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Chapter 5 : Ratios, Rates & Proportions Section 4

Chapter 5 : Ratios, Rates & Proportions Section 4. Solving Proportions. Anticipatory Set. California Standards. Number Sense 1.2: Interpret and use ratios in different contexts.

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Chapter 5 : Ratios, Rates & Proportions Section 4

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  1. Chapter 5: Ratios, Rates & Proportions Section 4 Solving Proportions

  2. Anticipatory Set

  3. California Standards • Number Sense 1.2: Interpret and use ratios in different contexts. • Number Sense 1.3: Use proportions to solve problems. Use Cross-Multiplication as a method for solving such problems.

  4. Language of the Discipline • PROPORTION: An equation stating that two RATIOS are EQUAL. • Examples: 1/2 =2/4 a/b = c/d, where b and d CANNOT equal ZERO • UNIT RATE: The RATE of ONE UNIT for a given quantity. • CROSS PRODUCTS PROPERTY: When given two ratios, this property states that the CROSS PRODUCTS will EQUAL each other. If the two ratios have EQUAL cross products, they form a PROPORTION. • Example: 6/7 = 12/14 Are these a PROPORTION? -Using CROSS PRODUCTS, we take opposing values and multiply. *Remember to use the Numerators and Denominators on the diagonal from each other. a/b = c/d mean (a)(d) = (b)(c) (6)(14) = (7)(12) 84 = 84 -CROSS PRODUCTS proves that these two RATIOS are a PROPORTION

  5. What is a PROPORTION?(Input) • PROPORTION: • A PROPORTION is an EQUATION stating that 2 RATIOS are EQUAL. • Some people think of EQUIVALENT Fractions as PROPORTIONAL. • Another way to test for PROPORTIONALITY is to use the Cross Products Property. • Here, 2 ratios are set equal, values are multiplied diagonally, if BOTH resulting products are EQUAL you have a PROPORTION. • If not EQUAL, the ratios are NOT PROPORTIONAL.

  6. Solving Proportions Using Unit Rate(Input/Modeling) • Once again, we realize that new skills build off of previous ones. In this case, we are revisiting UNIT RATES & applying them to a new situation. • When given a proportion, you can use UNIT RATE to solve a proportion. • First, find the UNIT RATE, then MULTIPLY to solve the problem. • Example: A store sells 4 Champion Candy bars for $3.00. You plan on purchasing 10. How much will the candy bars cost you? • 4 Champion Candy Bars cost $3.00. $3.00 for 4. $3.00/4 = $0.75 a candy bar. • The UNIT RATE is $0.75 for ONE Champion Candy Bar. • You plan on purchasing 10 candy bars. (Unit Rate)(Number of Bars) = COST. • ($0.75)(10 Candy Bars) = $7.50 • 10 Champion Candy Bars will cost you $7.50.

  7. Solving Proportions Using Unit Rate(Input/Modeling) • Example #2 • A store charges $8.40 for a dozen Hello Kitty pencils. • You only want to purchase 7 pencils. What is your total cost? • $8.40 for a DOZEN pencils means $8.40/12. • Unit Rate is $0.70 a pencil. • (Unit Rate)(Number) = COST • ($0.70)(7) = $4.90 • 7 Hello Kitty Pencils will cost you EXACTLY $4.90. • Example #3 • At a bakery, you can purchase 20 croissants for $30.00. • You would like to purchase 6 croissants for your family. How much will you be charged? • $30.00 for 20 croissants means $30.00/20. • Unit Rate is $1.50 a croissant. • (Unit Rate)(Number) = COST • ($1.50)(6) = $9.00 • The bakery will charge you $9.00 for the 6 croissants.

  8. CROSS PRODUCTS PROPERTY(Input/Modeling) • With RATIOS and PROPORTIONALITY, a Mathematic Property will come in handy. Remember that properties come in handy because that give the RULE or GUIDELINE on how to attack a problem. • The CROSS PRODUCTS PROPERTY states that if two ratios form a proportion, the CROSS PRODUCTS are EQUAL. If two ratios have EQUAL Cross Products, they form a PROPORTION. • There are two ways to look at PROPROTIONS. • ARITHMETIC: 5/7 = 25/35 (5)(35) = (7)(25) 175 = 175 • ALGEBRAIC: a/b = c/d b and d CANNOT equal ZERO (0). ad = bc

  9. Solving Using the Cross Products Property(Input/Modeling) • Referring back to CROSS PRODUCTS PORPERTY, you can use PROPORTIONS and ALGEBRA to solve. • Use PROPORTIONAL setup & Cross Products Property to solve the problem. • Example: Solve 28/35 = X/175 • Remember the Cross Products Property. You use proportion and the property to create a math equation where Algebra can solve for the One unknown value. • 28/35 = X/175 • (28)(175) = (35)(X) • 4,900 = 35X • 4,900/35 = 35X/35 • 140 = X • 28/35 = 140/175 BOTH simplify down to 4/5. CPP yields 4,900 on BOTH SIDES

  10. Solving Using the Cross Products Property(Input/Modeling) • Example #2: • Solve 2/5 = E/86.5 • 2/5 = E/86.5 • Use Cross Products • (2)(86.5) = (5)(E) • 173 = 5E • 173/5 = 5E/5 • 34.6 = E • DOUBLE CHECK • 2/5 = 34.6/86.5 • (2)(86.5) = (34.6)(5) • 173 = 173 • Answer is CORRECT • Example #3 • Solve 1.4/5.7 = 28/H • 1.4/5.7 = 28/H • Use Cross Products • (1.4)(H) = (5.7)(28) • 1.4H = 159.6 • 1.4H/1.4 = 159.6 • H = 114 • DOUBLE CHECK • 1.4/5.7 = 28/114 • (1.4)(114) = (5.7)(28) • 159.6 = 159.6 • Answer is CORRECT

  11. The Big Idea • PROPORTIONS • A pair of ratios that equal one another. • Proportions can be solved using multiple methods. • Using UNIT RATES to Solve • Use the original rate to determine a UNIT RATE. • Multiply the UNIT RATE by the NUMBER of Units to determine the Cost. • Using CROSS PRODUCTS PROPERTY to Solve • Cross Products Property states that a pair of Ratios are a PROPORTION when their cross products equal the same value. • Remember that you are taking the NUMERATOR from one Ratio and MUTLIPLYING it by the DENOMINATOR of the other. • Use this property and ALGEBRA to solve the missing value. • Once the missing cross product is determined, DOUBLE CHECK to make certain it works in the original proportion.

  12. Check for Understanding • Please determine the BEST answer for the following expression. • Carry out ALL work and calculations in your NOTES for later reference • Please write your answer on your wipe boards and wait for the teacher’s signal. • On the count of 3, hold up your wipe boards.

  13. Checking for Understanding Question #1 • Question #1: • A store sells 5 pairs of socks for only $15.00. What is the Unit Rate? Select the BEST answer: A. $15.00 a pair B. $1.50 a pair C. $3.00 a pair D. $5.00 a pair

  14. Checking for UnderstandingQuestion #2 • Question #2: • A 36 candies cost $10.80. How much would you be charged for 11 pieces of candy? Select the BEST answer: A. $39.60 B. $64.80 C. $118.80 D. $10.80

  15. Checking for UnderstandingQuestion #3 • Question #3: • A boutique sells 5 pairs of DESIGNER jeans for $650. How much would 3 pairs of jeans cost? The pair of ratios can be simplified down to: A. $1.950.00 B. $195.00 C. $675.00 D. $390.00

  16. Checking for UnderstandingQuestion #4 • Question #4: • Solve 45/55 = D/440 The pair of ratios can be simplified down to: A. 180 B. 360 C. 280 D. 320

  17. Checking for Understanding Question #5 • Question #5: • Solve 2.7/10.8 = R/75.6 Select the BEST answer: A. 18.9 B. 22.8 C. 21.7 D. 19.4

  18. Guided Practice/Independent Practice • Guided Practice: • Textbook on pg. • Work carefully, show your problem solving process, and double check all calculations. • Use scratch paper to carry out your work. • Once you have completed the assigned problems, please raise your pencil. • When you get a stamp from Ms. Graham, continue on to Independent Practice. • If you receive an “R” on your paper go to the back table. • Independent Practice • Textbook pg.

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