Chapter 7: Ratios and Similarity

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Chapter 7 Ratios and Similarity

• Regular Math

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Section 7.1 Ratios and Proportions

• A ratio is a comparison of two quantities by
  division.
• Ratios that make the same comparison are
  equivalent ratios.
• Ratios that are equivalent are said to be
  proportional, or in proportion.

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Example 1 Finding Equivalent Ratios

• Find two ratios that are equivalent to each given
  ratio.
• 6/8
• 12/16
• 3/4
• 48/27
• 96/54
• 16/9

• Try these on your own
• 9/27
• 18/54
• 1/3
• 64/24
• 128/48
• 8/3

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Example 2 Determining Whether Two Ratios are in
Proportion

• Simplify to tell whether the ratios form a
  proportion.
• 7/21 and 2/6
• 7/21 1/3
• 2/6 1/3
• The ratios are proportional.
• 9/12 and 16/24
• ¾
• 2/3
• The ratios arent proportional.

• Try these on your own
• 12/15 and 27/36
• The ratios are not proportional.
• 3/27 and 2/18
• The ratios are proportional.

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Example 3 Earth Science Application

• At 4 degrees Celsius, two cubic feet of silver
  has the same mass as 21 cubic feet of water. At 4
  degrees Celsius, would 126 cubic feet of water
  have the same mass as 6 cubic feet of silver?
• 2/21
• 2/21
• 6/126
• 1/21
• One hundred twenty six cubic feet of water would
  not have the same mass as 6 cubic feet at 4
  degrees celsius because the proportions are not
  equal.

• Try this one on your own
• At 4 degrees Celsius, four cubic feet of silver
  has the same mass as 42 cubic feet of water. At 4
  degrees Celsius, would 210 cubic feet of water
  have the same mass as 20 cubic feet of silver?
• 4/42
• 20/210
• Yes Two hundred ten feet of water would have
  the same mass as 20 cubic feet of silver at 4
  degrees celsius.

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Section 7.2 Ratios, Rates, and Unit Rates

• A rate is a comparison of two quantities that
  have different units.
• Unit rates are rates in which the second quantity
  is 1.
• Unit price is a unit rate used to compare costs
  per item.

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Example 1 Entertainment Application

• By design, movies can be viewed on screens with
  varying aspect ratios. The most common ones are
  43, 3720, 169, and 4720.
• Order the width-to-height ratios from least
  (standard tv) to greatest (widescreen tv).
• 43 4/3 1/3
• 3720 37/20 1.85
• 169 16/9 1.7
• 4720 47/20 2.35
• 43, 169, 3720, 4720
• A wide-screen television has screen width 32
  inches and height 18 inches. What is the aspect
  ratio of this screen?
• 3218
• 169

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Try these on your own

• Order the ratios 43, 2310, 139, and 4720 from
  least to greatest.
• 43, 139, 2310, 4720

• A television has screen width 20 in. and height
  15 in. What is the aspect ratio of this screen?
• 2015
• 43

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Example 2 Using a Bar Graph to Determine Rates

• The number of acres destroyed by wildfires in
  2000 is shown for the states with the highest
  totals. Use the bar graph to find the number of
  acres, to the nearest acre, destroyed in each
  state per day.
• Nevada 640,000/365 1753 acres per day
• Alaska 750,000/365 2055 acres per day
• Montana 950,000/365 2603 acres per day
• Idaho 1,400,000/365 3836 acres per day

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Try this one on your own

• Use the bar graph to find the number of acres, to
  the nearest acre, destroyed in Nevada and Alaska
  per week.
• Hint There are 52 weeks in a year.
• Alaska 750,000/52 14,423 acres per week
• Nevada 640,000/52 12,308 acres per week

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Example 3 Finding Unit Prices to Compare Costs

• Blank videotapes can be purchased in packages of
  3 for 4.99, or 10 for 15.49. Which is a better
  buy?
• Price / Quantity
• 4.99 / 3 1.66
• 5.49 / 10 1.55
• It is a better deal to buy the 10 videotapes.
• Leron can buy a 64 oz carton of orange juice for
  2.49 or a 96 oz carton for 3.99. Which is a
  better buy?
• Price / Quantity
• 2.49 / 64 0.0389
• 3.99 / 96 0.0416
• It is a better deal to buy the 64 oz carton of
  orange juice.

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Try these on your own

• Pens can be purchased in a 5-pack for 1.95 or a
  15-pack for 6.20. Which is a better buy?
• 1.95 / 5 0.39
• 6.20 / 15 0.41
• The 5-pack is a better buy.
• Jamie can buy a 15 oz jar of peanut butter for
  2.19 or a 20 oz jar for 2.78. Which is a better
  buy?
• 2.19 / 15 0.146
• 2.78 / 20 0.139
• The 20 oz jar is a better buy.

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Section 7.3 Analyze Units

• To convert units, multiply by one or more ratios
  of equal quantities called conversion factors.

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Example 1 Finding Conversion Factors

• Find the appropriate factor for each conversion.
• Quarts to Gallons
• There are 4 quarts in 1 gallon.
• 1 gal/4 qts
• Meters to Centimeters
• There are 100 centimeters in 1 meter.
• 100 cm/ 1 m

• Try these on your own
• Feet to Yards
• There are 3 feet in 1 yard.
• 1 yd/ 3 ft
• Pounds to Ounces
• There are 16 ounces in 1 pound.
• 16 oz / 1 lb

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Example 2 Using Conversion Factors to Solve
Problems

• The average American eats 23 pounds of pizza per
  year. Find the number of ounces of pizza the
  average American eats per year.
• (23 pounds / 1 year) x (16 ounces / 1 pound)
• 368 ounces per year

• Try this one on your own
• The average American uses 580 pounds of paper
  each year. Find the number of pounds of paper the
  average American use per month, to the nearest
  tenth.
• (580 pounds / 1 year) x (1 year / 12 months)
• 48.3 pounds per month

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Example 3 Problem Solving Application

• A car traveled 900 feet down a road in 15
  seconds. How many miles per hour was the car
  traveling?
• There is 5,280 feet in 1 mile.
• There are 60 minutes in 1 hour.
• There are 60 seconds in 1 minute.
• The car was traveling 45 miles per hour.

• Try this one on your own
• A car traveled 60 miles on a road in 2 hours. How
  many feet per second was the car traveling?
• The car was traveling 44 feet per second.

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Example 4 Physical Science Application

• A strobe lamp can be used to measure the speed of
  an object. The lamp flashes every 1/1000 second.
  A camera records the object moving 7.5 cm between
  flashes. How fast is the object moving in meters
  per second?

• 7.5 cm / 1/1000 sec.
• (1000 x 7.5) / (1/1000 x 1000)
• 7500 cm / 1 sec
• 7500cm/1 sec x 1 m/100 cm
• The object is traveling 75 meters per second.

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Try this one on your own

• A strobe lamp can be used to measure the speed of
  an object. The lamp flashes every 1/100 of a
  second. A camera records the object moving 52 cm
  between flashes. How fast is the object moving in
  meters per second?
• The object is moving 52 meters per second.

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Example 5 Transportation Application

• The rate of one knot equals one nautical mile per
  hour. One nautical mile is 1852 meters. What is
  the speed in meters per second of a ship
  traveling at 20 knots?
• 20 knots 20 nautical miles per hour
• 20 nm/1 hr x 1852 mi/1 nm x 1 h/3600 sec
• The ship is traveling about 10.3 meters per
  second.

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Try this one on your own

• The rate 1 knot equals 1 nautical mile per hour.
  One nautical mile is 1852 meters. What is the
  speed in kilometers per hour of a ship traveling
  at 5 knots?
• 5 knots 5 nautical miles
• 5 nm /1 hr x 1.852 km /1 nm
• 9.26 km/hr

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Section 7.4 Solving Proportions

• Cross products in proportion are equal. If the
  ratios are not in proportion, the cross products
  are not equal.

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Example 1 Using Cross Products to Identify
Proportions

• Tell whether the ratios are proportional
• 5/6 15/21
• 5x21 15x6
• 105 90
• No
• A shade of paint is made by mixing 5 parts red
  paint with 7 parts blue paint. If you mix 12
  quarts of blue paint with 8 quarts of red paint,
  will you get the correct shade?
• 5/7 8/12
• 5x12 7x8
• 60 56
• No, you will not get the correct shade.

• Try these on your own
• 6/15 4/10
• yes
• A mixture of fuel for a certain small engine
  should be 4 parts gasoline to 1 part oil. If you
  combine 5 quarts of oil with 15 quarts of
  gasoline, will the mixture be correct?
• 4/1 15/5
• no

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Example 2 Solving Proportions

• Solve the proportion.
• 12/d 4/14
• 12x14 4d
• 168 4d
• 168 4d
• 4 4
• d 42

• Try this one on your own
• p/12 5/6
• p 10
• 15/m 5/7
• m 21

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Example 3 Physical Science Application

• Two masses can be balanced on a fulcrum when
  mass1/length 2 mass 2/length1. The green box
  and the blue box are balanced. What is the mass
  of the blue box?
• 2/4 m/10
• 2x10 4m
• 20 4m
• 20 4m
• 4 4
• m 5

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Try this one on your own

• Allyson weighs 55 pounds and sits on a seesaw 4
  feet away from its center. If Marco sits on the
  seesaw 5 feet away from the center and the seesaw
  is balanced, how much does Marco weigh?
• 55 / 5 w / 4
• 220 5w
• 44 pounds

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Section 7.5 Dilations

• A dilation is a transformation that changes the
  size, but not the shape, of a figure.
• A scale factor describes how much a figure is
  enlarged or reduced.

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Example 1 Identifying Dilations
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Try these on your own

• Tell whether each transformation is a dilation.

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Example 2 Dilating a Figure

• Dilate the figure by a scale factor of 0.4 with P
  as the center of dilation.

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Try this one on your own

• Dilate the figure by a scale factor of 1.5 with P
  as the center of dilation.

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Example 3 Using the Origin as the Center of
Dilation

• Dilate the figure by a scale factor of 1.5. What
  are the vertices of the new image?
• The original vertices of the figure are A(4,8),
  B(3,2), and C(5,2).
• Dilate the figure by a scale factor of 2/3. What
  are the vertices of the new image?
• The original vertices of the figure are A(3,9),
  B(9,6), and C(6,3).

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Try these on your own

• Dilate the figure in Example 3A by a scale factor
  of 2. What are the vertices of the new figure?
• A(8,16)
• B (6,4)
• C (10,4)

• Dilate the figure in Example 3B by a scale factor
  of 1/3. What are the vertices of the new figure?
• A (1,3)
• B (3,2)
• C (2,1)

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Section 7.6 Similar Figures

• Similar figures have the same shape, but not
  necessarily the same size.

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Example 1 Using Scale Factor to Find Missing
Dimensions

• A picture 4 in. tall and 9 in. wide is to be
  scaled to 2.5 in. tall to be displayed on a Web
  page. How wide should the picture be on the Web
  page for the two pictures to be similar?
• 4 in. tall / 9 in. wide
• 2.5 in. tall / x in. wide
• 4x 9(2.5)
• 4x 22.5
• x 5.625 inches wide

• Try this one on your own
• A picture 10 inches tall and 14 inches wide is to
  be scaled to 1.5 inches tall to be displayed on a
  Web page. How wide should the picture be on the
  Web page for the two pictures to be similar?
• 10 in tall / 14 in wide
• 1.5 in tall / x in. wide
• 2.1 inches

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Example 2 Using Equivalent Ratios to Find
Missing Dimensions

• A companys logo is in the shape of an isosceles
  triangle with two sides that are each 2.4 in.
  long and one side that is 1.8 in long. On a
  billboard, the triangle in the logo has two sides
  that are each 8 feet long. What is the length of
  the third side of the triangle on the billboard?
• 2.4 inches / 8 feet 1.8 inches / x feet
• 2.4(x) 8(1.8)
• 2.4x 14.4
• x 6 inches

• Try this one on your own
• A T-shirt design includes an isosceles triangle
  with side lengths 4.5 in., 4.5 in., and 6 in. Ad
  advertisement shows an enlarged version of the
  triangle with sides that are each 3 ft long. What
  is the length of the third side of the triangle
  in the advertisement?
• 4.5 in / 3 ft 6 in. / x ft
• 4.5(x) 3(6)
• 4.5x 18
• X 4 feet

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Example 3 Identifying Similar Figures

• Triangle A compared to Triangle B
• 2/3 ¾
• 2(4) 3(3)
• 8 9
• NO

• Triangle A compared to Triangle C
• 2/3 4/6
• 2(6) 3(4)
• 12 12
• yes

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Try this one on your own

• Rectangle J compared to Rectangle L
• 10/4 12/5
• 10(5) 4(12)
• 50 48
• no
• Rectangle J compared to Rectangle K
• 10/4 5/2
• 10(2) 4(5)
• 2020
• yes

• Which rectangles are similar?