Chapter 8 Ratios and Proportions

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Chapter 8 Ratios and Proportions

• This Chapter will correspond with the novel Dr.
  Franklins Island

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Lesson 8.1 Daily Puzzler

• Two backhoes can dig 2 trenches in 2 hours. How
  long will it take 10 backhoes working at the same
  rate to dig 10 trenches?

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Lesson 8.1 Ratios

• Ratio-uses division to compare two numbers. There
  are 3 ways to write a ratio.
• a to b, where b is a nonzero number
• a/b, where b is a nonzero number
• ab, where b is a nonzero number
• Equivalent ratios-ratios that have the same value

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Writing Ratios

• In the Real World-Baseball
• How can you compare a baseball teams wins to its
  losses during spring training?
• Can you make a comparison about games played by
  the Cubs?
• Wins to losses
• Wins to games played

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Writing Ratios in Simplest Form

• A ride on a roller coaster lasts 2 minutes.
  Suppose you wait in line for 1 ½ hrs. to ride the
  coaster. Follow the steps to find the ratio of
  the time spent in line to time spent on the ride.
• Write the hrs. as minutes so units are the
  same. 1 h ½ h 60 min 30 min
• Write the ratio of time spent in line to time
  spent on the ride.
• time in line 90 451
• time on ride 2

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Comparing Ratios

• Luis and Amber compared their CD collections. To
  determine who has a greater ratio of rock CDs to
  pop CDs write the ratios.

Does Luis or Amber have a greater ratio of pop
CDs to hip-hop CDs? Does Luis or Amber have a
greater ratio of hip-hop CDs to rock CDs?
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8.1 Concept Check

• The ratio of male student to female student of an
  Outing Club trip was 54. Does this mean that
  there were 4 female students and 5 male students
  on the trip? EXPLAIN using complete sentences!

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Lesson 8.2 Daily Puzzler

• A drawer contains 8 black socks and 8 brown
  socks. You reach into the drawer and randomly
  pull out a sock. What is the maximum number of
  times you must repeat this action before you have
  two socks of the same color?

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Lesson 8.2 Rates

• Activity
• Use a watch to count the number of times your
  heart beats in 10 seconds. Record your result as
  a fraction. N heartbeats/10 seconds
• Ask your partner to count his/her pulse for 15
  seconds. Record the result as a fraction.
• Whose pulse is greater? Explain how you decided.

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• Rate-is a ratio of two quantities measured in
  different units.
• Unit rate-is a rate that has a denominator of 1
  unit. The following are examples of writing unit
  rates.
• 15 mi 15mi/1h, 15 miles per hour
• 1 h

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Finding a Unit Rate

• To find a unit rate you must rewrite the fraction
  so that the denominator is 1.
• During a peak growing season, the kudzu vine can
  grow 6 inches in 12 hours. What is the growth
  rate of kudzu in inches per hour?
• The growth rate of bamboo can be as great as 120
  cm in 2 days. Express this growth rate in cm per
  day.

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Finding Average Speed

• If you know distance traveled, travel time for
  a moving object, you can find the average rate,
  or average speed by diving the distance by the
  time or
• Average rate distance
• time

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Finding Average Speed Cont.

• Ex. 1 A skater took 2 minutes 30 seconds to
  complete a 1500 m race. What was the skaters
  average speed?
• First change the units so they are the same. 2
  min 30 sec 120 sec 30 sec 150 sec
• 1500 m 10m
• 150sec 1 sec
• It takes you 1 ½ hrs to jog 12 miles. What is
  your average jogging speed?

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Comparing Unit Rates

• A store sells the same pasta the following two
  ways 10 lbs. of bulks pasta for 15 and 2 lbs.
  for 3.98. To determine which is the better buy,
  find the unit price for both types.
• An amusement park sell a book of 15 tickets for
  26.25. It also sell a book of 5 tickets for 10.
  Determine which is the better buy by comparing
  the unit price for each book of tickets.

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8.2 Concept Check

• How are a rate and a unit rate related to a
  ratio? EXPLAIN using complete sentences.

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Lesson 8.3 Daily Puzzler

• Train A travels at 150 mph. Train B travels at
  120 mph. Each train travels at whole number of
  hours. The two train traveled a total of 1920
  miles. How many hours did each train travel?
  (Hint there are 3 correct answers.)

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Lesson 8.3 Slope

• Activity
• Draw an x-axis and a y-axis on graph paper.
• Start at the origin. Move 2 units to the right
  and 3 units up. Plot this point and label it A.
• Start at A. Move 4 units to the right and 6 units
  up. Plot this point and label it B.
• Draw a line through A, B, and the origin.
• Find the ratio of rise to run for each of the
  movements described in Steps 2 and 3. What do you
  notice?

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• Slope-of a nonvertical line is the ratio of the
  rise (vertical change) to the run (horizontal
  change) between any two points on the line. A
  line has a constant slope. (You must rise out of
  bed before you can run off to school)

A line that rises from left to right has a
positive slope.
A line that falls from left to right has a
negative slope.
Horizontal line has a slope of 0.
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Finding Slope of a Line pg 379

• To find the slop of a line, find the ratio of the
  rise to the run between two points on the line.
• Slope rise
• run

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Interpreting Slope as a Rate pg 379

• When the graph of a line represents a real-world
  situation, the slope of the line can often be
  interpreted as a rate.
• Volcanoes-The graph represents the distance
  traveled by lava flow over time. Find the slope
  of the line.

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Using Slope to Draw a Line

• Plot the given points
• Write the slope as a fraction
• Check the slope between the points.
• Draw a line through the two points
• Complete Getting Ready to Practice pg. 380

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Concept Check

• The ratio of rise to run between point A and B on
  line m is 2/3. What is the ratio of rise to run
  between point B and C on the same line?

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Lesson 8.4 Daily Puzzler

• Explain two methods you could use to solve the
  proportions 5/6 x/36. Then solve the
  proportion.

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Lesson 8.4 Writing Solving Proportions

• Proportion-an equation that states two ratios are
  equivalent.
• 3 6
• 5 10
• Using Equivalent Ratios when one is unknown in
  a proportion, you can find the by solving the
  proportion. One way to solve a proportion is to
  use mental math to find an equivalent ratio.

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Using Equivalent Ratios

• To find the number C of calories the person would
  burn while skateboarding for 60 min., solve the
  proportion 150 C
• 30 60

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Solving Proportions Using Algebra

• Solve the proportion
• 6 x
• 10 25

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Writing solving a proportion

• Empire State Building-At maximum speed, the
  elevators in the Empires State Building can pass
  80 floors in 45 sec. Find the of floors that
  the elevators can pass in 9 sec.

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8.4 Concept Check

• Explain two methods you could use to solve the
  proportion 5/6 x/36

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Lesson 8.5 Daily Puzzler

• Your team is in a tournament with 15 other teams.
  When a team loses a game, it is eliminated. How
  many games must be played to determine the winner?

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8.5 Solving Proportions Using Cross Products

• Cross products- in a proportion a/b c/d where b
  ? 0 and d ? 0, the cross products are ad and bc.