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

Anticipatory Set

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.

Language of the Discipline  PROPORTION: An equation stating that two RATIOS are EQUAL.  Examples: 1/2 =2/4a/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 = 12/14Are 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

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.

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.

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 $  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.

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/db and d CANNOT equal ZERO (0). ad = bc

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

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 =  1.4H/1.4 =  H = 114  DOUBLE CHECK  1.4/5.7 = 28/114  (1.4)(114) = (5.7)(28)  =  Answer is CORRECT

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.

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.

Checking for Understanding Question #1  Question #1:  A store sells 5 pairs of socks for only $ 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

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

Checking for Understanding Question #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. $ B. $ C. $ D. $390.00

Checking for Understanding Question #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

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

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.