262-263 C6LO-825200 3/7/03 9:03 AM Page 262 Ratio, Proportion, and Percent • Lesson 6-1 Write ratios as fractions and find Key Vocabulary unit rates. • ratio (p. 264) • Lessons 6-2 and 6-3 Use ratios and • rate (p. 265) proportions to solve problems, including scale • proportion (p. 270) drawings. • percent (p. 281) • Lesson 6-4 Write decimals and fractions as • probability (p. 310) percents and vice versa. • Lessons 6-5, 6-6, 6-7, and 6-8 Estimate and compute with percents. • Lesson 6-9 Find simple probability. The concept of proportionality is the foundation of many branches of mathematics, including geometry, statistics, and business math. Proportions can be used to solve real-world problems dealing with scale drawings, indirect measurement, predictions, and money. You will solve a problem about currency exchange rates in Lesson 6-2. 262 Chapter 6 Ratio, Proportion, and Percent 262-263 C6LO-865108 11/3/03 10:58 AM Page 263 Prerequisite Skills ToTo bebe successfulsuccessful inin thisthis chapter,chapter, you’llyou'll needneed toto mastermaster these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 6. X. For Lesson 6-1 Convert Measurements Complete each sentence. (For review, see pages 718–721.) 1. 2 ft ϭ ? in. 2. 4 yd ϭ ? ft 3. 2 mi ϭ ? ft 4. 3 h ϭ ? min 5. 8 min ϭ ? s 6. 4 lb ϭ ? oz 7. 2 T ϭ ? lb 8. 5 gal ϭ ? qt 9. 3 pt ϭ ? c 10. 3 m ϭ ? cm 11. 5.8 m ϭ ? cm 12. 2 km ϭ ? m 13. 5 cm ϭ ? mm 14. 2.3 L ϭ ? mL 15. 15 kg ϭ ? g For Lessons 6-2 and 6-3 Multiply Decimals Find each product. (For review, see page 715.) 16. 7(3.4) 17. 6.1(8) 18. 2.8 ϫ 5.9 19. 1.6 ϫ 8.4 20. 0.8 ϫ 9.3 21. 0.6(0.3) 22. 12.4(3.8) 23. 15.2 ϫ 0.2 For Lesson 6-9 Write Fractions in Simplest Form Simplify each fraction. If the fraction is already in simplest form, write simplified. (For review, see Lesson 4-5.) 24. ᎏ4ᎏ 25. ᎏ5ᎏ 26. ᎏ6ᎏ 27. ᎏ1ᎏ2 8 15 10 25 28. ᎏ2ᎏ2 29. ᎏ1ᎏ5 30. ᎏ3ᎏ6 31. ᎏ3ᎏ6 20 16 42 48 Fractions, Decimals, and Percents Make this Foldable to help you organize your notes. Begin with a piece of notebook paper. Fold in Thirds Label Fold in thirds Draw lines Fraction Decimal Percent lengthwise. along folds and label as shown. Reading and Writing As you read and study the chapter, complete the table with the commonly-used fraction, decimal, and percent equivalents. Chapter 6 Ratio, Proportion,Chapter 3 andEquations Percent 263263 264-269 C6L1-825200 3/3/03 9:26 AM Page 264 Ratios and Rates • Write ratios as fractions in simplest form. • Determine unit rates. Vocabulary are ratios used in paint mixtures? • ratio The diagram shows a gallon of paint that is • rate made using 2 parts blue paint and 4 parts • unit rate yellow paint. a. Which combination of paint would you use to make a smaller amount of the same shade of paint? Explain. Combination A Combination B b. Suppose you want to make the same shade of paint as the original mixture? How many parts of yellow paint should you use for each part of blue paint? WRITE RATIOS AS FRACTIONS IN SIMPLEST FORM Aratio is a comparison of two numbers by division. If a gallon of paint contains 2 parts blue paint and 4 parts yellow paint, then the ratio comparing the blue paint to the yellow paint can be written as follows. 2 2 to 4 2:4 ᎏᎏ 4 Recall that a fraction bar represents division. When the first number being compared is less than the second, the ratio is usually written as a fraction in simplest form. Ϭ 2 Study Tip The GCF of 2 1 2 1 ᎏᎏ ϭ ᎏᎏ The simplest form of ᎏᎏ is ᎏᎏ. Look Back 2 and 4 is 2. 4 2 4 2 To review how to write a fraction in simplest form, Ϭ see Lesson 4-5. 2 Example 1 Write Ratios as Fractions Express the ratio 9 goldfish out of 15 fish as a fraction in simplest form. Ϭ 3 9 3 ᎏᎏ ϭ ᎏᎏ Divide the numerator and denominator by the GCF, 3. 15 5 Ϭ 3 The ratio of goldfish to fish is 3 to 5. This means that for every 5 fish, 3 of them are goldfish. 264 Chapter 6 Ratio, Proportion, and Percent 264-269 C6L1-825200 3/3/03 9:27 AM Page 265 When writing a ratio involving measurements, both quantities should have the same unit of measure. Example 2 Write Ratios as Fractions Express the ratio 3 feet to 16 inches 3 ft as a fraction in simplest form. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3233 34 35 36 3 feet 36 inches ᎏᎏ ϭ ᎏᎏ Convert 3 feet to inches. 16 in. 16 inches 16 inches 9 inches ϭ ᎏᎏ Divide the numerator and denominator by the GCF, 4. 4 inches Written in simplest form, the ratio is 9 to 4. Concept Check Give an example of a ratio in simplest form. FIND UNIT RATES Arate is a ratio of two measurements having different kinds of units. Here are two examples of rates. Miles and hours are Dollars and pounds are different kinds of units. different kinds of units. 65 miles in 3 hours $16 for 2 pounds When a rate is simplified so that it has a denominator of 1, it is called a unit rate. An example of a unit rate is $5 per pound, which means $5 per 1 pound. Example 3 Find Unit Rate Study Tip SHOPPING A package of 20 recordable CDs costs $18, and a package of Alternative Method 30 recordable CDs costs $28. Which package has the lower cost per CD? Another way to find the Find and compare the unit rates of the packages. unit rate is to divide the Ϭ cost of the package by 20 the number of CDs in the package. ᎏ18 dolᎏlars ϭ ᎏ0.9 doᎏllars Divide the numerator and denominator 20 CDs 1 CD by 20 to get a denominator of 1. Ϭ 20 For the 20-pack, the unit rate is $0.90 per CD. Ϭ 30 – ᎏ28 dolᎏlars ϭ ᎏ0.93 dᎏollars Divide the numerator and denominator 30 CDs 1 CD by 30 to get a denominator of 1. Ϭ 30 For the 30-pack, the unit rate is $0.93 per CD. So, the package that contains 20 CDs has the lower cost per CD. Concept Check Is $50 in 3 days a rate or a unit rate? Explain. www.pre-alg.com/extra_examples Lesson 6-1 Ratios and Rates 265 264-269 C6L1-825200 3/7/03 8:14 AM Page 266 To convert a rate such as miles per hour to a rate such as feet per second, Study Tip you can use dimensional analysis. Recall that this is the process of carrying Look Back units throughout a computation. To review dimensional analysis, see Lesson 5-3. Example 4 Convert Rates ANIMALS A grizzly bear can run 30 miles in 1 hour. How many feet is this per second? 30 mi ᭿ ft You need to convert ᎏᎏ to ᎏᎏ. There are 5280 feet in 1 mile and 1 h 1s 30 mi 3600 seconds in 1 hour. Write 30 miles per hour as ᎏᎏ. 1 h 30 mi 30 mi 5280 ft 3600s ᎏᎏ ϭ ᎏᎏ и ᎏᎏ Ϭ ᎏᎏ Convert miles to feet and hours to seconds. 1 h 1 h 1 mi 1 h 30 mi 5280 ft 1 h 3600s 1 h ϭ ᎏᎏ и ᎏᎏ и ᎏᎏ The reciprocal of ᎏᎏ is ᎏᎏ. 1 h 1 mi 3600s 1 h 3600s 1 44 30 mi 5280 ft 1 h ϭиᎏ ᎏ иᎏ Divide the common factors and units. 1 h 1 mi 3600 s 120 1 44 ft ϭ ᎏᎏ Simplify. s So, 30 miles per hour is equivalent to 44 feet per second. Concept Check 1. Draw a diagram in which the ratio of circles to squares is 2:3. 2. Explain the difference between ratio and rate. 3. OPEN ENDED Give an example of a unit rate. Guided Practice Express each ratio as a fraction in simplest form. GUIDED PRACTICE KEY 4. 4 goals in 10 attempts 5. 15 dimes out of 24 coins 6. 10 inches to 3 feet 7. 5 feet to 5 yards Express each ratio as a unit rate. Round to the nearest tenth, if necessary. 8. $183 for 4 concert tickets 9. 9 inches of snow in 12 hours 10. 100 feet in 14.5 seconds 11. 254.1 miles on 10.5 gallons Convert each rate using dimensional analysis. 12. 20 mi/h ϭ ft/min 13. 16 cm/s ϭ m/h Application GEOMETRY For Exercises 14 and 15, refer to the figure below. 14. Express the ratio of width to length as 6 cm a fraction in simplest form. 15. Suppose the width and length are each increased by 2 centimeters. Will the ratio 10 cm of the width to length be the same as the ratio of the width to length of the original rectangle? Explain.