Example Convert –4 ᎏ 5 6 ᎏ to an improper fraction. Temporarily ignore the negative sign and perform the conversion: 4 ᎏ 5 6 ᎏ = ᎏ 4 ϫ 6 6 +5 ᎏ = ᎏ 2 6 9 ᎏ . The final answer includes the negative sign: – ᎏ 2 6 9 ᎏ . ■ To convert from an improper fraction to a mixed number, simply treat the fraction like a division prob- lem, and express the answer as a fraction rather than a decimal. Example Convert ᎏ 2 7 3 ᎏ to a mixed number. Perform the division: 23 ÷ 7 = 3 ᎏ 2 7 ᎏ . Percents Percents are always “out of 100.”45% means 45 out of 100. Therefore, to write percents as decimals, move the dec- imal point two places to the left (to the hundredths place). 45% = ᎏ 1 4 0 5 0 ᎏ = 0.45 3% = ᎏ 1 3 00 ᎏ = 0.03 124% = ᎏ 1 1 2 0 4 0 ᎏ = 1.24 0.9% = ᎏ 1 . 0 9 0 ᎏ = ᎏ 1,0 9 00 ᎏ = 0.009 Here are some conversions you should be familiar with: Fraction Decimal Percentage ᎏ 1 2 ᎏ .5 50% ᎏ 1 4 ᎏ .25 25% ᎏ 1 3 ᎏ .333 . . . 33.3 –– % ᎏ 2 3 ᎏ .666 . . . 66.6 –– % ᎏ 1 1 0 ᎏ .1 10% ᎏ 1 8 ᎏ .125 12.5% ᎏ 6 1 ᎏ .1666 . . . 16.6 –– % ᎏ 1 5 ᎏ .2 20% – THEA MATH REVIEW– 105 Absolute Value The absolute value of a number is the distance of that number from zero. Distances are always represented by pos- itive numbers, so the absolute value of any number is positive. Absolute value is represented by placing small ver- tical lines around the value: |x|. Examples The absolute value of seven: |7|. The distance from seven to zero is seven, so |7| = 7. The absolute value of negative three: |–3|. The distance from negative three to zero is three, so |–3| = 3. Exponents P OSITIVE EXPONENTS A positive exponent indicates the number of times a base is used as a factor to attain a product. Example Evaluate 2 5 . 2 is the base and 5 is the exponent. Therefore, 2 should be used as a factor 5 times to attain a product: 2 5 = 2 ϫ 2 ϫ 2 ϫ 2 ϫ 2 = 32 ZERO EXPONENT Any non-zero number raised to the zero power equals 1. Examples 5 0 = 1 70 0 = 1 29,874 0 = 1 NEGATIVE EXPONENTS A base raised to a negative exponent is equivalent to the reciprocal of the base raised to the positive exponent (absolute value of the exponent). – THEA MATH REVIEW– 106 Examples 5 –1 = ᎏ 1 5 ᎏ 7 –2 = ( ᎏ 1 7 ᎏ ) 2 = ᎏ 4 1 9 ᎏ ( ᎏ 2 3 ᎏ ) –2 = ( ᎏ 3 2 ᎏ ) 2 = ᎏ 9 4 ᎏ EXPONENT RULES ■ When multiplying identical bases, you add the exponents. Examples 2 2 ϫ 2 4 ϫ 2 6 = 2 12 a 2 ϫ a 3 ϫ a 5 = a 10 ■ When dividing identical bases, you subtract the exponents. Examples ᎏ 2 2 7 3 ᎏ = 2 4 ᎏ a a 9 4 ᎏ = a 5 ■ If a base raised to a power (in parentheses) is raised to another power, you multiply the exponents together. Examples (3 2 ) 7 = 3 14 (g 4 ) 3 = g 12 PERFECT SQUARES 5 2 is read “5 to the second power,” or, more commonly, “5 squared.” Perfect squares are numbers that are second powers of other numbers. Perfect squares are always zero or positive, because when you multiply a positive or a negative by itself, the result is always positive. The perfect squares are 0 2 ,1 2 ,2 2 ,3 2 Perfect squares: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 . . . P ERFECT CUBES 5 3 is read “5 to the third power,” or, more commonly, “5 cubed.” (Powers higher than three have no special name.) Perfect cubes are numbers that are third powers of other numbers. Perfect cubes, unlike perfect squares, can be both positive or negative. This is because when a negative is multiplied by itself three times, the result is negative. The perfect cubes are 0 3 ,1 3 ,2 3 ,3 3 Perfect cubes: 0, 1, 8, 27, 64, 125 . . . ■ Note that 64 is both a perfect square and a perfect cube. – THEA MATH REVIEW– 107 SQUARE ROOTS The square of a number is the product of the number and itself. For example, in the statement 3 2 = 3 ϫ 3 = 9, the number 9 is the square of the number 3. If the process is reversed, the number 3 is the square root of the number 9. The symbol for square root is ͙3 ෆ and is called a radical. The number inside of the radical is called the radi- cand. Example 5 2 = 25, therefore ͙25 ෆ = 5 Since 25 is the square of 5, it is also true that 5 is the square root of 25. The square root of a number might not be a whole number. For example, the square root of 7 is 2.645751311. It is not possible to find a whole number that can be multiplied by itself to equal 7. Square roots of non-per- fect squares are irrational. CUBE ROOTS The cube of a number is the product of the number and itself for a total of three times. For example, in the state- ment 2 3 = 2 ϫ 2 ϫ 2 = 8, the number 8 is the cube of the number 2. If the process is reversed, the number 2 is the cube root of the number 8. The symbol for cube root is the same as the square root symbol, except for a small three ͙ 3 34 ෆ . It is read as “cube root.” The number inside of the radical is still called the radicand, and the three is called the index. (In a square root, the index is not written, but it has an index of 2.) Example 5 3 = 125, therefore ͙ 3 125 ෆ = 5. Like square roots, the cube root of a number might be not be a whole number. Cube roots of non-perfect cubes are irrational. Probability Probability is the numerical representation of the likelihood of an event occurring. Probability is always repre- sented by a decimal or fraction between 0 and 1; 0 meaning that the event will never occur, and 1 meaning that the event will always occur. The higher the probability, the more likely the event is to occur. A simple event is one action. Examples of simple events are: drawing one card from a deck, rolling one die, flipping one coin, or spinning a hand on a spinner once. SIMPLE PROBABILITY The probability of an event occurring is defined as the number of desired outcomes divided by the total number of outcomes. The list of all outcomes is often called the sample space. – THEA MATH REVIEW– 108 P(event) = Example What is the probability of drawing a king from a standard deck of cards? There are 4 kings in a standard deck of cards. So, the number of desired outcomes is 4. There are a total of 52 ways to pick a card from a standard deck of cards, so the total number of outcomes is 52. The probability of drawing a king from a standard deck of cards is ᎏ 5 4 2 ᎏ . So, P(king) = ᎏ 5 4 2 ᎏ . Example What is the probability of getting an odd number on the roll of one die? There are 3 odd numbers on a standard die: 1, 3, and 5. So, the number of desired outcomes is 3. There are 6 sides on a standard die, so there are a total of 6 possible outcomes. The probability of rolling an odd number on a standard die is ᎏ 3 6 ᎏ . So, P(odd) = ᎏ 3 6 ᎏ . Note: It is not necessary to reduce fractions when working with probability. P ROBABILITY OF AN EVENT NOT OCCURRING The sum of the probability of an event occurring and the probability of the event not occurring = 1. Therefore, if we know the probability of the event occurring, we can determine the probability of the event not occurring by subtracting from 1. Example If the probability of rain tomorrow is 45%, what is the probability that it will not rain tomorrow? 45% = .45, and 1 – .45 = .55 or 55%. The probability that it will not rain is 55%. P ROBABILITY INVOLVING THE WORD “OR” Rule: P(event A or event B) = P(event A) + P(event B) – P(overlap of event A and B) When the word or appears in a simple probability problem, it signifies that you will be adding outcomes. For example, if we are interested in the probability of obtaining a king or a queen on a draw of a card, the number of desired outcomes is 8, because there are 4 kings and 4 queens in the deck. The probability of event A (drawing a king) is ᎏ 5 4 2 ᎏ , and the probability of drawing a queen is ᎏ 5 4 2 ᎏ . The overlap of event A and B would be any cards that are both a king and a queen at the same time, but there are no cards that are both a king and a queen at the same time. So the probability of obtaining a king or a queen is ᎏ 5 4 2 ᎏ + ᎏ 5 4 2 ᎏ – ᎏ 5 0 2 ᎏ = ᎏ 5 8 2 ᎏ . # of desired outcomes ᎏᎏᎏ total number of outcomes – THEA MATH REVIEW– 109 Example What is the probability of getting an even number or a multiple of 3 on the roll of a die? The probability of getting an even number on the roll of a die is ᎏ 3 6 ᎏ , because there are three even numbers (2, 4, 6) on a die and a total of 6 possible outcomes. The probability of getting a multiple of 3 is ᎏ 2 6 ᎏ , because there are 2 multiples of three (3, 6) on a die. But because the outcome of rolling a 6 on the die is an overlap of both events, we must subtract ᎏ 1 6 ᎏ from the result so we don’t count it twice. P(even or multiple of 3) = P(even) + P(multiple of 3) – P(overlap) = ᎏ 3 6 ᎏ + ᎏ 2 6 ᎏ – ᎏ 1 6 ᎏ = ᎏ 4 6 ᎏ COMPOUND PROBABILITY A compound event is performing two or more simple events in succession. Drawing two cards from a deck, rolling three dice, flipping five coins, having four babies, are all examples of compound events. This can be done “with replacement”(probabilities do not change for each event) or “without replacement” (probabilities change for each event). The probability of event A followed by event B occurring is P(A) ϫ P(B). This is called the counting principle for probability. Note: In mathematics, the word and usually signifies addition. In probability, however, and signifies multi- plication and or signifies addition. Example You have a jar filled with 3 red marbles, 5 green marbles, and 2 blue marbles. What is the probability of getting a red marble followed by a blue marble, with replacement? “With replacement” in this case means that you will draw a marble, note its color, and then replace it back into the jar. This means that the probability of drawing a red marble does not change from one simple event to the next. Note that there are a total of 10 marbles in the jar, so the total number of outcomes is 10. P(red) = ᎏ 1 3 0 ᎏ and P(blue) = ᎏ 1 2 0 ᎏ so P(red followed by blue) is ᎏ 1 3 0 ᎏ ϫ ᎏ 1 2 0 ᎏ = ᎏ 1 6 00 ᎏ . If the problem was changed to say “without replacement,” that would mean you are drawing a marble, not- ing its color, but not returning it to the jar. This means that for the second event, you no longer have a total num- ber of 10 outcomes, you only have 9 because you have taken one red marble out of the jar. In this case, P(red) = ᎏ 1 3 0 ᎏ and P(blue) = ᎏ 2 9 ᎏ so P(red followed by blue) is ᎏ 1 3 0 ᎏ ϫ ᎏ 2 9 ᎏ = ᎏ 9 6 0 ᎏ . – THEA MATH REVIEW– 110 . always “out of 100. 45 % means 45 out of 100. Therefore, to write percents as decimals, move the dec- imal point two places to the left (to the hundredths place). 45 % = ᎏ 1 4 0 5 0 ᎏ = 0 .45 3% =. standard deck of cards? There are 4 kings in a standard deck of cards. So, the number of desired outcomes is 4. There are a total of 52 ways to pick a card from a standard deck of cards, so the. draw of a card, the number of desired outcomes is 8, because there are 4 kings and 4 queens in the deck. The probability of event A (drawing a king) is ᎏ 5 4 2 ᎏ , and the probability of drawing