Basic Math Review Courtesy of the Student Success Center Revised 2016 Sharon Stiehm Computation/Arithmetic Natural Numbers These are the numbers used for counting, sometimes called “Counting Numbers.” They are … 1, 2, 3, 4, 5, 6, 7, 8, 9, 10… There is no largest natural number and the smallest natural number is one (1). Whole Numbers To answer questions such as, “How many” and “How much,” we generally use whole numbers. The set or collection of whole numbers is 0, 1, 2, 3, 4, 5, 6, 7, … There is no largest whole number, and the smallest whole number is zero (0). Integers Consist of all Natural Numbers, Whole Numbers, and Negative Numbers. These numbers can best be illustrated on a “number line.” For each Natural Number, 1, 2, 3, 4, 5, …, there is an opposite number to the left of zero, -1, -2, -3, -4, -5, and so on. Negative numbers are considered to be the “opposites” of positive numbers. Rational Numbers This is a larger number system, which includes the number systems mentioned above and “quotients of integers with nonzero divisors.” Quotients of integers are commonly expressed as “Fractions and Decimals.” Examples of these are… 2/3, -2/3, 7/1, -0.17, 0/8, -8.75, .3334 Every rational number has a point on the number line. However, there are some points on the line for which there is no rational number. These points correspond to what is called “irrational numbers.” Fractions Fractions are a portion of a whole number and they consist of a numerator, denominator, and fraction bar. In the fraction ½, the (1) one is the numerator (2) two is the denominator (/ or --) is the fraction bar that separates the numerator and denominator. Fractions are another way of showing division. One-half (1/2) can be expressed as 1 ÷ 2. When evaluating this expression we get the decimal five-tenths (.5). Revised 2016 Sharon Stiehm Decimals Decimals are fractions whose denominators are powers of ten (10). The number of decimal places equals the number of zeros in the denominator: 1/4 or 25/100 in decimal form is .25 ¾ or 75/100 in decimal form is .75 When adding or subtracting decimal numbers, place the numbers in a column with the decimal points lined up vertically. Percentage “Of all wood harvested, 35% is used for paper production.” This means for every 100 tons of wood harvested, 35 tons is used to produce paper. 35% is also considered a ratio of 35 to 100, or 35:100. Using a percentage is a more convenient way of showing this relationship. A percentage is changed to a decimal number by dropping the percent sign and moving the decimal two places to the left. (Example: 37.5% = .375 and 45% = .45) Irrational Numbers As mentioned before, irrational numbers represent points on the number line for which there is no rational number. Examples of irrational numbers are… π (read as “pi”) and √2 (square root of “2”). π is used in finding the area and circumference of a circle, the approximate number for π is 3.141592654… √2 is the length of the diagonal of a square with sides of length one (1). There is no rational number that can be multiplied by itself to get 2. 1.4 is an approximation of √2 because (1.4)2 = 1.96 1.4142 is an even better approximation because (1.4142)2 = 1.99996164 Real Numbers Real Numbers are the set of all numbers corresponding to points on the number line and include… Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Applications An application is one of the main uses of mathematics. It is how we use our math skills in everyday life. (The previous explanation for percentage is a good example). Problem Solving Problem solving is the other main use of mathematics. Look at a situation and try to translate the problem into mathematical terms. Revised 2016 Sharon Stiehm Five Steps for Problem Solving 1. Familiarize yourself with the situation. If the problem is described in words, read it carefully. Draw a picture whenever possible. Choose a letter like a, b, c, n, x, y, z, etc. to represent the unknown quantity. In math, we call this letter the “variable.” 2. Translate the problem into an equation. 3. Solve for the variable. 4. Check the answer in the original word problem. 5. State the answer to the problem with appropriate units. Order of Operations Rules for “Order of Operations”: 1. Begin with the innermost expression, do all calculations within parentheses ( ), brackets [ ], and braces { } before doing expressions outside these symbols. 2. Evaluate all exponential expressions. 3. Do all multiplications and divisions in order from left to right. 4. Do all additions and subtractions in order from left to right. Revised 2016 Sharon Stiehm Whole Number Problems 1. True or False: -1, 2, 4, -3, 0, and 12 are all whole numbers. 2. True or False: 3, 4, 7, 78, and 1,000,000 are all whole numbers. 3. What digit is in the ones place? 2356 4. What digit is in the tens place? 2356 5. Write a sentence in which the number 250,000,000 is used. 6. Connor is 6’4” tall and weighs 170 lbs. If Connor wants to weigh 190 lbs. by the next basketball season, how many pounds per month does he need to gain? The basketball season starts in 4 months. 7. What is the sum of these numbers: 123, 556, 2, and 2? 8. Find the product of these whole numbers: 23 x 54. 9. Round to the nearest whole number: 86.256 10. Write a word name for the number in the sentence: In a recent year, the average salary of a player in the NBA was $1,867,000. 11. Kyle has to be at school by 8:00 a.m. He drives 30 mph and lives 5 miles from school. a. How long will it take him to get to school? b. What time must Kyle leave for school to arrive five minutes early? 12. Dr. Schindler orders 20 mg. of Paxil in the morning and 20 mg. at night. He wants his patient to be on this medication for 30 days. How many 20 mg. pills will the patient receive from the pharmacist? 13. Amie needs to buy a graphing calculator for college. The one she wants costs $110. Amie only has $65. If she budgets $15 from her paycheck each week, how long will Amie need to save before she can buy the calculator? Revised 2016 Sharon Stiehm Whole Number Problems - Answer Sheet 1. False. We use whole numbers to count objects. It is impossible to count objects using negative numbers. Therefore, negative numbers are not included and zero is the smallest whole number. 2. True. Whole numbers increase to infinity. 3. 6 (2356) 4. 5 (2356) 5. Answers will vary; however, the number 250,000,000 is written as two-hundred- fifty million. 6. Connor needs to gain 20 pounds and he has 4 months to do it. Divide the number of pounds by the number of months to find how many pounds he needs to gain each month. 20 ÷ 4 = 5. Answer: Connor will need to gain five pounds per month. 7. 702 123 556 2 + 21 702 8. 1,242 23 92 x 54 + 1150 92 1242 9. 86 To round to the nearest tenth, find the tenths place (86.256). Look at the number to the right. If the number is five or more, round up. If it is four or less, round down by dropping the decimal numbers. 10. One-million, eight-hundred sixty-seven thousand dollars. 11. a. Ten minutes. Use the distance formula (D = r t) Distance is equal to rate multiplied by the amount of time. If his distance is 5 miles and his rate is 30 mph, solve for “t” (time). (Express 30 mph as 30 ÷ 60 or 30/60) 30/60 t = 5 miles 1/2 t = 5 (Multiply both sides by 2 to eliminate the fraction) t = 10 Answer: It takes Kyle 10 minutes to get to school. b. 7:45 A.M. If it takes Kyle ten minutes to get to school by 8:00, then he must leave by 7:45 to arrive five minutes early at 7:55. Revised 2016 Sharon Stiehm 12. 60 20-mg pills. If the patient is taking two pills a day for 30 days, multiply the number of pills per day by the number of days. 13. Three weeks. Amie already has $65, subtract this amount from $110 and she only needs $45. If Amie budgets $15 per week, divide the amount needed by the amount budgeted each week, 45 ÷ 15 = 3. In three weeks, Amie will have enough money to buy the graphing calculator. Revised 2016 Sharon Stiehm Reading Fractions Fractions are a useful way to express remainders in math problems. The number .25 can also be written as 1/4. Five-tenths written in fraction form is 5/10 or when reduced to lowest terms is 1/2. Reducing Fractions To reduce a fraction, follow these steps: 1. Find the “largest common factor” (LCF) for the numerator and denominator. 2. Then divide the numerator and denominator by this number. For example in the fraction 21/49, the LCF is 7. 21 ÷ 7 = 3 and 49 ÷7 = 7. Therefore, 21/49 reduced to lowest terms is 3/7. Mixed Numbers In a mixed number, a whole number is written next to a proper fraction. Examples are 1½, 3¾, and 1¼.