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Ratios and Rates

Ratios and Rates. LESSON 4-1. Problem of the Day. How many miles are in 21,120 yd? (Hint: 1 mi = 5,280 ft). 12 mi. 4-1. Ratios and Rates. LESSON 4-1. Check Skills You’ll Need. (For help, go to Lesson 2-3.). 1. Vocabulary Review What is the least common

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Ratios and Rates

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  1. Ratios and Rates LESSON 4-1 Problem of the Day How many miles are in 21,120 yd? (Hint: 1 mi = 5,280 ft) 12 mi 4-1

  2. Ratios and Rates LESSON 4-1 Check Skills You’ll Need (For help, go to Lesson 2-3.) 1. Vocabulary Review What is the least common denominator of two rational numbers? Determine which rational number is greater. 3 9 1 6 15 25 4 5 45 54 2 3 4 7 7 12 2. , 3. , 4. , 5. , Check Skills You’ll Need 4-1

  3. Ratios and Rates LESSON 4-1 Check Skills You’ll Need Solutions 1. The least common denominator is the smallest multiple the denominators have in common. 3 9 4 5 45 54 7 12 2. 3. 4. 5. 4-1

  4. Convert minutes to seconds so that both measures are in the same units. Divide the common units. 36 s 12 min 36 s 720 s = Divide the numerator and denominator by the GCF, 36. 36 720 36 ÷36 720 ÷ 36 = 1 20 = Simplify. 1 20 The ratio of 36 seconds: 12 minutes is . Ratios and Rates LESSON 4-1 Additional Examples Write the ratio 36 seconds to 12 minutes in simplest form. Quick Check 4-1

  5. cost number of minutes $4.50 30 min Write a rate comparing cost to minutes. = Divide. = $.15/min Ratios and Rates LESSON 4-1 Additional Examples Computer time costs $4.50 for 30 min. What is the unit rate? The unit rate is $.15 per minute. Quick Check 4-1

  6. Vanessa Keneesha Write the rates comparing miles to gallons. miles gallons 267 mi 11 gal miles gallons 210 mi 9 gal = = Divide. 23.33333333 mi/gal 24.27272727 mi/gal Round to the nearest tenth. 24.3 mi/gal 23.3 mi/gal Ratios and Rates LESSON 4-1 Additional Examples Keneesha drove her car 267 mi using 11 gal of gas. Vanessa drove her car 210 mi using 9 gal. Give the unit rate for each. Which car got more miles per gallon of gas? Keneesha’s car got more miles per gallon. 4-1

  7. Check for Reasonableness 24.3 • 11 = 267.3 and 267.3 267. Also, 23.3 • 9 = 209.7 and 209.7 210. The answers are reasonable. Ratios and Rates LESSON 4-1 Additional Examples (continued) Quick Check 4-1

  8. 1 15 Ratios and Rates LESSON 4-1 Lesson Quiz Express each ratio in simplest form. 1. 27 laps : 81 minutes 2. 12 minutes : 3 hours 3. Carli walked 16 miles in 5 hours. Find the unit rate. 4. A 21-oz bottle of shampoo costs $2.80. A 12-oz bottle costs $1.35. Which has the better unit rate? 1 3 3.2 mi/h 12-oz bottle 4-1

  9. 1 2 1 8 3 4 7 8 , , , Converting Units LESSON 4-2 Problem of the Day Write a fraction in lowest terms, with a single digit numerator, that is about the same as each decimal: 0.52, 0.13, 0.74, 0.88. 4-2

  10. Converting Units LESSON 4-2 Check Skills You’ll Need (For help, go to Lesson 2-5.) 1. Vocabulary Review What is the product of a number and its reciprocal? Find each product. Write the answer in simplest form. 10 3 1 4 4 6 5 6 • 3. • 4. 5. • • 6 7 8 3 4 9 3 2 • • Check Skills You’ll Need 4-2

  11. 2 1 2 4 • 3 9 • 2 4 • 3 9 • 2 2 3 6 • 8 7 • 3 6 • 8 7 • 3 16 7 2 7 = = = = = 2 1 1 3 Converting Units LESSON 4-2 Check Skills You’ll Need Solutions 1. 1 2. 3. 4. 5. 5 10 • 1 3 • 4 10 • 1 3 • 4 5 6 = = 2 2 4 • 5 6 • 6 4 • 5 6 • 6 10 18 5 9 = = = 3 4-2

  12. 5,280 ft. 1 mi Since 5,280 ft = 1 mi, use the conversion factor Multiply by a conversion factor . 0.7 mi 1 5,280 ft 1 mi 0.7 = • 5,280 ft 1 mi (0.7)(5,280) ft 1 = Simplify. Divide. = 3,696 ft Converting Units LESSON 4-2 Additional Examples Convert 0.7 mi to ft. There are 3,696 feet in 0.7 miles. Quick Check 4-2

  13. Estimate 6.84 7. Then, 7 • 60 ÷ 1000 = 0.42. Multiply by two ratios that each equal one. 6.84 m 1 s 6.84 m 1 s 1 km 1000 m 60 s 1 min = • • Divide by the common units. (6.48)(1)(60) km (1)(1,000)(1) min = Simplify. Use a calculator. = 0.4104 Converting Units LESSON 4-2 Additional Examples Quick Check A rowing team completed a 2000-m course at a rate of 6.84 m/s. Convert this rate to kilometers per minute. The team rowed at a rate of 0.4104 km/min. Check for Reasonableness The answer 0.4104 km/min is close to the estimate 0.42. The answer is reasonable. 4-2

  14. Round to the nearest number divisible by 4. 33 qt 32 qt 32 qt 1 1 gal 4 qt Multiply by the conversion factor. = • Divide by the common units. 32 4 = gallons Simplify. Divide. = 8 gallons Converting Units LESSON 4-2 Additional Examples Use compatible numbers to estimate the number of gallons in 33 quarts. 1 gal 4 qt The conversion factor for changing gallons to quarts is . Quick Check There are about 8 gallons in 33 quarts. 4-2

  15. Multiply by the conversion factor . 650 g 1 1 oz 28.4 g 650 g = • 1 oz 28.4 g (650)(1) oz 22.9 oz  28.4 Simplify. Divide using = a calculator. Converting Units LESSON 4-2 Additional Examples Convert 650 g to ounces. There are about 22.9 oz in 650 g. Quick Check 4-2

  16. Converting Units LESSON 4-2 Lesson Quiz 1. Convert 0.75 hours to seconds. 2. $150 per hour is how much per minute? 3. 69.2 cm is about how many meters? 4. Convert 12 qt to liters. 2,700 seconds $2.50 per min 0.7 m about 11.3L 4-2

  17. Solving Proportions LESSON 4-3 Problem of the Day Write each word phrase as an algebraic expression. a. 12 times a number b. 8 less than a number c. twice the sum of 5 and a number 12n n – 8 2(5 + n) 4-3

  18. Solving Proportions LESSON 4-3 Check Skills You’ll Need (For help, go to Lesson 2-2.) a + 2 b + 2 1. Vocabulary Review Is the fraction in simplest form? Explain. Write each fraction in simplest form. 2. 3. 4. 5. 30 99 42 12 132 602 70 25 Check Skills You’ll Need 4-3

  19. Solving Proportions LESSON 4-3 Check Skills You’ll Need Solutions 1. Yes; there is no common factor between the numerator and denominator. 2.3. 4.5. 1 1 30 99 3 • 10 3 • 33 10 33 42 12 6 • 7 6 • 2 7 2 1 2 = = = = = 3 1 1 1 1 70 25 5 • 14 5 • 5 14 5 4 5 132 602 2 • 66 2 • 301 66 301 = = = 2 = = 1 1 4-3

  20. 8 18 4 9 Write as a proportion. gallons gallons Use number sense to find a common multiplier. 8 18 4 9 Since = ,they form a proportion. Solving Proportions LESSON 4-3 Additional Examples 8 18 4 9 Do and form a proportion? Explain. Quick Check 4-3

  21. 0.7876 1 125 p Write the proportion . = Irish pounds euros Write the cross products. 0.7876 • p = 1 • 125 0.7876 • p 0.7876 125 0.7876 Divide each side by 0.7876. = Use a calculator. 125 0.7876 Solving Proportions LESSON 4-3 Additional Examples The fixed rate of conversion is 1 euro = 0.7876 Irish pounds. How many euros would you receive for 125 Irish pounds? Let p = the number of euros. You would receive 158.71 euros. Quick Check 4-3

  22. Solve each proportion. 2. = 3. = 4. Suppose the exchange rate for dollars to Indian rupees is 0.02. How many rupees should you receive for $100? w 12 3 4 4 5 20 r Solving Proportions LESSON 4-3 Lesson Quiz 5 8 10 24 1. Is proportional to ? Explain. No; the fractions are not equal. 9 25 5,000 rupees 4-3

  23. Similar Figures and Proportions LESSON 4-4 Problem of the Day A football team scored 38 points in a game. They scored 3 points for a field goal and 7 points for each touchdown with an extra point. How many field goals did they make? How many touchdowns? 1 field goal and 5 touchdowns or 2 touchdowns and 8 field goals 4-4

  24. Similar Figures and Proportions LESSON 4-4 Check Skills You’ll Need (For help, go to Lesson 4-3.) • Vocabulary Review What are the cross products for 10 15 2 3 = ? Solve each proportion. 2. = 3. = 4. = k 50 16 25 21 t 22 10 324 m 7 13 Check Skills You’ll Need 4-4

  25. 7t = 273 = t = 39 10k = 1,100 = k = 110 7t 7 273 7 10k 10 1,100 10 1 4 Similar Figures and Proportions LESSON 4-4 Check Skills You’ll Need Solutions 4.16m = 8,100; m = 506 4-4

  26. First, check to see if corresponding angles are congruent. AR BSAll right angles are 90°. CTDU Similar Figures and Proportions LESSON 4-4 Additional Examples Is rectangle ABCD similar to rectangle RSTU? Explain why or why not. 4-4

  27. AB RS DA UR AB corresponds to RS. DA corresponds to UR. 6 48 3 24 Substitute. Write the cross products. 6 • 24 48 • 3 Simplify. 144 = 144 Similar Figures and Proportions LESSON 4-4 Additional Examples (continued) Next, check to see if corresponding sides are in proportion. The corresponding sides are in proportion, so rectangle ABCD is similar to rectangle RSTU. Quick Check 4-4

  28. 22 in. x 2.75 in. 5 in. Set up a proportion. = Write the cross products. 2.75 • x = 5 • 22 2.75 x = 110 Simplify. 2.75x 2.75 110 2.75 Divide each side by 2.75. = x = 40 Simplify. Similar Figures and Proportions LESSON 4-4 Additional Examples A stonemason’s sketch of a carving to be made on a building includes the letter “E” shown below. If the width of the actual letter in the arrangement is 22 in., what is the height? The height of the letter is 40 inches. Quick Check 4-4

  29. 14 d 12 21 Write a proportion. = Write the cross products. 12 • d = 21 •14 12d = 294 Simplify. 12d 12 294 12 Divide each side by 12. = d = 24.5 Simplify. Similar Figures and Proportions LESSON 4-4 Additional Examples RST ~ PSU. Find the value of d. The value of d is 24.5. Quick Check 4-4

  30. Similar Figures and Proportions LESSON 4-4 Lesson Quiz 1. Are the triangles similar? Explain. 2. A model of a building is 18 in. tall and 24 in. wide. The building is 30 ft tall. How wide is the building? No; their sides are not proportional. 40 ft 4-4

  31. Similar Figures and Proportions LESSON 4-4 Lesson Quiz 3. In the figure at the right,  MNO ~ LNP. Find the value of a. 18 4. If all the lengths in Exercise 3 are doubled, are the triangles still similar? Explain why or why not. Yes; corresponding values are multiplied by the same factor. 4-4

  32. Similarity Transformations LESSON 4-5 Problem of the Day There are three different 1-digit numbers greater than zero and all odd. Their sum is 15. What are the numbers? 3, 5, 7 or 1, 5, 9 4-5

  33. Similarity Transformations LESSON 4-5 Check Skills You’ll Need (For help, go to Lesson 3-4.) 1.Vocabulary Review The first coordinate in an ordered pair is the ? -coordinate. Graph each point on a coordinate plane. 2. A(3, 6) 3.B(–2, 7) 4. C(5, –1) 5. D(–3, 0) Check Skills You’ll Need 4-5

  34. Similarity Transformations LESSON 4-5 Check Skills You’ll Need Solutions 1.2-5. x 4-5

  35. A C is 3 times AC. Since A is the center of dilation A = A . A = A A B C is the image of ABC after a dilation with a scale factor of 3. A B is 3 times AB. ABC ~ A B C Similarity Transformations LESSON 4-5 Additional Examples Quick Check Find the image of ABC after a dilation with center A and a scale factor of 3. 4-5

  36. Step 1 Multiply the x- and y-coordinates of each point by . Step 2 Graph the image. 1 2 1 2 K (–2, –1) K (–1, – ) L (0, 2) L (0, 1) M (4, 2) M (2, 1) N (4, –1) N (2, – ) 1 2 Similarity Transformations LESSON 4-5 Additional Examples Quick Check Find the coordinates of the image of quadrilateral KLMN after a dilation with a scale factor of . Quadrilateral KLMN has vertices K (–2, –1), L (0, 2), M (4, 2), and N (4, –1). 1 2 4-5

  37. 6 4 3 2 PQ PQ image original = = = 1.5 Similarity Transformations LESSON 4-5 Additional Examples The figure below PQR shows the outline of a playing field. A city planner dilates the design to show the area available for community youth to play sports. Find the scale factor. Is it an enlargement or a reduction? The scale factor is 1.5. The dilation is an enlargement. Quick Check 4-5

  38. A (0, 0), B (2, 0), C (1, 1) A (0, 0), B (40, 0), C (20, 20) 1 3 , reduction Similarity Transformations LESSON 4-5 Lesson Quiz ABC has coordinates A(0, 0), B(10, 0), and C(5, 5). Find the coordinates of the image of ABC after a dilation with each scale factor. 1. 2. 4 3. 1 5 Figure ABCD shows the outline of a porch. The figure A′B′C′D′ is the outline of a table formed by dilating ABCD. Find the scale factor. Is it an enlargement or a reduction? 4-5

  39. Scale Models and Maps LESSON 4-6 Problem of the Day Mirror primes are pairs of prime numbers in which the digits are reversed, such as 13 and 31. Find all the mirror primes less than 100. 13 and 31, 17 and 71, 37 and 73, and 79 and 97; 11 is its own mirror image. 4-6

  40. Scale Models and Maps LESSON 4-6 Check Skills You’ll Need (For help, go to the Skills Handbook page 632.) 1. Vocabulary Review A product is the result of which operation? Multiply. 2. 4  3.2 3. 7.6 5.9 4. 1.8 22 5. 13 6.5 Check Skills You’ll Need 4-6

  41. Scale Models and Maps LESSON 4-6 Check Skills You’ll Need Solutions 1. multiplication 2. 12.8 3. 44.84 4. 39.6 5. 84.5 4-6

  42. 1 2 Let = the actual length of the cellar. 1 2 blueprint measure (in.) actual measure (ft) blueprint length (in.) actual length (ft) 4 = 8 Scale Models and Maps LESSON 4-6 Additional Examples On a blueprint, the cellar is 4 in. by 3 in. The scale is in. = 8 ft. What are the length and width of the actual cellar? First, find the actual length of the cellar. 4-6

  43. 1 2 • = 8 • 4 Write the cross products. 1 2 Simplify. = 32 1 2 32 Divide each side by . = 1 2 1 2 1 2 Simplify. = 64 Scale Models and Maps LESSON 4-6 Additional Examples (continued) 4-6

  44. 1 2 blueprint measure (in.) actual measure (ft) blueprint length (in.) actual length (ft) 3 = 8 w Scale Models and Maps LESSON 4-6 Additional Examples (continued) The length of the actual room is 64 ft. Next, find the actual width of the cellar. Let w = the actual width of the cellar. 4-6

  45. 1 2 • w = 8 • 3 Write the cross products. 1 2 Simplify. w = 24 1 2 w 24 Divide each side by . = 1 2 1 2 1 2 Simplify. w = 48 Scale Models and Maps LESSON 4-6 Additional Examples (continued) Quick Check The width of the actual room is 48 ft. 4-6

  46. map (cm) actual (km) 1 map (cm) actual (km) 7.5 Set up a proportion. = d 50 1 • d = 50 • 7.5 Write the cross products. d = 375 Simplify. Scale Models and Maps LESSON 4-6 Additional Examples The map distance from El Paso, Texas, to Chihuahua, Mexico, measures about 7.5 cm. The scale is 1 cm = 50 km. What is the actual distance? Let d be the actual distance from El Paso, Texas to Chihuahua, Mexico. The actual distance from El Paso, Texas to Chihuahua, Mexico is 375 kilometers. Quick Check 4-6

  47. Scale Models and Maps LESSON 4-6 Lesson Quiz 1. A 6-ft man is designing a new chair that would make him feel like a 2.5-ft child. The seat of a normal chair is 1.5 ft high. How high should he make the seat in his new chair? 2. A map scale shows 4 cm to represent 6 km. Two intersections measure 1 cm apart on the map. What is the actual distance? 3.6 ft 1.5 km 4-6

  48. Scale Models and Maps LESSON 4-6 Lesson Quiz For Exercises 3–4, use the diagram. 3. A tennis court is 36 ft wide. A drawing of the court is 2 in. long and 1 in. wide. Find the scale used. 1 4 1 in. = 36 ft 4. Find the actual length of the court. 81 ft 4-6

  49. Similarity and Indirect Measurement LESSON 4-7 Problem of the Day A rectangular field is 120 yd long and 53 yd 1 ft wide. How much longer is the field than it is wide? 66 yd 2 ft 4-7

  50. Similarity and Indirect Measurement LESSON 4-7 Check Skills You’ll Need (For help, go to Lesson 4-4.) 1. Vocabulary Review Similar figures have the same ? but not necessarily the same size. 2. If ABC ~ XYZ, which angle is congruent to B? Check Skills You’ll Need 4-7

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