COMMISSION John earns 4.5% commission on all of his sales. What is his commission if his sales total $235.12? To find the part of the sales John earns, set up a proportion: Cross multiply. RATE OF CHANGE If a pair of shoes is marked down from $24 to $18, what is the percent of decrease? To solve the percent, set up the following proportion: Cross multiply. Note that the number 6 in the proportion setup represents the discount, not the sale price. SIMPLE INTEREST Pat deposited $650 into her bank account. If the interest rate is 3% annually, how much money will she have in the bank after 10 years? x ϭ 25% decrease in price 24x 24 ϭ 600 24 24x ϭ 600 6 24 ϭ x 100 24 Ϫ 18 24 ϭ x 100 part whole ϭ change original cost ϭ % 100 x ϭ 10.5804 Ϸ $10.58 100x 100 ϭ 1058.04 100 100x ϭ 1058.04 x 235.12 ϭ 4.5 100 part whole ϭ change original cost ϭ % 100 – ARITHMETIC– 333 Interest = Principal (amount invested) × Interest rate (as a decimal) × Time (years) or I = PRT. Substitute the values from the problem into the formula I = (650)(.03)(10). Multiply I = 195 Since she will make $195 in interest over 10 years, she will have a total of $195 + $650 = $845 in her account. Exponents The exponent of a number tells how many times to use that number as a factor. For example, in the expres- sion 4 3 , 4 is the base number and 3 is the exponent,or power. Four should be used as a factor three times: 4 3 = 4 × 4 × 4 = 64. Any number raised to a negative exponent is the reciprocal of that number raised to the positive expo- nent: Any number to a fractional exponent is the root of the number: Any nonzero number with zero as the exponent is equal to one: 140° = 1. Square Roots and Perfect Squares Any number that is the product of two of the same factors is a perfect square. 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, Knowing the first 20 perfect squares by heart may be helpful. You probably already know at least the first ten. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 256 1 4 ϭ 4 2 256 ϭ 4 27 1 3 ϭ 3 2 27 ϭ 3 25 1 2 ϭ 2 25 ϭ 5 3 Ϫ2 ϭ 1 1 3 2 2 ϭ 1 9 – ARITHMETIC– 334 Radicals A square root symbol is also known as a radical sign. The number inside the radical is the radicand. To simplify a radical, find the largest perfect square factor of the radicand ͙ෆ32 = ͙ෆ16 × ͙ ෆ 2 Take the square root of that number and leave any remaining numbers under the radical. ͙ෆ32 = 4͙ ෆ 2 To add or subtract square roots, you must have like terms. In other words, the radicand must be the same. If you have like terms, simply add or subtract the coefficients and keep the radicand the same. Examples 1. 3͙ ෆ 2 + 2͙ ෆ 2 = 5͙ ෆ 2 2. 4͙ ෆ 2 – ͙ ෆ 2 = 3͙ ෆ 2 3. 6͙ ෆ 2 + 3͙ ෆ 5 cannot be combined because they are not like terms. Here are some rules to remember when multiplying and dividing radicals: Multiplying: ͙ ෆ x × ͙ ෆ y = ͙ ෆෆෆ xy ͙ ෆ 2 × ͙ ෆ 3 = ͙ ෆ 6 Dividing: Counting Problems and Probability The probability of an event is the number of ways the event can occur, divided by the total possible outcomes. The probability that an event will NOT occur is equal to 1 – P(E). P1E2ϭ Number of ways the event can occur Total possible outcomes B 25 16 ϭ 2 25 2 16 ϭ 5 4 B x y ϭ 2 x 2 y – ARITHMETIC– 335 The counting principle says that the product of the number of choices equals the total number of pos- sibilities. For example, if you have two choices for an appetizer, four choices for a main course, and five choices for dessert, you can choose from a total of 2 × 4 × 5 = 40 possible meals. The symbol n! represents n factorial and is often used in probability and counting problems. n! = (n) × (n – 1) × (n – 2) × × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Permutations and Combinations Permutations are the total number of arrangements or orders of objects when the order matters. The formula is , where n is the total number of things to choose from and r is the number of things to arrange at a time. Some examples where permutations are used would be calculating the total number of dif- ferent arrangements of letters and numbers on a license plate or the total number of ways three different peo- ple can finish first, second, and third in a race. Combinations are the total number of arrangements or orders of objects when the order does not mat- ter. The formula is , where n is the total number of objects to choose from and r is the size of the group to choose. An example where a combination is used would be selecting people for a commit- tee. Statistics Mean is the average of a set of numbers. To calculate the mean, add all the numbers in the set and divide by the number of numbers in the set. Find the mean of 2, 3, 5, 10, and 15. The mean is 7. Median is the middle number in a set. To find the median, first arrange the numbers in order and then find the middle number. If two numbers share the middle, find the average of those two numbers. Find the median of 12, 10, 2, 3, 15, and 12. First put the numbers in order: 2, 3, 10, 12, 12, and 15. Since an even number of numbers is given, two numbers share the middle (10 and 12). Find the aver- age of 10 and 12 to find the median. The median is 11. 10 ϩ 12 2 ϭ 22 2 2 ϩ 3 ϩ 5 ϩ 10 ϩ 15 5 ϭ 35 5 n C r ϭ n! r!1n Ϫ r 2! n P r ϭ n! 1n Ϫ r 2! 2 – ARITHMETIC– 336 Mode is the number that appears the most in a set of numbers and is usually the easiest to find. Find the mode of 33, 32, 34, 99, 66, 34, 12, 33, and 34. Since 34 appears the most (three times), it is the mode of the set. NOTE: It is possible for there to be no mode or several modes in a set. Range is the difference between the largest and the smallest numbers in the set. Find the range of the set 14, –12, 13, 10, 22, 23, –3, 10. Since –12 is the smallest number in the set and 23 is the largest, find the difference by subtracting them. 23 – (–12) = 23 + (+12) = 35. The range is 35. – ARITHMETIC– 337 . after 10 years? x ϭ 25% decrease in price 24x 24 ϭ 600 24 24x ϭ 600 6 24 ϭ x 100 24 Ϫ 18 24 ϭ x 100 part whole ϭ change original cost ϭ % 100 x ϭ 10. 5804 Ϸ $10. 58 100 x 100 ϭ 105 8.04 100 100 x ϭ 105 8.04 x 235.12 ϭ 4.5 100 part whole ϭ change original. 12, 10, 2, 3, 15, and 12. First put the numbers in order: 2, 3, 10, 12, 12, and 15. Since an even number of numbers is given, two numbers share the middle (10 and 12). Find the aver- age of 10. PRT. Substitute the values from the problem into the formula I = (650)(.03) (10) . Multiply I = 195 Since she will make $195 in interest over 10 years, she will have a total of $195 + $650 = $845 in her account. Exponents The