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BASIC COLLEGE MATH MT102 Whole Number Objectives Standard and expanded form Write a standard numeral in words Determine if a given number is less than or greater than another. Round a number to the nearest ten, hundred, or thousand Be able to add, subtract, multiply and divide Determine if a number is prime or composite Find the prime factors of a number Write a number as the product of primes Order of operations Solving simple equations Standard and expanded form Examples of standard form: 406 5143 98 Examples of expanded form: 400 +0+6 5000 + 100 + 40 + 3 90 + 8 Write a standard numeral in words using place value Know your place value chart! Pg. 7 The “and” is only used to hold the decimal point The place values behind the decimal end with a “th” Determine if a given number is less than or greater than another. A < B (A is less than B) Example: 5<9 A > B (A is greater than B) Example: 12 > 5 Round a number to the nearest ten, hundred, or thousand Rules for rounding: Step 1: Underline the place to which you are rounding. Step 2: If the first number to the right of the underlined place is 5 or more, add one to the underlined number. Otherwise do not change the underline number. Step 3: Change all the numbers to the right of the underlined number to zeros. Be able to add, subtract, multiply and divide The definitions: Addend = numbers to be added Sum = the result or total in an addition problem Minuend = the larger of the numbers in a subtraction problem Subtrahend = the number being subtracted Difference = the result in a subtraction problem Multiplier, multiplicand, factor = numbers being multiplied Product = the result in a multiplication problem Dividend = the number being divided Divisor = the number used to divide Quotient = the result in a division problem Be able to add, subtract, multiply and divide Practice, practice, practice Determine if a number is prime or composite A prime number is a number having exactly two different factors, itself and 1. examples: 2, 3, 5, 7, and 11 A composite number is any number that is greater than one and not prime. example: 4, 6, 8, 9, and 12 Find the prime factors of a number Take the number and divided by the first few primes 2, 3, 5, 7, 11, and so on For help look at the rules of divisibility Pg. 73 & 74 Write a number as the product of primes You can write in the form of a factor tree or in base and exponent form. Pg. 75 Order of operations PEMDAS Do all calculations inside grouping symbols like parentheses, brackets, or braces Evaluate all exponential expressions Do all multiplications and divisions in order from left to right Do additions and subtractions in order left to right. Solving simple equations The solution of an equation is the replacement that makes the equation a true statement. When we find the solution of an equation we say that we have solved the equation. Examples: X – 7 = 14 X + 13 = 26 Fractions p.111 Diagrams http://www.visualfractions.com/index.htm Proper and Improper Fractions p.111 A proper fraction is a fraction in which the numerator is less than the denominator. An improper fraction is a fraction in which the numerator is equal to or greater than the denominator. Improper Fractions as Mixed Numbers p.112 A mixed number is a number representing the sum of a whole number and a proper fraction. To write an improper fraction as a mixed number: Divide the numerator by the denominator. This will be your whole number in your mixed number The fractional part uses the remainder as the numerator and the same denominator as the original fraction. Example: 17/2 = 8 1/2 Mixed Numbers as Improper Fractions p.1113 To write a mixed number as an improper fraction: Multiply the denominator by the whole number and add the numerator. Use the number obtained above as the numerator and the same denominator as the denominator. Equivalent Fractions p.121 Equivalent fractions Fraction A can 15/30 be reduced to equal fraction B =½ To obtain an equivalent fraction you multiply the numerator and denominator by the same number. 3/5 = 3(2)/5(2) = 6/10 Reducing Fractions p. 123 A fraction is reduced to lowest terms when there are no common factors in the numerator and denominator. You can usually start by dividing the numerator and denominator by 2 or 3 (primes) This is done quickly by finding the GCF (greatest common factor) Comparing Fractions p.124 To compare fractions write both fractions with the same denominator. Easiest: product of both denominators Example: 3/5, 10/30 3(30)/5(30) = 90/150, 10(5)/30(5) = 50/150 Which fraction is larger? 3/5 > 10/30 Multiplication of Fractions and Mixed Numbers p.134 Multiply fractions across Products of numerators Products of denominators Multiply mixed numbers turn all mixed numbers into improper fractions Multiply away Fractions and exponents p.134 (2/3)2 2/3(2/3) = 4/9 Division of Fractions and Mixed Numbers p.135 Flip and multiply! Flip and multiply! Flip and multiply! 3/5 divided by 2/3 (3/5)3/2 = 9/10 Dividing mixed numbers turn all mixed numbers into improper fractions Divide away Adding and Subtracting Fractions When adding/subtracting fractions: The fractions must have the same denominator Finding the lowest common denominator You can list multiples p. 145 You can divide each denominator by primes (factor tree) Finding the greatest common denominator Multiply the denominators REDUCE! Adding and Subtracting Mixed Numbers Convert mixed numbers in to improper fractions Find a common denominator Add or subtract away! Order of operations Please Excuse My Dear Aunt Sally Decimals p.192 Write the word name for a decimal Write a decimal in expanded form Add two or more decimals Subtract one decimal from another Multiply two or more decimals Divide one decimal by another Round decimals Write a fraction as an equivalent decimal Write repeating decimal as a fraction Compare two or more decimals Compare fractions to decimals Order of operations for decimals Write the word name for a decimal p.192 Remember the decimals point means “and” Example 567.89 Five hundred sixty seven and eighty nine hundredths Write a decimal in expanded form p.193 Example 245.34 200 + 40 + 5 + 3/10 + 4/100 Add two or more decimals p.193 The decimal points need to be lined up Remember to add from right to left Example 245.67 + 32.23 277.9 Subtract one decimal from another p.196 The decimal points need to be lined up Remember to subtract from right to left Example 145.67 - 22.20 123.47 Multiply two or more decimals p.204 Multiply the two decimal numbers as if they were whole numbers. The number of decimal digits in the product is the sum if the number of decimals digits in the factors Example 2.29 → two decimal digits x 3.5 → + one decimal digit 1145 687 8.015 → three decimal digits Divide one decimal by another p.206 Write the problem in long division form Move the decimal point in the divisor until a whole number is obtained Move the decimal point in the dividend the same number of places as you moved in the divisor Place a decimal point directly above the decimal point in the dividend Divide Divide one decimal by another p.206 6.5 √52 65 √52 65 √520. 65 √520. Round decimals p.208 Same principle as rounding whole numbers Example Round 45.68 45.6783 to the nearest hundredth Write a fraction as an equivalent decimal p.215 You need to do the division in the fraction to get the equivalent fraction Example 2 ⅜ = 19/8 = 2.375 Write repeating decimal as a fraction p.218 Write the repeating part as the numerator of the fraction The denominator consist of as many nines as there are digits in the repeating decimal Example .¯9 9/9 Another example .¯324 324/999 Compare two or more decimals p.224 To compare decimals make sure all decimals have the same number of decimal digits (use extra 0s to the right of the decimal point if necessary) Compare corresponding digits starting at the left until two of the digits are different. The number with the larger digit is the larger of the decimals Example 9.234 > 9.210 Compare fractions to decimals p.225 Write both in the same form (decimals or fractions) Order of operations for decimals p.227 Again Please Excuse My Dear Aunt Sally Ratio, Rate & Proportions Ratio = a quotient of two numbers Rate = is a ratio used to compare two different kinds of measures Proportions = a equation that states that two ratios are equal. 3 ways to write ratios 3 to 8 3:8 ⅜ * ratios can be reduced, but should not be written as mixed numbers! Reducing ratios If ratio has decimals, move the decimal point to make the numbers whole numbers. Remember you have to move the decimal point the same number of digits on the top and bottom. Example 7.26 to 2.31 726 to 231 22 to 7 Using mixed numbers in Ratios Write 2⅓ days to 2 days as a ratio in lowest terms. 2⅓ 2 Change mixed number to an improper number 7/3 2 7X 1 =7 3 2 6 Solving for an unknown using proportions. 5=x 8 12 5(12) = 8(x) 60 = 8x 60 = 8x/8 60/8 = x X = 7.5 Solving for a unit rate Unit rate = a rate in which the denominator is 1 is called a unit rate. Solving for a unit rate If my car drove for 450 miles on one tank of gas (15 gallons), what is the unit rate? 450 miles for 15 gallons X miles for 1 gallon 450 = x 15 1 450 = 15(x) 450/15 = x X = 30 Remember to ALWAYS write your units of measurement My car gets 30 miles per gallon!