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Basic College Math

Basic College Math. John Moore john_moore@heald.edu 916.414.2777. 3,000,000 + 500,000 + 70,000 + 5,000 + 100 + 4. Expanded Notation. 3,575,104. Standard Notation. Standard & Expanded Notation. 3. 5. 7. 5. 1. 0. 4. Properties of Addition. 1. Associative Property of Addition

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Basic College Math

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  1. Basic College Math John Moore john_moore@heald.edu 916.414.2777

  2. 3,000,000 + 500,000 + 70,000 + 5,000 + 100 + 4 Expanded Notation 3,575,104 Standard Notation Standard & Expanded Notation 3 5 7 5 1 0 4

  3. Properties of Addition 1. Associative Property of Addition When we add three numbers, we can group them in any way. (2 + 4) + 3 = 2 + (4 + 3) 6 + 3 = 2 + 7 9 = 9 2. Commutative Property of Addition Two numbers can be added in either order with the same result. 8 + 4 = 4 + 8 12 = 12 3. Identity Property of Zero When zero is added to a number, the sum is that number. 7 + 0 = 7 0 + 12 = 12

  4. Properties of Multiplication 1. Associative Property of Multiplication When we multiply three numbers, we can group them in any way. (2  4)  3 = 2  (4  3) 8  3 = 2  12 24 = 24 2. Commutative Property of Multiplication Two numbers can be multiplied in either order with the same result. 8  4 = 4  8 32 = 32 3. Identity Property of One When one is multiplied by a number, the result is that number. 7  1 = 7 1  12 = 12 4. Distributive Property of Multiplication Multiplication can be distributed over addition without changing the result. 3  (2 + 4) = (3  2) + (3  4) 3  6 = 6 + 12 18 = 18

  5. Exponents Examples: 34 = 3  3  3  3 = 81 53 = 5  5  5 = 125 35 = 3  3  3  3  3 = 243 122 = 12  12 = 144 74 = 7  7  7  7 = 2401

  6. Order of Operations • Order of Operations • Parentheses. • Exponents. • Multiply or Divide from left to right. • Add or Subtract from left to right. Do first Do last Example: 24 ÷2 – 4  2 24 ÷2 – 4  2 = 12 – 8 = 4

  7. Order of Operations Example: Work inside the parentheses. = 4 + (16 – 13)4 – 3 Evaluate the exponent. = 4 + 34 – 3 Add or subtract. = 4 + 81 – 3 = 85 – 3 Add or subtract. = 82

  8. Fractions Chapter 2

  9. Divisibility Rules • A number is divisible by 2 if the last digit is: • 0, 2, 4, 6, or 8. • A number is divisible by 3 if: • the sum of the digits is divisible by 3. • A number is divisible by 5 if: • the last digit is 0 or 5.

  10. Multiplying Fractions To multiply fractions, multiply numerators across and denominators across.

  11. Dividing Fractions When fractions are divided, invert the second fraction and multiply.

  12. Adding Fractions Fractions must have common denominators before they can be added or subtracted. + =

  13. Creating Equivalent Fractions Fractions with unlike denominators cannot be added. In order to add fractions with different denominators: 1) Find a Common Denominator 2) Build into equivalent fractions with the Common Denominator.

  14. Example: Creating Equivalent Fractions If fractions have different denominators, find the Common Denominator and build up each fraction so that its denominators are the same.

  15. Example: Creating Equivalent Fractions

  16. Chapter 3 Decimals

  17. Place Values • Digits in a decimal number have a value dependant on the place of the digits. • Adding extra zeros to the right of the last decimal digit does not change the value of the number. • 1.2345670 • 1.234567 Ones position1.234567 Tenths1.234567 Hundredths1.234567 Thousandths1.234567 Ten thousandths1.234567 Hundred Thousandths1.234567Millionths

  18. 0 Place holder Adding Decimals Example: 718.97 + 496.5 718.97 + 496.5 1215.47 Line up decimal points

  19. 13 13 1 00 Place holders Subtracting Decimals Example: 243.967 – 84.2 243.967 – 84.2 159.767 Line up decimal points

  20. Multiplying Decimals Example: Multiply 0.17  0.4 2 decimal places 1 decimal place .068 3 decimal places in product (2 + 1 = 3)

  21. . . Place the decimal point of the answer directly above the caret.   366 549 549 Dividing by a Decimal Example: Divide 4.209 ÷ 1.83 Move each decimal point to the right two places. 2.3 Mark the new position by a caret ().

  22. .000 36 140 repeating remainder 126 140 Converting a Fraction to a Decimal Example: Write as an equivalent decimal .277

  23. Ratios, Rates and Proportions Chapter 4

  24. Equality Test for Proportions A variable is a letter used to represent a number we do not yet know. An equation has an equal sign. This indicates that the values on each side are equivalent.

  25. Solving for a Variable Example: Solve for n. Check: 5  11.4 = 57

  26. Basic College MathematicsChapter 5 – Percentages John J. Moore

  27. Changing a Percent to a Decimal • Drop the % symbol. • Convert to a fraction. • Change to decimal form. • Move decimal point two places to the left.

  28. Writing Percents as Decimals • Example: Write 19% as a decimal. 19% = 19. = 0.19 • Example: Write 2.67% as a decimal. 2.67% = 2.67 = 0.0267 Add an extra zero to the left of the 2.

  29. Fractions to Percents • Write the fraction as a decimal then convert to a percent. Example: Write as a percent. Divide. Write as a decimal. Convert to a percent.

  30. Solving Percent ProblemsUsing Equations • Use the following table to translate from a written problem to a mathematical equation.

  31. Percent Problems into Equations • Example: Translate into an equation. • Example: 24 is what percent of 144? 24 = n x 144 What is 9% of 65? n = 9% x 65

  32. Percent Proportion 10% of 500 is 50 The amount, a, is the part compared to the whole. p is the percent number. The base, b, is the entire quantity (usually follows the word of ).

  33. Markup Problems • Example: • Mark and Peggy are out to dinner. They have $66 to spend. They want to tip the server 20%, how much can they afford to spend on the meal? • n = the cost of the meal Cost of meal n + tip at 20% of meal cost $66 = 100% of n + 20% of n = $66 $66 120% of n =

  34. Simple Interest Problems • Interest is money paid for borrowing money. • Principalis the amount deposited or borrowed. • Interest rateis per year, unless otherwise stated. • If the interest rate is in years, the timeis also in years. Interest = principal  rate  time I = PR T Example: Find the simple interest on a loan of $3600 borrowed at 6% for 8 years. = 3600  0.06  8 = $1728

  35. Signed Numbers Chapter 9

  36. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 – 4.8 1.5 Negative numbers Positive numbers The Number Line A number line is a line on which each point is associated with a number. The set of positive numbers, negative numbers, and zero make up all Signed Numbers.

  37. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 “greater than” “less than” Ordering Numbers Signed numbers are listed in order on the number line. – 4 < – 1 2 > 1

  38. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 Distance of 4 Distance of 5 Absolute Value The absolute value of a number is the distance between that number and zero on a number line. | –4| = 4 |5| = 5

  39. –3 + –11 Adding Two Numbers with Same Signs • Add the absolute value of the numbers. • Use the common sign in the answer. Example: Add (– 3) + (– 11) Add the absolute values of the numbers 3 and 11. – 14

  40. 5 + – 9 Adding Two Numbers with Different Signs • Subtract the absolute value of the numbers. • Use the sign of the number with the larger absolute value. Example: Add 5 + (– 9) Subtract the absolute values of the numbers 5 and 9. – 4

  41. – 24 + 38 – 36 + 4 Adding Two Numbers with Different Signs Example: Add (–24) + (38) Subtract the absolute values of the numbers 24 and 38. 14 Example: (– 36) + 4 – 32

  42. Adding Two Numbers with Different Signs Commutative Property of Addition a + b = b + a.

  43. – 50 – 70 – 64 – 64 Adding Three or More Signed Numbers Example: (–56) + 6 + (–14) Because addition is commutative, the numbers can be added in any way. (–56) + 6 + (–14) (–56) + (–14) + 6 or + (–14) + 6

  44. The opposite of 4 is – 4. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 The sum of a number and its opposite is zero. 10 – 4 = 6 10 + (–4) = 6 Subtracting is the same as adding the opposite. 15 – 8 = 7 15 + (–8) = 7 12 – 2 = 10 12 + (–2) = 10 Opposite Numbers The opposite of a positive number is a negative number with the same absolute value. 4 + (– 4) = 0

  45. The opposite of 14 is –14. Change the subtraction to addition. Perform the addition of the two negative numbers. Subtraction of Signed Numbers To subtract signed numbers, add the opposite of the second number to the first number. Example: Subtract – 6 – 14 –6 + (–14) –20

  46. The opposite of –13 is 13. Perform the addition. Change the subtraction to addition. Subtraction of Signed Numbers Example: Subtract –21 – (–13) –21 + (13) –8 • When subtracting two signed numbers: • The first number does not change. • The subtraction sign is changed to addition. • Write the opposite of the second number. • Find the result of the addition problem.

  47. Multiplying and Dividing Signed Numbers

  48. Multiplication with Different Signs Notice the following pattern when multiplying numbers with different signs. Note that when we multiply a positive number by a negative number, we get a negative number.

  49. Multiplication with Different Signs To multiply two numbers with different signs, multiply the absolute value. The result is negative. Example: Multiply –6 (4) –6 (4) = –24 Example: Multiply 12 (–9) The result will always be negative. 12 (–9) = –108

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