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3. Lesson 3-6 Ratios and Proportions Lesson 3-7 Percent of Change
Lesson 11-6 Similar Triangles
Contents

4. Example 1 Determine Whether Ratios Form a Proportion
Example 2 Use Cross Products
Example 3 Solve a Proportion
Example 4 Use Rates
Example 5 Use a Scale Drawing
Lesson 6 Contents

5. Determine whether the ratios and form a proportion.
Answer: The ratios are equal. Therefore, they form a proportion.
Example 6-1a

6. Do the ratios and form a proportion?
Answer: The ratios are not equal. Therefore, they do not form a proportion.
Example 6-1b

7. Find the cross products.
Use cross products to determine whether the pair of ratios below forms a proportion.
Write the equation.
Find the cross products.
Simplify.
Answer: The cross products are not equal, so . The ratios do not form a proportion.
Example 6-2a

8. Find the cross products.
Use cross products to determine whether the pair of ratios below forms a proportion.
Write the equation.
Find the cross products.
Simplify.
Answer: The cross products are equal, so . Since the ratios are equal, they form a proportion.
Example 6-2b

9. Use cross products to determine whether the pair of ratios below forms a proportion.
Answer: The cross products are equal. Therefore, the ratios do form a proportion.
Example 6-2c

10. Use cross products to determine whether the pair of ratios below forms a proportion.
Answer: The cross products are not equal. Therefore, the ratios do not form a proportion.
Example 6-2d

11. Find the cross products.
Solve the proportion .
Original equation
Find the cross products.
Simplify.
Divide each side by 8.
Answer:
Simplify.
Example 6-3a

12. Solve the proportion .
Answer: 6.3
Example 6-3b

13. Plan Write a proportion for the problem.
Bicycling The gear on a bicycle is 8:5. This means that for every eight turns of the pedals, the wheel turns five times. Suppose the bicycle wheel turns about times during a trip. How many times would you have to turn the pedals during the trip?
Explore Let p represent the number of times needed to crank the pedals.
Plan Write a proportion for the problem.
turns of the pedals
wheel turns
Example 6-4a

14. Find the cross products.
Solve
Original proportion
Find the cross products.
Simplify.
Divide each side by 5.
Answer: 3896 = p
Simplify.
Example 6-4b

15. Examine If it takes 8 turns of the pedal to make the wheel turn 5 times, then it would take 1.6 turns of the pedal to make the wheel turn 1 time. So, if the wheel turns 2435 times, then there are 2435  1.6 or 3896 turns of the pedal. The answer is correct.
Example 6-4c

16. Answer: About 60,480 cels were drawn to produce Snow White.
Before 1980, Disney created animated movies using cels. These hand drawn cels (pictures) of the characters and scenery represented the action taking place, one step at a time. For the movie Snow White, it took 24 cels per second to have the characters move smoothly. The movie is around 42 minutes long. About how many cels were drawn to produce Snow White?
Answer: About 60,480 cels were drawn to produce Snow White.
Example 6-4d

17. Explore Let d represent the actual distance.
Maps In a road atlas, the scale for the map of Connecticut is 5 inches = 41 miles. The scale for the map of Texas is 5 inches = 144 miles. What are the distances in miles represented by 2.5 inches on each map?
Explore Let d represent the actual distance.
Plan Write a proportion for the problem.
scale
actual
Connecticut:
Example 6-5a

18. Texas:
scale
actual
Example 6-5b

19. Find the cross products.
Solve Connecticut:
Find the cross products.
Simplify.
Divide each side by 5.
Simplify.
or 20.5
Example 6-5c

20. Find the cross products.
Solve Texas:
Find the cross products.
Simplify.
Divide each side by 5.
Simplify.
or 72
Example 6-5d

21. Answer:. The actual distance in Connecticut represented by 2
Answer: The actual distance in Connecticut represented by 2.5 inches is 20.5 miles. The actual distance in Texas represented by 2.5 inches is 72 miles.
Examine: 2.5 inches is of 5 inches. So 2.5 inches represents (41) or 20.5 miles in Connecticut and (144) or 72 miles in Texas. The answer is correct.
Example 6-5f

22. The scale on a map of the United States is. inches = 750 miles
The scale on a map of the United States is inches = 750 miles. The distance, on the map, between Los Angeles and Washington, D.C., is about inches. What is the distance in miles between the two locations?
Answer: The distance in miles between Los Angeles and Washington, D.C., is about 2,114 miles.
Example 6-5g

23. End of Lesson 6

24. Example 1 Find Percent of Change Example 2 Find the Missing Value
Example 3 Find Amount After Sales Tax
Example 4 Find Amount After Discount
Lesson 7 Contents

25. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.
original: 32 new: 40
Find the amount of change. Since the new amount is greater than the original, the percent of change is a percent of increase.
40 – 32 = 8
Example 7-1a

26. Find the percent using the original number, 32, as the base.
change
original amount
percent change
100 percent
Find the cross products.
Simplify.
Divide each side by 32.
Simplify.
Answer: The percent of increase is 25%.
Example 7-1b

27. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.
original: 20 new: 4
Find the amount of change. Since the new amount is less than the original, the percent of change is a percent of decrease.
20 – 4 = 16
Example 7-1c

28. Find the percent using the original number, 20, as the base.
change
original amount
percent change
100 percent
Find the cross products.
Simplify.
Divide each side by 20.
Simplify.
Answer: The percent of decrease is 80%.
Example 7-1d

29. Answer: The percent of change is a decrease of 28%.
a. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.
original: 20 new: 18
Answer: The percent of change is a decrease of 28%.
Example 7-1e

30. Answer: The percent of change is an increase of 300%.
b. State whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change.
original: 12 new: 48
Answer: The percent of change is an increase of 300%.
Example 7-1f

31. Sales The price a used-book store pays to buy a book is $5
Sales The price a used-book store pays to buy a book is $5. The store sells the book for 28% above the price that it pays for the book. What is the selling price of the $5 book?
Let s = the selling price of the book. Since 28% is the percent of increase, the amount the used-book store pays to buy a book is less than the selling price. Therefore, s – 5 represents the amount of change.
Example 7-2a

32. Find the cross products.
change
book store cost
percent change
100 percent
Find the cross products.
Distributive Property
Add 500 to each side.
Simplify.
Divide each side by 100.
Simplify.
Answer: The selling price of the $5 book is $6.40.
Example 7-2b

33. Answer: The price of jeans at the second store is $31.72.
At one store the price of a pair of jeans is $ At another store the same pair of jeans has a price that is 22% higher. What is the price of jeans at the second store?
Answer: The price of jeans at the second store is $31.72.
Example 7-2c

34. The tax is 5% of the price of the meal.
Sales Tax A meal for two at a restaurant costs $ If the sales tax is 5%, what is the total price of the meal?
The tax is 5% of the price of the meal.
Use a calculator.
Round $ to $1.64. Add this amount to the original price.
Answer: The total price of the meal is $34.49.
Example 7-3a

35. Answer: The total price of the CD player is $74.71.
A portable CD player costs $ If the sales tax is 6.75%, what is the total price of the CD player?
Answer: The total price of the CD player is $74.71.
Example 7-3b

36. The discount is 20% of the original price.
Discount A dog toy is on sale for 20% off the original price. If the original price of the toy is $3.80, what is the discounted price?
The discount is 20% of the original price.
Subtract $0.76 from the original price.
Answer: The discounted price of the dog toy is $3.04.
Example 7-4a

37. Answer: The discounted price of the cap player is $16.99.
A baseball cap is on sale for 15% off the original price. If the original price of the cap is $19.99, what is the discounted price?
Answer: The discounted price of the cap player is $16.99.
Example 7-4b

38. End of Lesson 7

39. Example 1 Determine Whether Two Triangles Are Similar
Example 2 Find Missing Measures
Example 3 Use Similar Triangles to Solve a Problem
Lesson 6 Contents

40. The ratio of sides XY to AB is
Determine whether the pair of triangles is similar. Justify your answer.
The ratio of sides XY to AB is
The ratio of sides YZ to BC is
Example 6-1a

41. The ratio of sides XZ to AC is
Answer: Since the measures of the corresponding sides are proportional, triangle XYZ is similar to triangle ABC.
Example 6-1a

42. Determine whether the pair of triangles is similar. Justify your answer.
Answer: Since the corresponding angles have equal measures, the triangles are similar.
Example 6-1b

43. Find the missing measures if the pair of triangles is similar.
Since the corresponding angles have equal measures, The lengths of the corresponding sides are proportional.
Example 6-2a

44. Corresponding sides of similar triangles are proportional.
and
Find the cross products.
Divide each side by 18.
Example 6-2a

45. Corresponding sides of similar triangles are proportional.
and
Find the cross products.
Divide each side by 18.
Answer: The missing measures are 27 and 12.
Example 6-2a

46. Find the missing measures if the pair of triangles is similar.
Corresponding sides of similar triangles are proportional.
and
Example 6-2b

47. Find the cross products.
Divide each side by 4.
Answer: The missing measure is 7.5.
Example 6-2b

48. Find the missing measures if each pair of triangles is similar. a.
Answer: The missing measures are 18 and 42.
Example 6-2c

49. Find the missing measures if each pair of triangles is similar. b.
Answer: The missing measure is 5.25.
Example 6-2c

50. Shadows Richard is standing next to the General Sherman Giant Sequoia three in Sequoia National Park. The shadow of the tree is 22.5 meters, and Richard’s shadow is 53.6 centimeters. If Richard’s height is 2 meters, how tall is the tree?
Since the length of the shadow of the tree and Richard’s height are given in meters, convert the length of Richard’s shadow to meters.
Example 6-3a

51. Let the height of the tree.
Simplify.
Let the height of the tree.
Richard’s shadow
Tree’s shadow
Richard’s height
Tree’s height
Cross products
Answer: The tree is about 84 meters tall.
Example 6-3a

52. Answer: The length of Trudie’s shadow is about 0.98 meter.
Tourism Trudie is standing next to the Eiffel Tower in France. The height of the Eiffel Tower is 317 meters and casts a shadow of 155 meters. If Trudie’s height is 2 meters, how long is her shadow?
Answer: The length of Trudie’s shadow is about meter.
Example 6-3b

53. End of Lesson 6

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