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Ratios and Rates

Ratios and Rates. Wednesday, December 12, 2012. Warm up (Go ahead and get your calculator):. 1. Sally’s elbow macaroni sells for $3.49 for 15 ounces. How much would it be for 8 ounces? Be sure to round to the nearest penny. (hundredths) 2. -12 + 3(8) - (5 – 6) =.

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Ratios and Rates

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  1. Ratios and Rates Wednesday, December 12, 2012

  2. Warm up (Go ahead and get your calculator): 1. Sally’s elbow macaroni sells for $3.49 for 15 ounces. How much would it be for 8 ounces? Be sure to round to the nearest penny. (hundredths) 2. -12 + 3(8) - (5 – 6) =

  3. What have we been doing this week? • Learned how to write and reduce ratios • Learned what a rate is and how to calculate unit rates • Learned that equivalent ratios (like in our tables) can be graphed and form a straight line(linear relationship) • Learned that equivalent ratio graphs start at (0,0) and in order to graph a line you need two points.

  4. By the end of class you should be able to…. • Explain what a proportion is • Decide if two ratios are proportional by either reducing each ratio or comparing the cross products • Create a proportion when given a ratio • Find the missing value in a proportion

  5. 5-2 Identifying and Writing Proportions 25 15 10 6 = Reading Math Read the proportion by saying “ten is to six as twenty-five is to fifteen.” Course 2 An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios and are equivalent. 10 6 25 15 25 15 10 6 =

  6. 5-2 Identifying and Writing Proportions 8 17 24 ÷ 3 51 ÷ 3 24 51 Simplify . = 72 ÷ 8 72 128 9 16 = Simplify . 128 ÷ 8 9 16 8 17 Since = , the ratios are not proportional. Course 2 Additional Example 1A: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 24 51 72 128 A. ,

  7. 5-3 Solving Proportions 6 15 2 5 = CROSS PRODUCT RULE a b c d In the proportion = , the cross products, a · d and b · c are equal. Course 2 5 · 6 = 30 2 · 15 = 30 You can use the cross product rule to solve proportions with variables.

  8. 5-2 Identifying and Writing Proportions 150 105 10 7 150 ÷ 15 105 ÷ 15 Simplify . = 90 ÷ 9 90 63 10 7 = Simplify . 63 ÷ 9 10 7 10 7 Since = , the ratios are proportional. Course 2 Additional Example 1B: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 150 105 B. , 90 63

  9. 5-2 Identifying and Writing Proportions 6 7 54 ÷ 9 63 ÷ 9 54 63 Simplify . = 72 ÷ 72 1 2 72 144 = Simplify . 144 ÷ 72 1 2 6 7 Since = , the ratios are not proportional. Course 2 Try This: Example 1A Determine whether the ratios are proportional. 54 63 72 144 A. ,

  10. 5-2 Identifying and Writing Proportions Course 2 You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number.

  11. 5-2 Identifying and Writing Proportions Course 2 Insert Lesson Title Here Additional Example 3: Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. 3 5 A. 3 5 3 · 2 5 · 2 6 10 Multiply both the numerator and denominator by any number such as 2. = = 6 10 3 5 = Write a proportion. 28 16 B. Divide both the numerator and denominator by any number such as 4. 28 16 28 ÷ 4 16 ÷ 4 7 4 = = 7 4 28 16 = Write a proportion.

  12. 5-2 Identifying and Writing Proportions 3 10 9 21 4 7 9 21 3 10 2 3 , ; proportional , ; proportional , ; not proportional 3 8 2 5 , 15 40 16 40 , ; not proportional Course 2 Insert Lesson Title Here Lesson Quiz Determine whether the rates are proportional by writing them in simplest form and comparing them. 9 30 12 40 12 21 10 15 , , 1. 2. Determine if the ratios are proportional by finding a common denominator. 3 7 9 21 4. , 3.

  13. Solving for missing values in Proportions

  14. 5-3 Solving Proportions 6 15 2 5 = CROSS PRODUCT RULE a b c d In the proportion = , the cross products, a · d and b · c are equal. Course 2 5 · 6 = 30 2 · 15 = 30 You can use the cross product rule to solve proportions with variables.

  15. 5-3 Solving Proportions 9 15 m 5 = Course 2 Additional Example 1: Solving Proportions Using Cross Products Use cross products to solve the proportion. 15 · m = 9 · 5 The cross products are equal. 15m = 45 Multiply. 15m 15 = 45 15 Divide each side by 15 to isolate the variable. m = 3

  16. 5-3 Solving Proportions 6 7 m 14 = Course 2 Insert Lesson Title Here Try This: Example 1 Use cross products to solve the proportion. 7 · m = 6 · 14 The cross products are equal. 7m = 84 Multiply. 7m 7 84 7 Divide each side by 7 to isolate the variable. = m = 12

  17. 5-3 Solving Proportions Course 2 When setting up a proportion to solve a problem, use a variable to represent the number you want to find. In proportions that include different units of measurement, either the units in the numerators must be the same and the units in the denominators must be the same or the units within each ratio must be the same. 16 mi 4 hr 8 mi x hr 16 mi 8 mi 4 hr xhr = =

  18. 5-3 Solving Proportions 1 Understand the Problem Course 2 Try This: Example 2 John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level? Rewrite the question as a statement. • Find the number of pints of coolant required to raise the level to the 25 inch level. List the important information: • 6 pints is the 10 inch mark.

  19. 5-3 Solving Proportions Make a Plan 2 Course 2 Try This: Example 2 Continued Set up a proportion using the given information. Let x be the unknown amount. 6 pints 10 inches x 25 inches =

  20. 5-3 Solving Proportions 3 Solve Course 2 Try This: Example 2 Continued 6 10 x 25 Write the proportion. = 10 · x = 6 · 25 The cross products are equal. 10x = 150 Multiply. Divide each side by 10 to isolate the variable. 10x 10 150 10 = x = 15 15 pints of coolant will fill the radiator to the 25 inch level.

  21. 5-3 Solving Proportions x 9 19 57 = 2. Course 2 Insert Lesson Title Here Lesson Quiz: Part 1 Use cross products to solve the proportion. 45 t 25 20 1. = t = 36 x = 3 2 3 r 36 = 3. r = 24 n 10 28 8 n = 35 4. =

  22. Wrap up • Today you learned how to identify a proportion. (Is this a true proportion? 2/3 = 10/15) • Create a proportion when given a ratio (Given: create a proportion with 6/7) • Find the missing value in a proportion (m/54 = 5/12)

  23. Homework Workbook pages 95 and 96

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