1 divided by the number ). For example: the reciprocal of 4 is 1/4; the reciprocal of the variable x is 1 / x Multiplication notation: Multiplication can be represented in various forms using the symbols x, * ; when two quantities are shown together and are not separated by the +, -, / or signs, multiplication is implied. For example: 4 = 3 x 4 = ( 3 4 = ( 4 ) = ( 3 ( 4 ) = ( 3 ) ( 4 ) Likewise, 2 x a = a = 2a TIP: When the letter x is used to represent a variable, use the raised to indicate multiplication. Coefficient: When a number and a letter are multiplied, the numerical portion of the answer is called the coefficient and is always written in front of the letter. For example: ( x ( 3 ) = 3x Write T for true or F for false next to each expression. Exercise ] The opposite of - 6 is 6. 2] The reciprocal of 4 is ] - 14 is a positive number. 4] 17 is a positive number. 5] ( 3 + 4) ( 2 ) = 72 ELPT

11 Basic Properties of Addition Commutative Property: The result is the same regardless of the order of addition. For example: = ; likewise, x + y = y + x Associative Property: The whole is the sum of the parts. For example: = ( ) + ( ) = 3 + ( ) = 20 Identity Property: Adding zero to a number does not change the number. Similarly, subtracting zero from a number does not change the number. For example: = 3, a + 0 = a, 4-0 = 4 and x - 0 = x. Inverse Property: Adding the opposite, negative or additive inverse of a number to the number equals zero. For example: 3 + ( - 3 ) = 0 and a + ( - a ) = 0. TIP: The Commutative, Associative and Inverse Properties are not valid for subtraction. Consider the following examples: ( 4-2 ) ( 12-4 ) ( - 4 ) 0 Write T for true or F for false next to each expression. Exercise ] x + y = y + x 2] ( ) = ( ) + ( ) 3] 12-0 = 12 4] 15-3 = ] 5 + ( - 5 ) = 0 Basic Properties of Multiplication Commutative Property: The result is the same regardless of the order of multiplication. For example: 4 = 3 ; likewise, ab = ba Associative Property: The whole product is the product of the parts. For example: 5 = ( 4 ( 5 ) = ( 5 ) = 480 Identity Property: Multiplying a number by one does not change the number. Similarly, dividing a number by one does not change the number. ELPT

12 For example: 1 = 3, a ( 1 ) = a, 4 1 = 4 and x / 1 = x. Inverse Property: Multiplying a number by its reciprocal or multiplicative inverse equals one. For example: ½ = 1 and a ( 1 / a ) = 1. TIP: The Commutative, Associative and Inverse Properties are not valid for division. Consider the following examples: / ( 4 / 2 ) ( 12 / 4 ) / 2 2 / ( 1/2 ) 1 Exercise Write T for true or F for false next to each expression. 1] 4 = 6 2] 3 ( 1/3 ) = 0 3] 15 5 = ] x ( 1 / x ) = 1 5] 10 ( 10 ( 10 ) = ( 4 ( 10 ) Distributive Property of Addition, Subtraction and Multiplication This property allows the distribution of the quantity outside the parenthesis to the elements inside it. For example: 3 ( ) = = = 27 Likewise, a ( b + c ) = ab + ac and a ( b - c ) = ab - ac TIP: The distributive property is not valid for multiplication inside the parenthesis. Consider the following example: ( 5 ) 5 Exercise Use the distributive property to perform the following operations. For example: Given 3 ( ) write = = 27 1] 4 ( ) = 2] 5 ( 8-3 ) = 3] 3 ( 7-4 ) = 4] 3 ( a + b ) = 5] 10 ( ) 6] x ( ) = ELPT

13 7] a ( b + c ) = 8] a ( b - c ) = 9] 4 ( ) = 10] 0 ( ) = Operations with ZERO Addition: Adding zero to any number does not change the number. For example: = 3 and x + 0 = x Subtraction: Subtracting zero from any number does not change the number. For example: 5-0 = 5 and a - 0 = a Multiplication: Any number multiplied by zero equals zero. For example: 0 = 0 and 0 = 0 Division: Zero divided by any number equals zero. For example: 0 9 = 0 and 0 / x = 0 Division by zero is not allowed. No number can be divided by zero. For example: the expressions 9 0 and x / 0 are not defined. Be assured that the answer is not ZERO! Exercise Perform the following operations: 1] = 2] 0 ( 3 ) = 3] 7-0 = 4] 0 = 5] ( 0 ) ( 4 ) = 6] 0 10 = 7] 0 / 6 = 8] a + 0 = 9] = 10] 3 0 = 1.4 Basic Rules of the Decimal System Digits: There are ten (10) digits or different symbols used to express any number in the decimal numbering system: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Thus, 0 is the smallest digit and 9 is the largest. Base: The base used in the decimal numbering system is 10. The weight or place value of digits is determined ELPT

14 by this base. Weight: The weight or place value of a digit is determined by its location within the number. For whole numbers, going from right to left, the first digit to the right (the right-most position) has a weight of one (1), the second digit has a weight of ten (10), the third digit has a weight of one hundred (100), and so on. Thus, in the number 472, the 4 has a weight of 100, the 7 has a weight of 10 and the 2 has a weight of 1; so the number 472 can be written: 4 hundreds 7 tens 2 ones or, 4 x x x 1 or Likewise, the number 8,306 can be expressed as: 8 x x x x 1 or For decimal numbers, the whole number portion (appearing to the left of the decimal point) follows the same rules for whole numbers outlined above. The fractional part (appearing to the right of the decimal point), going from left to right, the first digit (to the right of the decimal point) has a weight of one-tenth (1/10 or 0.1), the second digit has a weight of one-hundredth (1/100 or 0.01), the third digit has a weight of one-thousandth (1/1000 or 0.001) and so on. Thus, in the number 59.32, the 5 has a weight of 10, the 9 has a weight of 1, the 3 has a weight of 0.1 and the 2 has a weight of 0.01; so the number can be written: 5 x x x x 0.01 or Likewise, the number can be expressed as: 0 x x x x Decimal numbers will be covered in more detail in Chapter 4. TIP: To get used to the idea of weights, it is helpful to think in terms of money. For example; $ 214 is equal to 2 one-hundred dollar bills plus 1 ten dollar bill plus 4 one dollar bills. Thus, the 2 has a weight of 100, the 1 has a weight of a 10 and the 4 has a weight of 1. ELPT

15 Most and Least Significant Digits Most significant digit (MSD) : The digit to the left-most position (the one with the largest weight) is the most significant digit. Example: In the number 563, 5 is the most significant digit (MSD); likewise, in the number 0.45, 4 is the most significant digit. Least significant digit (LSD) : The digit to the right-most position (the one with the smallest weight) is the least significant digit. Example: In the number 563, 3 is the least significant digit (LSD); likewise, in the number 0.45, 5 is the least significant digit. Exercise Write each number in terms of weights and indicate which digit is the MSD and which one is the LSD. Example: Given 32.5 write 3 x x x 0.1, MSD = 3, LSD = 5 1] 76 2] 809 3] 439 4] 1,642 5] 37 6] 14 7] ] 4 9] ] ELPT

16 1.5 Measurements and Units Some areas of mathematics study expressions, operations and properties dealing with pure numbers. In science and technology, however, we deal with numbers and variables representing measurable quantities such as length, weight, electric current and temperature, to mention a few. When you measure a quantity such as length, the measurement contains two ( 2 ) parts: a number value and a unit; for example you say the length of a room is 30 feet. An expression such as the weight of an object is 50 is not proper; it could mean 50 pounds, 50 kilograms or something else. It is important that you develop the habit of including units when expressing all measurements. Quantity: A quantity is anything that can be measured. For example: length, area, volume, voltage, pressure, velocity are quantities. Unit: A unit is the standard used to measure a given quantity. The same quantity can be measured using several units. For example: length can be measure in inches, feet, meters, etc. Symbol for Quantity: A symbol, usually a letter, is used to represent quantities in expressions and formulas. For example: L can represent length, T can represent temperature and E can represent voltage. Symbol for Unit: An abbreviation, a letter or special symbols are used to represent units of measurement. For example: A represents Amperes ( unit of electric current ), ft. and represent feet ( unit of length ) and C represents degrees centigrade ( unit of temperature ). A measurement can be expressed in several ways depending on the units used; you can specify the length of an object in feet, inches, meters, etc. and the units must be included when expressing the measurement. In some instances some units are more suitable than others; for example, instead of using inches, it is better to used miles or kilometers to describe the distance between two cities. ELPT

17 The following chart shows some common quantities along with acceptable symbols and units. Quantity Symbol for Quantity Unit ( s ) of Measurement Symbol for Unit Angle Ê Degree Area A square foot square inch square meter square millimeter Current I Ampere or Amp milliampere or milliamp ft 2 in 2 m 2 mm 2 A ma Energy e British Thermal Unit calorie Joule Kilowatt-hour Force F pound force Newton Length L inch feet yard mile meter millimeter kilometer Power P Watt KiloWatt Horsepower Pressure p pounds per square inch Pascal KiloPascal Resistance R Ohm Kilo-Ohm Temperature T degree Centigrade ( Celsius ) degree Farenheit Time t second hour Velocity v feet per second miles per hour meters per second kilometers per hour Voltage E Volt millivolt KiloVolt BTU cal J KWh lb-force N in. or ft. or yd. mi m mm km W KW HP psi Pa KPa S KS C F s h ft / s mph m / s km / h V mv KV ELPT

18 Volume V cubic feet cubic yard cubic meter ft 3 yd 3 m 3 Weight W ounce pound gram kilogram oz. lb. g Kg A few facts about quantities and units When letters are used to represent quantities, it is possible that more than one value of the same quantity may be present in a problem. For example there may be three different voltages in a circuit or two different pressures in a problem. To avoid confusion, letters are written with a subscript, a small number or letter below and to the right of the letter representing the quantity. For example: If E represents voltage and we several voltages in the circuit, we may write: E 1, E 2, E 3... and so on. Likewise, if P represents pressure, we write: P 1, P 2, P 3... and so on. Greek letters are often used to represent prefixes, quantities or units. Some common greek letters are: " lower case ALPHA $ lower case BETA ) upper case DELTA 8 lower case LAMBDA : lower case MU B lower case PI D lower case RHO E upper case SIGMA 1 upper case THETA 2 lower case THETA M upper case PHI N lower case PHI S upper case OMEGA T lower case OMEGA The International System of Units, universally abbreviated SI ( from the French Le Système International d'unités ), is the modern metric system of measurement. The SI was established in 1960 by the 11th General Conference on Weights and Measures (CGPM, Conférence Générale des Poids et Mesures). The CGPM is the international authority that ensures wide dissemination of the SI and modifies the SI as necessary to reflect the latest advances in science and technology. Units are often named after scientists. Here are a few examples: ELPT

19 ampere [A] The ampere is the basic unit of electric current. It is that current which produces a specified force between two parallel wires which are 1 meter apart in a vacuum. It is named after the French physicist Andre Ampere ( ). Hertz [Hz] The hertz is the SI unit of the frequency of a periodic phenomenon. One hertz indicates that 1 cycle of the phenomenon occurs every second. For most work much higher frequencies are needed such as the kilohertz [khz] and megahertz [MHz]. It is named after the German physicist Heinrich Rudolph Hertz ( ). joule [J] The joule is the SI unit of work or energy. One joule is the amount of work done when an applied force of 1 newton moves through a distance of 1 meter in the direction of the force. It is named after the English physicist James Prescott Joule ( ). newton [N] The newton is the SI unit of force. One newton is the force required to give a mass of 1 kilogram an acceleration of 1 meter per second per second. It is named after the English mathematician and physicist Sir Isaac Newton ( ). ohm [S] The ohm is the SI unit of resistance of an electrical conductor. Its symbol, shown here as S is the Greek letter known as 'omega'. It is named after the German physicist Georg Simon Ohm ( ). pascal [Pa] The pascal is the SI unit of pressure. One pascal is the pressure generated by a force of 1 newton acting on an area of 1 square meter. It is a rather small unit as defined and is more often used as a kilopascal [kpa]. It is named after the French mathematician, physicist and philosopher Blaise Pascal ( ). volt [V] The volt is the SI unit of electric potential. One volt is the difference of potential between two points of an electrical conductor when a current of 1 ampere flowing between those points dissipates a power of 1 watt. It is named after the Italian physicist Count Alessandro Giuseppe Anastasio Volta ( ). watt [W] The watt is used to measure power or the rate of doing work. One watt is a power of 1 joule per second. It is named after the Scottish engineer James Watt ( ). TEMPERATURE SCALES There have been five main temperature scales, each one being named after the person who invented it.of those five scales, Fahrenheit, Celsius, Rankine and Kelvin are still used. The Fahrenheit scale is commonly used in the United States; Celsius is the most widely used temperature scale throughout the world. G D FAHRENHEIT ( ) a German physicist, in about 1714 proposed the first practical scale. He called the freezing-point of water 32 degrees (so as to avoid negative temperatures) and the boiling-point 212 degrees. Anders CELSIUS ( ) a Swedish astronomer, proposed the 100-degree scale (from 0 to 100) in This was widely adopted as the centigrade scale. But since grades and centigrade were also measures of angle, in 1948 it officially became the Celsius scale. Also, the S I system of units (Système International) gives preference to naming units after people where possible. A brief comparison of the Celsius and Fahrenheit scales is shown on page 1-20 ELPT

20 Expressing Quantities Math is used in science and technology to perform calculations. Equations are used to compute quantities whose values depend on the values of other variables. Many of the problems encountered in real life will appear in the form of questions or statements. Typical Word Statement: The pressure is 60 pounds ( per square inch ) Typical Mathematical Expression: P = 60 psi Exercise Exercise Write a mathematical expression for each word statement. For example: Given The temperature is 78 degrees Centigrade write: T = 78 C 1] The pressure is 120 pounds (per square inch) 2] The voltage is 120 Volts. 3] The resistance is 350 ohms. 4] The temperature is 145 degrees Fahrenheit. 5] The energy is 465 BTU s. 6] The pressure is 250 Pascals. 7] The power is 12.5 horsepower. 8] The area is 378 square feet. ELPT

21 9] The current is 30 milliamps. 10] The length is 3.8 meters. Exercise Exercise Write a word statement for each mathematical expression. For example: Given E = 24 V write the voltage is 24 volts 1] P = 45 W 2] T = 167 F 3] R = 470 S 4] I = 4 ma 5] E = 110 V 6] p = 85 psi 7] P = 450 W 8] I = 4 A and E = 12 V 9] L = 23 10] T = 108 C L O H ACTIVITIES N F 7 1] What is a rational number? Give two examples of rational numbers. 2] How many whole numbers greater than or equal to 10 and less than 100 are there? 3] If a number is added to 8 we obtain 15, what is the number? 4] If a number is divided by 6 we obtain 12, what is the number? 5] Twice a number minus 3 equals 7; what is the number? ELPT

22 Review Problems 1-14 perform the operations indicated. Problems 15-20, use <, > or = to compare the quantities. 1] ( ) / ( ) = 2] 10 ( 7-2 ) = 3] 0 ( a ) = 4] 2 = 5] 12 0 = 6] 0 5 = 7] a = 8] ( 5-1 ) 9] 18 / 1 = 10] ( 1 ) ( ) 11] 6 ( 2 ) = 12] 9 0 = 13] 2 ( 3 + ) = 14] ( ) - ( 5-4 ) = 15] ] ] ] ] ] L NOTES O NOTES N NOTES Q NOTES P NOTES 7 ELPT