1. Do Now Find the unit rate. 1. 18 miles in 3 hours 2. 6 apples for $3.30 3. 3 cans for $0.87 4. 5 CD’s for $43 Course 2 5-4 Identifying and Writing Proportions 6 mi/h $0.55 per apple $0.29 per can $8.60 per CD Hwk: p 45 & 46

2. EQ: How do I find equivalent ratios and to identify proportions and solve proportions by using cross products? Course 2 5-4 Identifying and Writing Proportions M7N1.b Compare and order rational numbers, including repeating decimals; M7N1.d Solve problems using rational numbers; M7A2.a Given a problem, define a variable, write an equation, solve the equation, and interpret the solution; M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations

3. Vocabulary equivalent ratios proportion Course 2 5-4 Identifying and Writing Proportions

4. Course 2 5-4 Identifying and Writing Proportions 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 10 6 = 25 15 Read the proportion by saying “ten is to six as twenty-five is to fifteen.” Reading Math 10 6 = 25 15

5. Course 2 5-4 Identifying and Writing Proportions If two ratios are equivalent, they are said to be proportional to each other, or in proportion.

6. Determine whether the ratios are proportional. Additional Example 1A: Comparing Ratios in Simplest Forms Course 2 5-4 Identifying and Writing Proportions, 24 51 72 128 72 ÷ 8 128 ÷ 8 = 9 16 24 ÷ 3 51 ÷ 3 = 8 17 Simplify. 24 51 Simplify. 72 128 8 17 Since =, the ratios are not proportional. 9 16

7. Determine whether the ratios are proportional. Additional Example 1B: Comparing Ratios in Simplest Forms Course 2 5-4 Identifying and Writing Proportions, 150 105 90 63 90 ÷ 9 63 ÷ 9 = 10 7 150 ÷ 15 105 ÷ 15 = 10 7 10 7 Since =, the ratios are proportional. 10 7

8. Determine whether the ratios are proportional. Check It Out: Example 1A Course 2 5-4 Identifying and Writing Proportions, 54 63 72 144 72 ÷ 72 144 ÷ 72 = 1 2 54 ÷ 9 63 ÷ 9 = 6767 6767 Since =, the ratios are not proportional. 1212

9. Determine whether the ratios are proportional. Check It Out: Example 1B Course 2 5-4 Identifying and Writing Proportions, 135 75 9494 9 4 135 ÷ 15 75 ÷ 15 = 9 5 9595 Since =, the ratios are not proportional. 9494

10. Directions for making 12 servings of rice call for 3 cups of rice and 6 cups of water. For 40 servings, the directions call for 10 cups of rice and 19 cups of water. Determine whether the ratios of rice to water are proportional for both servings of rice. Additional Example 2: Comparing Ratios Using a Common Denominator Course 2 5-4 Identifying and Writing Proportions Write the ratios of rice to water for 12 servings and for 40 servings. Ratio of rice to water, 12 servings: 3636 Ratio of rice to water, 40 servings: 10 19 3636 = 3 · 19 6 · 19 = 57 114 10 19 = 10 · 6 19 · 6 = 60 114 57 114 60 114 Since =, the two ratios are not proportional. Servings of Rice Cups of Rice Cups of Water 1236 401019

11. Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans. Check It Out: Example 2 Course 2 5-4 Identifying and Writing Proportions Write the ratios of beans to water for 8 servings and for 35 servings. Ratio of beans to water, 8 servings: 4343 Ratio of beans to water, 35 servings: 13 9 4343 = 4 · 9 3 · 9 = 36 27 13 9 = 13 · 3 9 · 3 = 39 27 36 27 39 27 Servings of BeansCups of BeansCups of Water 843 35139 Since =, the two ratios are not proportional.

12. Course 2 5-4 Identifying and Writing Proportions You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number.

13. Additional Example 3: Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: Insert Lesson Title Here Course 2 5-4 Identifying and Writing Proportions A. 3535 3535 = 3 · 2 5 · 2 6 10 = 3535 = 6 10 B. 28 16 28 16 = 28 ÷ 4 16 ÷ 4 28 16 = 7474 = 7474

14. Check It Out: Example 3 Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: Insert Lesson Title Here Course 2 5-4 Identifying and Writing Proportions A. 2323 2323 = 2 · 3 3 · 3 6969 = 2323 = 6969 B. 16 12 16 12 = 16 ÷ 4 12 ÷ 4 16 12 = 4343 = 4343

15. Course 2 5-5 Solving Proportions For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If the cross products of the ratios are equal, then the ratios form a proportion. 5 · 6 = 30 2 · 15 = 30 = 2525 6 15

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

17. Use cross products to solve the proportion. Additional Example 1: Solving Proportions Using Cross Products Course 2 5-5 Solving Proportions 15 · m = 9 · 5 15m = 45 15m 15 = 45 15 m = 3 = m5m5 9 15

18. Check It Out: Example 1 Insert Lesson Title Here Course 2 5-5 Solving Proportions Use cross products to solve the proportion. 7 · m = 6 · 14 7m = 84 7m77m7 = 84 7 m = 12 6767 = m 14

19. Course 2 5-5 Solving Proportions 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 x hr

20. Additional Example 2: Problem Solving Application Course 2 5-5 Solving Proportions If 3 volumes of Jennifer’s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes?

21. Course 2 5-5 Solving Proportions 3434 = 26 x 3 · x = 4 · 26 3x = 104 3x 3 = 104 3 x = 34 2323 She needs 34 2323 inches for all 26 volumes. Additional Example 2 Continued

22. Check It Out: Example 2 Course 2 5-5 Solving Proportions 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?

23. Quiz Insert Lesson Title Here Course 2 5-4 Identifying and Writing Proportions Determine whether the ratios are proportional. 1. 9 30, 12 40 2. 12 21, 10 15 Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 3. 3838 4. 3737

24. Quiz Insert Lesson Title Here Course 2 5-4 Identifying and Writing Proportions 5. In pre-school, there are 5 children for every one teacher. In another preschool there are 20 children for every 4 teachers. Determine whether the ratios of children to teachers are proportional in both preschools. 20 4 5151 =

25. Lesson Quiz: Part I Insert Lesson Title Here Course 2 5-5 Solving Proportions Use cross products to solve the proportion. 6. 25 20 = 45 t 7. x9x9 = 19 57 8. 2323 = r 36 9. n 10 = 28 8

26. Lesson Quiz: Part II Insert Lesson Title Here Course 2 5-5 Solving Proportions 10.Carmen bought 3 pounds of bananas for $1.08. June paid $ 1.80 for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy?