Chapter 7: Ratios and Similarity

Regular Math

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.

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

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 aren’t

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.

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.

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.

Example 1: Entertainment Application

By design, movies can be viewed on screens with varying aspect ratios. The most common ones are 4:3, 37:20, 16:9, and 47:20.

– Order the width-to-height ratios from least (standard tv) to greatest (widescreen tv).

4:3 = 4/3 = 1/3 37:20 = 37/20 = 1.85 16:9 = 16/9 = 1.7 47:20 = 47/20 = 2.35

– 4:3, 16:9, 37:20, 47:20

– A wide-screen television has screen width 32 inches and height 18 inches. What is the aspect ratio of this screen?

32:18 16:9

Try these on your own…

Order the ratios 4:3, 23:10, 13:9, and 47:20 from least to greatest.

– 4:3, 13:9, 23:10, 47:20

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

– 20:15 4:3

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

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

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.

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.

Section 7.3: Analyze Units

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

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

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

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.

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.

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.

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.

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

Section 7.4: Solving Proportions

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

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

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

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

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

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.

Example 1: Identifying Dilations

Try these on your own…

Tell whether each transformation is a dilation.

Example 2: Dilating a Figure

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

Try this one on your own…

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

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).

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)

Section 7.6: Similar Figures

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

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

Example 2: Using Equivalent Ratios to Find Missing Dimensions

A company’s 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

Example 3: Identifying Similar Figures

Triangle A compared to Triangle C

– 2/3 = 4/6 2(6) =3(4) 12 = 12 yes

Triangle A compared to Triangle B

– 2/3 = ¾ 2(4) = 3(3) 8 = 9 NO

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) 20=20 yes

Which rectangles are similar?