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STUDY GUIDE NEGATIVE NUMBERS On a number line, 0 is the only number that is neither positive nor negative. All numbers to the right of 0 are positive numbers. All numbers to the left of 0 are negative numbers. Negative numbers have a negative sign in front of them: –4, –6, –12, etc. The number line below shows you what order the numbers go in. One important thing to remember is that the farther to the right a number is on the number line, the greater it is. Similarly, the farther to the left a number is on the number line, the smaller it is. Since positive numbers are farther to the right than negative numbers, positive numbers are always larger than negative numbers. Negative numbers are often used in the real world. For instance, a debt or loss is considered a negative quantity. In addition, all temperatures below zero degrees, such as –4° F, represent negative integers. Example 1: Which number is larger: –1 or –4? To find out which number is larger, we look at the number line. We see that –1 is farther to the right than –4. The farther to the right a number is, the larger it is. Answer: Therefore, –1 is larger than –4. Example 2: Michelle has a checking account. She wrote a check for $20 more than she had in her account, so now she owes the bank $20. What number represents the amount of money in her bank account? Answer: Michelle owes money to the bank. This is considered a debt, so it is a negative quantity. Normally, a balance in an account is positive. However, since Michelle owes the bank $20, her balance is –$20. COMPARE AND ORDER NUMBERS You need to know the following symbols: 1. < means "less than" 2. > means "greater than" 3. = means "equal to" Examples: To show that 1 is "less than" 2 write: 1 < 2 To show that 4 is "greater than" 2 write: 4 > 2 To show that 2 is "equal to" 2 write: 2 = 2 Tips when ordering numbers: 1. Convert fractions to decimals because they are easier to compare. 2. Double check your answer in order to avoid any mistakes. 3. Remember with symbols < and >, the symbol points to the smaller value. Ordering Decimal Numbers Align the ones digit for the numbers. Compare the place value for each digit starting from the left. Find the first difference, the number with the largest digit is the largest number. Example: Order the numbers 357.2153 and 357.241 357.2153 357.241 The hundredths digit is the first difference. Since 4 is larger than 1, the number 357.241 is larger than the number 357.2153. Ordering Fractions with a Common Denominator When the denominators are the same, then the proper/improper fraction with the largest numerator is the largest fraction. 7 11 Example: Order the numbers /13 and /13. The fractions have a common denominator of 13. Thus, compare the numerators. Since 7 < 11, 7 11 /13 < /13 Ordering Fractions with Different Denominators When the denominators are different rewrite the fractions using a common denominator. The proper/improper fraction with the largest numerator is the largest fraction. 3 4 Example: Order the numbers /5 and /9. The fractions have different denominators. Since 45 is the least common multiple of 5 and 9, rewrite the fractions using a common denominator of 45. 3 27 /5 = /45 4 20 /9 = /45 Now that the fractions are written with a common denominator, compare the numerators. Since 27 > 20, 3 27 20 4 /5 = /45 > /45 = /9 Alternatively, when the fractions have different denominators, find the decimal equivalent of each fraction by dividing the numerator by the denominator. Compare the decimal equivalents. PERCENTAGE A percent is a ratio whose second term is 100. Percent means parts per hundred. In mathematics, we use the symbol % for percent. Percents can easily be converted to a decimal by dividing by 100. Example: 20% = 0.2 Example: The price of a video game is $50. If the sales tax is 10%, what is the total cost of one video game? Answer: Sales tax is 10% of $50. To figure out the tax on $50, multiply $50 by 10%. $50 × 0.10 = $5 Now add the tax onto the original cost. $50 + $5 = $55 PERCENTS A percent is a ratio whose second term is 100. Percent means parts per hundred. In mathematics, we use the 20 symbol % for percent. Percents can be easily converted to a fraction by dividing by 100. Example: 20% = /100 Percents are seen in everyday life. For example, a whole pizza is 100% of a pizza. Half of a pizza is 50% of a pizza. In this case, a pizza is being used to model percents. In the same way, various types of figures can be used to model percents. 4 equal parts or fourths 1 One-fourth of the object is shaded: /4 In the figure above, what percent of the circle is shaded? We know that one-fourth, or one part out of four, is shaded. Remember, percent means parts per hundred, so we must convert one-fourth to a fraction with a denominator of 100. 4 times what number equals 100? 4 × 25 = 100 If we multiply the denominator by 25, we must do the same to the numerator: 1 × 25 = 25 1 25 25 So, /4 = /100, and /100 = 25%. CONVERTING FRACTIONS, DECIMALS, AND PERCENTS Converting from a decimal to a percent: MULTIPLY the decimal by 100. Converting from a percent to a decimal: DIVIDE the percent by 100. Examples: Convert 0.25 to a percent: (0.25 × 100) = 25% Convert 0.75 to a percent: (0.75 × 100) = 75% Convert 25% to a decimal: (25 ÷ 100) = 0.25 Convert 75% to a decimal: (75 ÷ 100) = 0.75 Fractions, decimals, and percents can all be used to represent part of a whole. It is important to know how to convert common fractions and common percents. See the table of common fractions and percents below. Fraction = Decimal = Percent 1 /10 = 0.1 = 10% 1 /5 = 0.2 = 20% 1 /2 = 0.5 = 50% 3 /4 = 0.75 = 75% 4 /5 = 0.8 = 80% 9 /10 = 0.9 = 90% The most common way to convert a fraction to a decimal is to divide its numerator by its denominator. Sometimes the division comes out exactly, and sometimes the division does not terminate. Converting a Fraction to a Decimal - Terminating Examples Converting a Fraction to a Decimal - Non-Terminating Examples To convert a decimal to a fraction, take the place value farthest to the right of the decimal and make it the denominator, and then remove the decimal point from the number and make it the numerator. Then, simplify the fraction to lowest terms. Example 1: Convert 0.6 to a fraction. Example 2: Convert 0.25 to a fraction. Example 3: Convert 0.125 to a fraction. To convert from a fraction to a percent, first convert the fraction to a decimal, and then convert to a percent. Example 1: Convert to a percent. Example 2: Convert 15% to a fraction. COMPUTE SOLUTIONS Below are some important tips to help you compute solutions with fractions. FRACTIONS Adding and Subtracting Multiplying Dividing Change mixed numbers to Change mixed numbers to Change mixed numbers to fractions. fractions. fractions. Find the least common Solve by multiplying the Flip the second fraction, and denominator. numerators together and the change the divide sign to a Solve. denominators together. multiply sign. Reduce when possible. Reduce when possible. Solve by multiplying the numerators and the denominators. Reduce when possible. 2 7 Example 1a (Adding): /5 + /35 = ?? Find the least common denominator, then solve: 2 7 14 7 21 /5 + /35 = /35 + /35 = /35 Reduce when possible: 21 3 /35 = /5 1 2 Example 1bAdding): 2 /4 + 3 /3 = ?? Change the mixed numbers to fractions, find the least common denominator, then solve: 1 2 9 11 27 44 71 11 2 /4 + 3 /3 = /4 + /3 = /12 + /12 = /12 or 5 /12 2 1 Example 2a (Subtracting): 3 /5 - 2 /4 = ?? Find the least common denominator, then solve: 2 1 8 5 3 3 /5 - 2 /4 = 3 /20 - 2 /20 = 1 /20 2 1 Example 2b (Subtracting): 3 /5 - 2 /4 = ?? First, change mixed numbers to fractions: 2 1 17 9 3 /5 - 2 /4 = /5 - /4 Next, find the least common denominator, then solve: 17 9 68 45 23 3 /5 - /4 = /20 - /20 = /20 or 1 /20 5 1 Example 3 (Multiplying): /6 × /5 = ?? When multiplying fractions, you do NOT need to find the least common denominator. Just multiply the numerators and the denominators. 5 1 5 /6 × /5 = /30 Reduce when possible: 5 1 /30 = /6 3 1 Example 4 (Dividing): 2 /4 ÷ 1 /2 = ?? First, change mixed numbers to fractions: 3 1 11 3 2 /4 ÷ 1 /2 = /4 ÷ /2 Next, to divide, flip the second fraction, and then multiply: 11 3 11 2 22 /4 ÷ /2 = /4 × /3 = /12 Reduce when possible: 22 11 /12 = /6 Adding Decimals Place the numbers so that the decimal points are aligned vertically. Add each column, starting on the right and working left. If the sum of a column is greater than ten, then carry the one to the next column to the left. Example 1: 2.34 + 6.9 = ? Solution: Place the numbers so that the decimal points are aligned vertically. Since 6.9 only has one number to the right of the decimal point and 2.34 has two, a 0 will be used as a place holder. Subtracting Decimals Place the numbers so that the decimal points are aligned vertically. Subtract each column, starting on the right and working left. If the number being subtracted is larger than the number it is being subtracted from, then add ten to the number and subtract one from the number in the next left column. This is called borrowing. Example 2: 7.008 - 3.996 = ? Solution: Place the numbers so that the decimal points are aligned vertically. Since the two numbers have the same number of digits to the right of the decimal, no place holders will be added. Multiplying Decimals Multiply the decimals like normal, and then move the decimal point. Move the decimal point according to how many numbers are to the right of the decimals in the problem. Example 3: 32.6 × 2.9 = ? Solution: Place one number above the other so that the last digits are lined up. Multiply the numbers together, ignoring the decimals until the very end. In the original numbers, there are two numbers after the decimals [6 and 9]; therefore, the decimal point in the answer is moved two places. Dividing Decimals Write the problem as a long division problem. Move the decimal up into the quotient. Finally, divide. Example 4: 2.56 ÷ 8 = ? Solution: Rewrite the problem as a long division problem. Bring the decimal up into the quotient, and then divide 256 by 8. Dividing a Decimal Number by a Decimal Number Start by moving the decimal place of the divisor and dividend to the right enough places to get rid of the decimal in the divisor. Then use normal long division. Example - Divide 39.44 by 2.9 Place the divisor (2.9) before the division bracket and place the dividend (39.44) in the division bracket. Since there is one decimal place in the divisor, move the decimal place in both the divisor and the dividend one place to the right. The division then becomes 394.4 divided by 29. Now Divide There are no more numbers in the dividend to bring down next to the 0. Therefore, 394.4 divided by 29 is 13.6 FACTORS AND MULTIPLES Factors A factor is a number that can divide into another number exactly without a remainder. Example: What are the factors of 24? Solution: 24 ÷ 1 = 24 24 ÷ 24 = 1 24 ÷ 2 = 12 24 ÷ 12 = 2 24 ÷ 3 = 8 24 ÷ 8 = 3 24 ÷ 4 = 6 24 ÷ 6 = 4 Therefore, 1, 2, 3, 4, 6, 8, 12, and 24 are factors of 24. Multiples A multiple of one number has the number as a factor. Example: Which is a multiple of 6? A. 26 B. 44 C. 18 D. 32 Solution: All the multiples of 6 have 6 as a factor. The multiples of 6 are below. {6, 12, 18, 24, 30, 36, 42, 48, ...} Of the choices, only 18 is a multiple of 6. PRIME FACTORIZATION A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. Prime Number - a number that only has two factors: 1 and itself. Prime Factorization - the expression of a positive integer as a product of prime numbers. Relatively Prime - numbers that have no common factors besides the number 1. (The numbers do not have to be prime numbers.) Example: Question 1: What is the prime factorization of 120? Answer: 120 ÷ 2 = 60 60 ÷ 2 = 30 30 ÷ 2 = 15 15 ÷ 3 = 5 3 120 = 2 × 2 × 2 × 3 × 5 = 2 × 3 × 5 SCIENTIFIC NOTATION Scientific notation is often used when working with really big or really small numbers. Scientific notation has three parts: a number greater than or equal to 1 and less than 10 a multiplication sign, and a power of 10. 23,000,000 written in scientific notation is: 7 2.3 × 10 0.0000064 written in scientific notation is: -6 6.4 × 10 MEASURING ANGLES Angles are measured in degrees with a protractor. The number of degrees tells you how wide open the angle is. The picture above shows how a protractor is used to measure angles. As you can see, the angle above measures 135°. There are three types of angles: 1. acute - angles that measure between 0° and 90° 2. right - angles that measure exactly 90° 3. obtuse - angles that measure between 90° and 180° PERIMETER AND AREA Perimeter is the distance around a flat (2-dimensional) shape. Perimeter = L + L + W + W = 2L + 2W The rectangle above has a width of W and a length of L. Example:Find the perimeter of a rectangle with a length of 8 units and a width of 10 units. Solution: Perimeter = 2L + 2W = 2(8 units) + 2(10 units) = 16 units + 20 units = 36 units Area is the amount of space taken up by a flat (2-dimensional) shape. The rectangle above has a width of W and a length of L. Area = L × W Example: Find the area of a rectangle with a length of 8 units and a width of 10 units. Solution: Area = L × W = 8 units × 10 units = 80 square units CIRCLES You need to know the relationship between the radius, diameter, and circumference of a circle. radius - the length of a line segment from the center of a circle to its edge. diameter - the length of a line segment from one side of a circle to the other passing through the center of the circle. circumference - the distance around a circle. chord - a line segment whose ends lie on a circle. A circle's circumference is about 3.14 times the diameter. Question: About what is the circumference of the circle above? Answer: The radius of the circle is 10 cm. First, find the diameter of the circle. diameter = 10 cm × 2 diameter = 20 cm Now since the circumference is about three times the diameter, the circumference is about (20 × 3.14) 62.8 cm. A circle's area is given by the following formula: 2 Area = r where r is the radius of the circle. Question: What is the area of the circle above? Answer: The radius of the circle is 10 cm. To find the area of the circle, plug this value into the formula. So we have the following: 2 2 Area = r = × (10) = × 100 = 100 = 314 COORDINATE GEOMETRY The coordinate plane is made up of a horizontal number line, the x-axis, and a vertical number line, y-axis. The x- axis and y-axis intersect at the origin, or point (0,0). Points in the coordinate plane are identified by combining the x coordinate of the point and the y coordinate of the point, (x,y). This is called an ordered pair and it is important to note that the x-coordinate always comes before the y-coordinate when writing a point as an ordered pair. When solving geometry problems on a coordinate plane, remember that the gridlines and intervals of the coordinate plane will provide the information needed to solve the problem. Example: What is the perimeter of the object on the coordinate plane above? Solution: 20 units Explanation: The perimeter is the distance around an object. Use the grid to measure the length of each side of the figure. The object has six sides, with lengths of 6, 4, 3, 3, 3, and 1. Add up the lengths of each side to find the perimeter: 6 + 4 + 3 + 3 + 3 + 1 = 20 units. OBJECT TRANSFORMATIONS ON THE COORDINATE PLANE Object transformations change a geometric figure by rotating it, reflecting it, or translating it. A reflection is a transformation in which the figure flips over a line of reflection. The objects are mirror images of each other. On the coordinate plane below, the figure is reflected over the x-axis. A rotation is a transformation in which the figure rotates around a point. In other words, it "turns" or "pivots" around a point. On the coordinate plane below, the figure is rotated around point (2,2). A translation is a transformation that "slides" a figure in any direction. It does not change the orientation or size of the figure. POSSIBLE OUTCOMES It is important to be able to figure out all possible outcomes (or combinations) of a probability experiment. There are a couple ways to solve these types of problems. 1. One way is to list all possible solutions. Example 1. You flip a quarter 2 times in a row. How many different outcomes could there be? List all possibilities: H-H H-T T-H T-T There are 4 possible outcomes (combinations). 2. Another way is to multiply to find the number of possible outcomes. Example 1. At a restaurant, you can order a soup, salad, & sandwich combination for lunch. They have 3 kinds of soup, 2 kinds of salad, and 4 kinds of sandwiches. How many possible combinations of these three items are there? Multiply: 3 (soups) × 2 (salads) × 4 (sandwiches) 3 × 2 × 4 = 24 There are 24 possible combinations. PROBABILITY Probability refers to the chance that an event will happen. Probability is presented as the ratio of the number of ways an event can occur relative to the number of possible outcomes. Number of ways an event can occur Probability of Event = Number of possible outcomes Example 1: If you roll a die, what’s the probability of rolling a four? Number of ways an event can occur: {4} P(4) = Number of possible outcomes: {1,2,3,4,5,or 6} outcomes 1 P(4) = /6 Example 2: If you roll a die, what’s the probability of rolling a number less than 4? Number of ways an event can occur: {1,2,or 3} P(Less Than 4) = Number of possible outcomes: {1,2,3,4,5,or 6} 3 1 P(Less than 4) = /6 = /2 Example 3: If you pick from a bag that contains 5 blue marbles, 2 green marbles, and 3 red marbles, what’s the probability of picking a red marble? Number of ways an event can occur: {3 red} P(red marble) = Number of possible outcomes: {5 blue + 2 green + 3 red} 3 P(red marble) = /10 Probability of A and B If event A and event B are independent, the probability of both event A and event B occurring is P(A) × P(B). Example 1: If you flip a coin twice, what’s the probability that heads comes up both times? 1 P(heads 1st flip) = /2 1 P(heads 2nd flip) = /2 Since P(heads 1st flip) and P(heads 2nd flip) are independent events: 1 1 1 P(heads both flips) = /2 × /2 = /4 Example 2: If you roll 2 dice, what’s the probability that 4 comes up on both dice? 1 P(4 on 1st roll) = /6 1 P(4 on 2nd roll) = /6 Since P(4 on 1st roll) and P(4 on 2nd roll) are independent events: 1 1 1 P(4 on both rolls) = /6 × /6 = /36 PROBABILITY Theoretical Probability of Event A # of ways A can occur P(A) = # of possible outcomes Example 1: A fair coin is tossed. Calculate the probability that the coin will land on heads. Solution: When tossing a coin, it is possible for it to land on heads or tails, so there are 2 possible outcomes. In only 1 of the 2 1 outcomes, the coin could land on heads. Therefore, the theoretical probability that the coin will land on heads is /2. Experimental Probability of Event A An experimental probability is based on data collected from an experiment. # of times A occurred P(A) = # of outcomes Example 2: Joey tossed a coin ten times and recorded his results in the table below. Toss Result 1 H 2 H 3 T 4 H 5 T 6 T 7 H 8 H 9 T 10 H What is the experimental probability of the coin landing on heads? Solution: There were a total of 10 outcomes and 6 of them were heads. 6 3 Therefore, the experimental probability of the coin landing on heads is /10 = /5 NUMBER PROPERTIES Commutative Property: An operation is commutative if you can change the order of the terms involved without changing the result. Addition and multiplication are both commutative. For any real numbers a, b, c, and d, Addition: a + b = b + a Multiplication: c × d = d × c Subtraction is not commutative: 4 - 1 is not equal to 1 - 4 Division is also not commutative: 6 ÷ 2 is not equal to 2 ÷ 6 Associative Property: An operation is associative if you can group numbers in any way without changing the answer. It doesn't matter how you combine them, the answer will always be the same. Addition and multiplication are both associative. For any real numbers a, b, and c, Addition: (a + b) + c = a + (b + c) Multiplication: (a × b) × c = a × (b × c) Subtraction is not associative: (4 - 3) - 2 is not equal to 4 - (3 - 2) Division is also not associative: (12 ÷ 2) ÷ 3 is not equal to 12 ÷ (2 ÷ 3) Distributive Property: When you distribute something, you give pieces of it to many different people. The most common distributive property is the distribution of multiplication over addition. It says that when a number is multiplied by the sum of two other numbers, the first number can be handed out or distributed to both of those two numbers and multiplied by each of them separately. For any real numbers a, b, and c, a × (b + c) = (a × b) + (a × c) (b - c) × a = (b × a) - (c × a) ORDER OF OPERATIONS Order of Operations 1. Perform any calculations inside parentheses or brackets. 2. Simplify exponents from left to right. 3. Perform all multiplications and divisions in order from left to right. 4. Perform all additions and subtractions in order from left to right. Here are examples of how you simplify an expression using the order of operations: Example 1: 2 90 ÷ 3 + 7 × 3 - 12 = 90 ÷ 9 + 7 × 3 - 12 Exponents 90 ÷ 9 + 7 × 3 - 12 = 10 + 7 × 3 - 12 Division 10 + 7 × 3 - 12 = 10 + 21 - 12 Multiplication 10 + 21 - 12 = 31 - 12 Addition 31 - 12 = 19 Subtraction Example 2: 2 90 ÷ (3 + 6) × 3 - 12 = 90 ÷ (9 + 6) × 3 - 12 Exponent within parentheses 90 ÷ (9 + 6) × 3 - 12 = 90 ÷ 15 × 3 - 12 Addition within parentheses 90 ÷ 15 × 3 - 12 = 6 × 3 - 12 Division 6 × 3 - 12 = 18 - 12 Multiplication 18 - 12 =6 Subtraction ALGEBRAIC EXPRESSIONS When evaluating an expression or solving a formula, make sure to follow the correct order of operations. Order of Operations 1. Perform any calculations inside parentheses or brackets. 2. Simplify exponents from left to right. 3. Perform all multiplications and divisions in order from left to right. 4. Perform all additions and subtractions in order from left to right. Example 1. If a = 5 and b = 2, what is the value of 10(a+3b) - 3? 10 (a + 3b) - 3 Substitute values in for a and b and then follow the correct order of operations. 10 (5 +3(2)) - 3 10 (5 + 6) - 3 10 (11) - 3 110 - 3 107 Example 2. If a = bc, a = 36 and b = 6, solve for c. a = bc Substitute values in for a and b. 36 = (6)c To solve for c, divide both sides by 6. 36 6 /6 = /6 c 6=c LINEAR EQUATIONS Operations can be performed on equations to simplify as long as the operations are performed on both sides of the equation. This keeps the equation balanced. Examples: 1. In the equation below, 8 is subtracted from both sides in order to simplify the equation. Equation: y+8 = 10 8 is subtracted from both sides: y + 8 - 8 = 10 - 8 The equation is simplified: y = 2 2. In the equation below, 3 is added to both sides in order to simplify the equation. Equation: y-3 = 7 3 is added to both sides: y-3+3 = 7+3 The equation is simplified: y = 10 3. In the equation below, both sides of the equation are multiplied by 4 in order to simplify the equation. y Equation: /4 = 3+2 y Both sides multiplied by 4: /4 × 4 = (3 + 2) × 4 The equation is simplified: y = 20 4. In the equation below, both sides of the equation are divided by 3 in order to simplify the equation. Equation: 3y = 7-1 Both sides are divided by 3: 3y ÷ 3 = (7 - 1) ÷ 3 The equation is simplified: y = 2 Remember to check the answers by substituting the value of the variable back into the original equation. INEQUALITIES An inequality is used to show the relationship between quantities. Inequalities are written using the following symbols. < - less than > - greater than < - less than or equal to > - greater than or equal to Solving inequalities is similar to solving equations except for one thing. When an inequality is multiplied or divided by a negative number, the inequality symbol switches directions. Example 1: Solve the following inequality for x. 2x - 4 < 10 Solution: First, add 4 to both sides of the inequality. 2x - 4 + 4 < 10 + 4 2x < 14 Then, divide both sides of the inequality by 2. 2x ÷ 2 < 14 ÷ 2 x <7 RATIOS, PROPORTIONS, AND PERCENTS A ratio represents a comparison between two values. A ratio of two numbers can be expressed in three ways. A ratio of "one to two" can be written as: 1 to 2 1:2 1 /2 Example: There are 7 boys and 8 girls in Mrs. Hamilton's math class. What is the ratio of girls to students? Solution: There are 8 girls and 7 + 8 = 15 total students. So, the ratio of girls to students is 8 to 15 or 8:15. A proportion is made up of two ratios with an "=" (equal) sign between them. For example: 1 4 = 2 8 3 Example: Billy is building a model of a 12 foot long truck. The model is /8 the size of the truck. How long is Billy's model? Solution: First make a proportion for the situation with one unknown: x 3 = 12 8 To solve for an unknown in a proportion: Cross-multiply: multiply the numbers that are diagonal to each other. Set the two products equal to each other. Divide both sides by the number next to the x. x 3 = 12 8 8(x) = 12(3) Cross multiply 8x = 36 36 x = /8 Divide by 8 x = 4.5 Therefore, Billy's model is 4.5 feet long. A percent is a ratio whose second term is 100. Percent means parts per hundred. In mathematics, we use the symbol % for percent. Percents can easily be converted to a decimal by dividing by 100. Example: 20% = 0.2 Example: The price of a hamburger $2.50. If the sales tax is 8%, what is the total cost of one hamburger? Solution: To figure out the tax on $2.50, multiply $2.50 by 8%. $2.50 × 0.08 = $0.20 Now add the tax onto the original cost. $2.50 + $0.20 = $2.70