Pre-Algebra Lesson 5-1 Comparing and Ordering Rational Numbers

Pre-Algebra Lesson 5-1 Today, the school’s baseball and soccer teams had games. The baseball team plays every 7 days. The soccer team plays every 3 days. When will the teams have games on the same day again? 7, 14, 21, 28, 35, 42,...List the multiples of 7. 3, 6, 9, 12, 15, 18, 21,...List the multiples of 3. The LCM is 21. In 21 days both teams will have games again. Comparing and Ordering Rational Numbers Additional Examples

Pre-Algebra Lesson 5-1 Find the LCM of 16 and 36. = 144Multiply. 16 = = Write the prime factorizations. The LCM of 16 and 36 is 144. LCM = Use the greatest power of each factor. Comparing and Ordering Rational Numbers Additional Examples

Pre-Algebra Lesson 5-1 Find the LCM of 5a 4 and 15a. 5a 4 = 5 a 4 15a = 3 5 aWrite the prime factorizations. = 15a 4 Multiply. The LCM of 5a 4 and 15a is 15a 4. LCM = 3 5 a 4 Use the greatest power of each factor. Comparing and Ordering Rational Numbers Additional Examples

Pre-Algebra Lesson 5-1 Graph and compare the fractions in each pair. is on the left, so < b. – 1313, – 1616 is on the right, so >.– 1616 – 1616 – 1313 a , – 1313 – 1616 Comparing and Ordering Rational Numbers Additional Examples

Pre-Algebra Lesson 5-1 The softball team won of its games and the hockey team won of its games. Which team won the greater fraction of its games? Step 1: Find the LCM of 7 and 9. 7 = 7 and 9 = 3 2 LCM = = 63 Step 2: Write equivalent fractions with a denominator of = = Step 3: Compare the fractions > >, so The softball team won the greater fraction of its games. Comparing and Ordering Rational Numbers Additional Examples

Pre-Algebra Lesson 5-1 Order,, and from least to greatest = = = = = = The LCM of 7, 4, and 3 is 84. Use 84 as the common denominator <<, so<<. Comparing and Ordering Rational Numbers Additional Examples

Pre-Algebra Fractions and Decimals Lesson 5-2

Pre-Algebra Fractions and Decimals Lesson 5-2 The fuel tank of Scott’s new lawn mower holds gal of gasoline. Scott poured 0.4 gal into the tank. Did Scott fill the tank? 1212 = 1 ÷ 2 = 0.5 Since = 0.5 and 0.5 > 0.4, Scott did not fill the tank Additional Examples

Pre-Algebra Fractions and Decimals Lesson 5-2 Write each fraction as a decimal. State the block of digits that repeats. a. b ÷ 6 = … Divide. Place a bar over the digit that repeats. = = 0.83; the digit that repeats is ÷ 11 = …Divide. = 0.63Place a bar over the block of digits that repeats. = 0.63; the block of digits that repeats is Additional Examples

Pre-Algebra Fractions and Decimals Lesson 5-2 Write the numbers in order, from least to greatest. –0.8,,, – –1.25 < –0.8 < < 0.25 Compare the decimals. 3 ÷ 12 = 0.25 Change the fractions to decimals. –5 ÷ 4 = –1.25 From least to greatest, the numbers are, –0.8, 0.125, and – 3 12 Additional Examples

Pre-Algebra Fractions and Decimals Lesson 5-2 Write 1.72 as a mixed number in simplest form. Keep the whole number 1. Write seventy-two hundredths as a fraction = Divide the numerator and denominator of the fraction by the GCF, ÷ ÷ 4 = 1 Simplify = 1 Additional Examples

Pre-Algebra Fractions and Decimals Lesson 5-2 Write 0.18 as a fraction in simplest form. n Let the variable n equal the decimal. = 0.18 As a fraction in simplest form, 0.18 = n= 18.18Because 2 digits repeat, multiply each side by 10 2, or n 99 = Divide each side by 99. n 18 ÷ 9 99 ÷ 9 = Divide the numerator and denominator by the GCF, = Simplify. 100n= n= 0.18 – 99n= 18 The Subtraction Property of Equality lets you subtract the same value from each side of the equation. So, subtract to eliminate Additional Examples

Pre-Algebra Adding and Subtracting Fractions Lesson 5-3 Find each sum or difference. Simplify if possible. a. b = Add the numerators =Simplify = 5b5b 12 b – – = 5b5b 12 b 12 – 5 b Subtract the numerators. 7b7b = Simplify. Additional Examples

Pre-Algebra Adding and Subtracting Fractions Lesson 5-3 Simplify each difference. a. b – 2y2y – – = Use the Order of Operations to simplify. –14 24 = Simplify. = –7 24 Simplify. 2y2y – = – 5 y y 16 Rewrite using a common denominator. = 32 – 5y 16y Simplify – = 1 4 – Use a common denominator. Additional Examples

Pre-Algebra Adding and Subtracting Fractions Lesson 5-3 Suppose one day you rode a bicycle for 3 hours, and jogged for 1 hours. How many hours did you exercise? You exercised for 4 hours = Write mixed numbers as improper fractions.= Rewrite using a common denominator. = Use the Order of Operations to simplify. = = Write as a mixed number. = Simplify. Additional Examples

Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 Find = 2 5 Multiply the numerators. = Simplify. Multiply the denominators. 3 7 Additional Examples

Pre-Algebra Multiplying and Dividing Fractions Lesson a.Find = Divide the common factors. = 1212 Multiply. b. 5w5w 3w 17 Find. 5w5w 3w 17 = 5w5w 3w Divide the common factors =Multiply. Additional Examples

Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 Keesha’s desktop is a rectangle 3 ft long and 1 ft wide. What is the area of her desktop? A Area of a rectangle = length width.= = Write 3 and 1 as improper fractions, and Multiply.= 21 4 Write as a mixed number = 5 The area of Keesha’s desk is 5 ft Additional Examples

Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 a. Find ÷ ÷ = 10 7 Multiply by the reciprocal of the divisor. = Divide the common factors. = 6767 Multiply. Additional Examples

Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 (continued) b. Find ÷. 94q94q = 3232 Simplify. 27 8q 94q94q ÷ = 4q94q9 Multiply by the reciprocal of the divisor. 27 8q 27 8q = 1 4q94q9 1 Divide the common factors. 27 8q = Write as a mixed number. Additional Examples

Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 Find 4 ÷ (–3 ) ÷ (–3 ) = ÷ (– ) Change to improper fractions = (– )Multiply by –, the reciprocal of – = – Divide the common factors = –, or –1Simplify Additional Examples

Pre-Algebra Using Customary Units of Measurement Lesson 5-5 Choose an appropriate unit of measure. Explain your choice. a. weight of a hummingbird b. length of a soccer field Measure its weight in ounces because a hummingbird is very light. Measure its length in yards because it is too long to measure in feet or inches and too short to measure in miles. Additional Examples

Pre-Algebra Using Customary Units of Measurement Lesson 5-5 Use dimensional analysis to convert 68 fluid ounces to cups. 68 fl oz = Use a conversion factor that changes fluid ounces to cups. 68 fl oz 1 1 c 8 fl oz 17 = Divide the common factors and units. 68 fl oz 1 c 8 fl oz = c Simplify. = 8 c Write as a mixed number There are 8 c in 68 fl oz Additional Examples

Pre-Algebra Using Customary Units of Measurement Lesson 5-5 Fred’s fruit stand sells homemade lemonade in 6 -pint bottles for $1.99. Jill’s fruit stand stand sells homemade lemonade in 3 -qt containers for the same price. At which stand do you get more lemonade for your money? Since 7 pints > 6 pints, you get more lemonade for your money at Jill’s stand qt = qt Use a conversion factor that changes quarts to pints pt 1 qt = Divide the common factors and units. 1 2 pt 1 qt 7 qt 2 1 = 7 ptMultiply. Additional Examples

Pre-Algebra Problem Solving Strategy: Work Backward Lesson 5-6 Your flight leaves the airport at 10:00 A.M. You must arrive 2 hours early to check your luggage. The drive to the airport takes about 90 minutes. A stop for breakfast takes about 30 minutes. It will take about 15 minutes to park and get to the terminal. At what time should you leave home? Move the hands of the clock to find the time you should leave home. Write the starting time for each event. Additional Examples

Pre-Algebra Problem Solving Strategy: Work Backward Lesson 5-6 (continued) You should leave home at 5:45 A.M. Additional Examples

Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 One school recycles about of its waste paper. The student council set a goal of recycling of the school’s waste paper by the end of the year. By how much does the school need to increase its paper recycling to reach the goal? Words plus is Let n = the increase. Equation + n = fraction school recycles the increase student goal Additional Examples

Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 (continued) + n = – + n = – Subtract from each side n = Use 3 4 as the common denominator. 3 3 – n = Use the Order of Operations. 9 – 4 12 n = Simplify To meet the student council goal, the school needs to recycle more of its waste paper Additional Examples

Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 (continued) Check:Is the answer reasonable? The present fraction of paper waste that is recycled plus the increase must equal the goal. Since + = + = =, the answer is reasonable Additional Examples

Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Solve x – = x – + = +Add to each side x – = x = Use 9 3 as the common denominator x = Use the Order of Operations x = Divide the common factors x = Simplify Additional Examples

Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Solve q – 6 = – q – 6 = – q = Use the Order of Operations. – q = Simplify q = – Use 5 2 as the common denominator. q = – + Write mixed numbers as improper fractions q = 4 Write as a mixed number q – = – 1 + 6Add 6 to each side Additional Examples

Pre-Algebra Lesson 5-8 Solve 7y = (7y) = Multiply each side by, the reciprocal of y = 1313 y = Simplify Solving Equations by Multiplying Fractions Additional Examples

Pre-Algebra Lesson w = Divide the common factors w = 2 Write as a mixed number Solve w = w = Multiply each side by, the reciprocal of w = w = Simplify Solving Equations by Multiplying Fractions Additional Examples

Pre-Algebra Lesson 5-8 Solve – c = – c = c = Divide common factors = – Simplify – – c = – Multiply each side by –, the reciprocal of – Solving Equations by Multiplying Fractions Additional Examples

Pre-Algebra Lesson 5-8 How many 2 -t trucks can you place on a rail car that has a carrying capacity of 15 t? 1212 Words times is Let n = the number of trucks. Equation 2 n = 15 weight of each truck the number of trucks carrying capacity 1212 Solving Equations by Multiplying Fractions Additional Examples

Pre-Algebra Lesson 5-8 (continued) n = 15 n = Divide common factors n = 15Write 2 as n = 15Multiply each side by, the reciprocal of = 6 Simplify. You can place 6 trucks on the rail car. Solving Equations by Multiplying Fractions Additional Examples

Pre-Algebra Powers of Products and Quotients Lesson 5-9 Simplify (3z 5 ) 4. (3z 5 ) 4 = 3 4 (z 5 ) 4 Raise each factor to the fourth power. = 3 4 z 5 4 Use the Rule for Raising a Power to a Power. = 3 4 z 20 Multiply exponents. = 81z 20 Simplify. Additional Examples

Pre-Algebra Powers of Products and Quotients Lesson 5-9 a. Simplify (–3a) 4. b. Simplify –(3a) 4. (–3a) 4 = (–3) 4 (a) 4 = 81a 4 –(3a) 4 = (–1)(3a) 4 = (–1)(3) 4 (a) 4 = –81a 4 Additional Examples

Pre-Algebra Powers of Products and Quotients Lesson 5-9 Find the area of a square with side length. x4x4 A = s 2 s = length of a side x242x242 = x 2 16 = The area of the square is square units. x 2 16 = x4x4 2 Additional Examples