Proportional Reasoning and Ratios for Solving Problems

1. Warm Up #1 0.375, 37.5% 125%, 1 1/4 1.) Convert 3/8 into a decimal and percent. 2.) Convert 1.25 into percent and fraction. 3.) Correna wants to wrap a present. She has a piece of ribbon that is 2 ¼ feet long and another that is 3.7 feet long. What is their combined length? 2 ¼ = 2.25 feet, so 2.25 ft. + 3.7 ft. = 5.95 ft.

2. Ratio and Proportion

3. Ratio and Proportion What am I Learning Today? How will I show that I learned it? Write ratios in three ways using the correct order to accurately represent quantitative relationships Write and solve problems using proportional reasoning

4. Vocabulary • Ratio: A comparison of two quantities by division and measured in the same unit Proportion: An equation that states that two ratios are equivalent

5. Questions Notes Questions Answers • 3 ways: • As a fraction: 1/2 • With the word “to”: 1 to 2 • - With a colon (:): 1:2 How do I write a ratio? What do ratios compare? Ratios can compare: - part to part (boys to girls) - part to whole (boys to total students) - whole to part (total students to boys) What are equivalent ratios? Ratios that name the same comparison How do I find equivalent ratios? Multiplying or dividing both terms in the ratio by the same number

6. 29 12 29 ___ ___ 12 12 29 For a time, the Boston Symphony Orchestra was made up of 95 musicians. For example, you can use a ratio to compare the number of violins (29) with the number of violas (12). This ratio can be written in three ways: 29 to 12, 29:12, or Notice that the ratio of violins to violas, is different from the ratio of violas to violins, . The order of the terms is important.

7. or 5 to 2 or 5:2 5 7 __ __ __ 2 5 14 14 or 7 to 14 or 7:14 or 14 to 5 or 14:5 Writing Ratios Use the table to write the ratio. cats to rabbits dogs to total number of pets Part to part Part to whole total number of pets to cats Whole to part

8. 3 9 1 1 3 3 9 3 __ __ __ __ __ __ __ __ 2 2 6 18 6 6 18 6 3 ÷ 3 ____ 6 ÷ 3 3 • 3 6 • 3 So , , and are equivalent ratios. Writing Equivalent Ratios Write three equivalent ratios to compare the number of diamonds to the number of spades in the pattern. number of diamonds There are 3 diamonds and 6 spades. = number of spades There is 1 diamond for every 2 spades. = = If you triple the pattern, there will be 9 diamonds for 18 spades. = =

9. TRY THIS: Use the table to write each ratio. 1.giraffes to monkeys 2. polar bears to all bears 3. monkeys to all animals 4. all animals to all bears Give two equivalent ratios for 4:7 2:17 4:7 17:26 26:7 8:14 12:21

10. Consider this… • When might you want to simplify a ratio? • When making a comparison out of a large/uncountable group like from a survey, so you want to know that 1 out of every 4 dentists recommend a particular product rather than 25,000 out of 100,000 • When might you not want to simplify a ratio? • When you have a finite number of things and it is important to know the total, so 3 out of the 19 CDs in her collection are bluegrass.

11. 3 4 2 n __ __ __ __ = 5 15 4 8 Questions Answers Two ratios are proportionateif their cross products are equal 1) If possible, treat like equivalent fractions. How do I know if ratios are proportionate? A proportion shown by the model is = . 4 divided by2 is 2 8 divided by 2 is 4 2) If not, use cross products to solve for the missing information. Find the cross products. 5 •n= 3 • 15 The cross products are equal. 5n= 45 Divide both sides by 5 to undo the multiplication. n = 9

12. 5 6n 90 n __ __ ___ ___ = 6 18 6 6 Using Cross Products to Complete Proportions Find the missing value in the proportion. Find the cross products. 6 •n= The cross products are equal. 5 • 18 6n= 90 n is multiplied by 6. Divide both sides by 6 to undo the multiplication. = n = 15

13. Try These: Solving Proportions • Find the missing value in each proportion. • n/5 = 4/10 • 3/9 = 2/n • 6/n = 3/7 n = 2 n = 6 n = 14

14. Using Proportion to solve word problems Understand: What is asked? Underline the question. What information is given? Highlight the information. Grandma can ride the 24 miles to Blackfoot in two hours on her bike. If she rides at the same rate, how long will it take her to ride her bike the 36 miles to Rexburg? Plan: Is a ratio given? If so, set up the problem. Put the given ratio on one side of the equal sign. Put the other piece of information on the same level as its label. Put an “x” in the empty space. Solve: Solve the proportion. Ask: Is the answer realistic? Replace the variable with the answer and solve again.

15. tbsp tbsp gal gal ___ ___ ___ ___ tbsp tbsp gal gal Caution! In a proportion, the units must be in the same order in both ratios. = or =

16. f f 1 tbsp 1 tbsp ____ _____ _____ ____ 4 gal 6 gal 6 gal 4 gal Measurement Application According to the label, 1 tablespoon of plant fertilizer should be used per 6 gallons of water. How many tablespoons of fertilizer would you use for 4 gallons of water? Let f be the amount of fertilizer for 4 gallons of water. = = Write a proportion. 6 •f= 1 • 4 The cross products are equal.

17. 2 2 6f 4 __ __ __ ___ 3 6 6 3 Measurement Application Continued f is multiplied by 6. 6f=4 Divide both sides by 6 to undo the multiplication. = Write your answer in simplest form. f = tbsp You would use tbsp of fertilizer for 4 gallons of water.

18. Solving Proportions: Try These! • Shane’s neighbor pledged $1.25 for every 0.5 miles that Shane swims in the charity swim-a-thon. If Shane swims 3 miles, how much money will his neighbor donate? • A cocoa recipe calls for 4 tbsp of cocoa mix to make an 8 oz. serving. How many tbsp of cocoa mix are needed to make a 15 oz. serving? • The label on a bottle of salad dressing states that there are 3 grams of fat per tablespoon. If you use 3 tablespoons, how many grams of fat would you be getting? $7.50 7.5 Tbsp 9 g