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Proportions in Pre-Algebra: Ratios, Similar Shapes, and Percents

Learn about proportions in pre-algebra, including ratios, similar shapes, and percents. Explore how to solve proportion problems and convert ratios and percents to decimals. Practice applying these concepts through real-life examples and activities.

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Proportions in Pre-Algebra: Ratios, Similar Shapes, and Percents

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  1. Chapter 6 Pre-Algebra T McDowell

  2. 1 4 = 1:3 = 3:9 2 8 Proportions 11/09 - If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures. Proportions - Similar describes things which have the same shape but are not the same size. Similar

  3. If a/b = c/d then ad = bc ac b d = Cross Multiplying ad = bc • 2/3 = 4/6 • 10/x = 6/3 • 5/6 = x/72 Examples

  4. 8 ft 4 ft = 2 ft 4 ft Ratio The ratio of the smaller figure to the larger figure is 1:2 (said “one to two”). This can also be written as a fraction of ½. Proportion A proportion can be made relating the height and the width of the smaller figure to the larger figure: 8 ft 4 ft 2 ft 4 ft

  5. 8 ft 4 ft = 2 ft x ft Solving Proportion Problems First, designate the unknown side as x. Then, set up an equation using proportions. What does the numerator represent? What does the denominator represent? height width 8 feet 4 feet Then solve for x by cross multiplying: 4x = 16 X = 4 2 feet ? feet

  6. Similar Shapes 11/10 Similar shapes are very important because if we know the dimensions of one shape and one of the dimensions of another shape similar to it, we can figure out the unknown dimensions.

  7. You Try These two stick figures are similar. • Write a proportion relating the similar shapes. • Find the missing width. 8 feet 12 feet 4 feet x feet

  8. You Try These two trapezoids are similar. • Write a proportion relating the similar shapes. • Find the missing sides. 15 a 10 40 24 x

  9. You Try • The scale of a map is 1 inch : 10 miles. Find the actual distance given the distance on the map. • 4 inches • 10 inches • 1 foot • 5.5 inches • 6.75 inches

  10. Leonardo da Vinci 1452 - 1519

  11. Write a ratio that represents each statement. The average adult human figure is about 7 to 7.5 heads tall. The arms' wingspan (measured from the tips of the middle fingers) is about equal to the body height. The length of the foot is about equal to the length of the forearm. 7 head heights 1 body height 1 wingspan 1 body height 1 foot length 1 forearm length

  12. da Vinci Proportions Activity Measure in inches

  13. The eyes are at the mid-height of the head. • The head also can be divided into thirds • top of the head to the bottom of the forehead • bottom of the forehead to bottom of the nose • bottom of nose to the bottom of the chin. • Width of head is between four and five eyes wide. • Height of the face is about equal to length of hand. • Eyes are apart by a distance of one eye width. • Bottom of the nose to the corner of the eye is equal to the height of the ear. • Width of base of nose is equal to width of the eye. • The width of the mouth is equal to the distance between pupils, or the width of two eyes. Draw like de Vinci Use these proportions to draw a head.

  14. Ratios, Decimals, and Percents 11/16 A ratio that compares a number to 100 Percent 54/100 = 54% 36/100 = 36% 4/25 = 16/100 = 16% Examples

  15. Percents as Fractions Percents can also be converted into fractions. Place the percent over 100 Reduce the fraction into simplest form. Example 88% 88 100  4  4 22 25

  16. You try • Write each fraction as a percent • 67/100 • ¾ • 32/50 • Write each percent as a fraction • 92% • 48% • 326% 7

  17. Since percents can be written as fractions, they can also be converted to decimals Percents as decimals The fastest way to convert a percent to a decimal is to move the decimal 2 hops left. Examples 37% 37.% 0.37

  18. Decimals as Percents Decimals can also be converted to percents The fastest way to convert a decimal to a percent is to move the decimal 2 hops right. Examples 3.45 345%

  19. You try Convert each percent into a decimal • 25% • 457% • 0.4% Convert each decimal into a percent • 0.89 • 0.056 • 9.97

  20. You try Workbook P 101 # all

  21. Turn in homework • Sharpen pencil • Sit down • Get ready for notes

  22. Proportions and Percents 11/17 One way to solve problems involving percents is to set up a proportion. Proportions What is 45% of 60? We know 45% is 45/100, but we don’t know what part of 60 we need so that is x/60 45 = x 100 60 Solve the proportion by cross multiplying

  23. You try Write a proportion and solve. • 23% of 158 • 15% of 24 • 345% of 106

  24. What percent of 86 is 4? You know you are looking for a percent or x/100 Finding the percent x = 4 100 86 Solve the proportion by cross multiplying

  25. You try Write a proportion and solve. • What percent is 56 of 109? • What percent is 3 of 9? • What percent is 150 of 80?

  26. 34 is 67% of what number You have the percent so write that as a ratio: 67/100 Finding the whole amount 34 is the numerator of the other ratio—we don’t know the denominator: 34/x 67 = 34 100 x Solve the proportion by cross multiplying

  27. You try Write a proportion and solve. • 12 is 56% of what number? • 54 is 120% of what number? • 21 is 5% of what number?

  28. A tile floor has 90 blue tiles, which is 15% of all the tiles in the floor. How many tiles are in the floor in all? Know what you are looking for You have the percent so write that as a ratio: 15/100 90 is only part of the whole floor so it is the numerator of the other ratio—we don’t know the denominator: 90/x 15 = 90 100 x Solve the proportion by cross multiplying

  29. You try Workbook P 103 # all

  30. Turn in homework • Get your workbook • Sharpen pencil • Sit down • Get ready for notes

  31. Percents and Equations 11/18 Of means to multiply Is means equal sign Math Words Finding the part What is 35% of 90? x = 35% x 90 x = 0.35 x 90 x = 31.5

  32. You try Write an equation to solve • What is 14% of 65? • What is 135% of 15? • What is 82% of 110?

  33. Of means to multiply Is means equal sign Finding the Percent 20 is what percent of 120? 20 = x • 120  120  120 0.167 = x 16.7% = x

  34. You try Write a proportion and solve. • 12 is what percent of 90? • 90 is what percent of 82? • 34 is what percent of 150?

  35. Of means to multiply Is means equal sign Finding The whole 15 is 45% of what number? 15 = 45% • x 15 = 0.45 • x  0.45  0.45 33.3 = x

  36. You try Write a proportion and solve. • 24 is 42% of what number? • 145 is 110% of what number? • 5 is 30% of what number?

  37. You try Workbook P 105 # all

  38. Binder Check • What was the topic for the notes given on 11/18? • What was the answer to number 55 from the homework assigned 11/16, p 313, # 55-70 • Write the calculator policy from the Classroom Guidelines and Procedures handout.

  39. Writing Proportions 11/30 Similar Shapes Write a proportion and solve for the unknown side. 34 10 26 x

  40. Review E 28 B Similar Shapes C 6 x 14 A D  ABC ~ EDC Since we are told that  ABC ~ EDC, we also know that AB ~ ED, BC ~ DC, and AC ~ EC

  41. Draw a picture/diagram • Make a list of what you know and what you are looking for • Solve the problem Solving Word Problems

  42. Word Problem At a given time of day, a building of unknown height casts a shadow that is 24 feet long. At the same time of day, a post that is 8 feet tall casts a shadow that is 4 feet long. What is the height of the building? Similar Shapes x 8 20 4

  43. You try Workbook p 189 # all p 190 # 1-4

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