Ratios and Proportions

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Ratios and Proportions
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Ratios

• A ratio is a comparison of two quantities.
• Its a fraction!

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Ratios

• A ratio can be written in a variety of ways
• Examples ½, ¾, 1 out of 10, 16, 30, 0.45

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Practice with Ratios

• A soccer team played 30 games and won 14. What
  is the ratio of the number of wins to the number
  of losses?

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Practice with Ratios

• In an 8th grade class, there are b boys and g
  girls. Express the ratio of the number of boys
  to the total number in the class.

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Practice with Ratios
Answer C E ½ or 12
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Practice with Ratios
Answer A D, 2/3 or 23
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Practice with Ratios
Answer C 3/2 or 32
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Practice with Ratioshttp//math.rice.edu/lanius/
proportions/rate.html
Answer D 2/1 or 21
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Practice with Ratios
Answer D 2/3 or 23
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Practice with Ratioshttp//math.rice.edu/lanius/
proportions/rate.html
Answer C 1 ½ or 32
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Proportions

• A proportion is a comparison of ratios.

A proportion is an equation that states that two
ratios are equal
Proportions always have an EQUAL sign!
Ratio
Ratio
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Proportions

• A proportion can be written in two ways
• or 510 12

Both are read as 5 is to 10 as 1 is to 2
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Proportions

• In simple proportions, if both fractions simplify
  to the same value, the proportion is true.

True Proportion Both fractions simplify to
1/5.
False Proportion Both fractions do not
simplify to 1/2.
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Proportions

• When solving for a missing part of a proportion,
  remember both ratios must simplify to the same
  value.

To solve this, we will use FISH!
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Proportions
30 1
30
30 6 5 6 x
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Practice with Proportions
2 60
120
120 40 3 40 x
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Rich can do 5 pull ups in 7 seconds. Yes, it
does make the veins pop out on his neck when he
does them, but, all the guys in his gym class
wish THEY could do it. Assuming he can keep
pulling at this gut wrenching rate, how many will
he complete in a minute?




Explore Underline what the question is asking.
Plan Highlight what we already know.
Convert minutes to seconds
Write your proportion. Make sure the units in the
numerator and denominator are the same.
Solve Use fish and simplify.
Check Is your answer reasonable?
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Rich can do 5 pull ups in 7 seconds. Yes, it
does make the veins pop out on his neck when he
does them, but, all the guys in his gym class
wish THEY could do it. Assuming he can keep
pulling at this gut wrenching rate, how many will
he complete in a minute?




Explore Underline what the question is asking.
how many will he complete in a minute
Plan Highlight what we already know.
5 pull ups in 7 seconds
There are 60 seconds in a minute.
Convert minutes to seconds
Write your proportion. Make sure the units in the
numerator and denominator are the same.
5 x 7 60
Solve Use fish.
x 42.9
7 x 43 300 301 300 It is reasonable to
do 43 pull ups in 1 minute.
Check Is your answer reasonable?
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While on a trip to the beach, Ray traveled 579
miles in 7.8 hours. If he had his ultra, turbo,
cruise control engaged and maintained a constant
rate of speed during those hours, how far did he
go in 2 hours, to the nearest tenth of a mile?


Explore Underline what the question is asking.
Mark
Plan Highlight what we already know.
Ask
Write your proportion. Make sure the units in the
numerator and denominator are the same.
Procedure
Solve Use fish.
Solve
Check Is your answer reasonable?
Check
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Practice Problem

• A proportion is shown. Which describes the
  relationship between x and 4?
• A. 4 is 5 less than x
• B. x is 5 less than 4
• C. 4 is five times as great as x
• D. x is 5 times as great as 4

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Proportions Assessment

• Proportions MAPS
• Use the problem solving steps provided.
• Empty squares will not be tolerated!
• Due Tomorrow!

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Ratio Activity from http//mathforum.org/alejan
dre/frisbie/math/student.leonardo.html
Vitruvius, a Roman engineer of the first century
B.C., influenced Leonardo da Vinci's work in
architecture and also his drawing of the human
figure. One of Leonardo's drawings is called the
Vitruvian Man. It is based on a model of ideal
proportions which Vitruvius established. The
drawing shows a square inscribed inside a circle.
There is a man with outstretched arms and legs,
in fact two pairs of each, which touch both the
circumference of the circle and the vertices of
the square. Upon viewing the drawing the
conclusion can be made that the length of a man's
arm span is equal to the height of the man. In
other words the ratio of the Vitruvian Man's arm
span to his height equals 1. How can we trust a
drawing? The following activity will investigate
if this is true or not.
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Question Is the ratio of our arm span to our
height really equal to 1?
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Using measuring devices work in pairs and measure
both your height and your arm span. As you
measure cut a piece of string equal to the two
measurements. This can be accomplished by
measuring first and then measuring out the string
OR by using the string as the measuring device,
cut it and then find the length of the string.
Either method should result in two pieces of
string per person and noted measurements per
person. As you and your partner are waiting for
the class to complete their measurements and
record them on the data sheet provided, look at
the difference between the lengths of string. As
you work with the data and calculate the ratios,
remember what you noticed about the two string
lengths.    
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Data

• Record results on the following chart.

Height Arm span Height Arm span

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Graphing

• Using the class data graph the two columns of
  data using the graph paper provided.
• Using the class data graph the third column of
  data using graph paper.

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Vitruviuss measurements

• Vitruvius, the architect, says in his work on
  architecture that the measurements of the human
  body are as follows
• 4 fingers made 1 palm
• 4 palms make 1 foot
• 6 palms make 1 cubit (1 cubit 18 inches or 45
  cm)
• 4 cubits make a man's height (or 72 inches)
• 4 cubits make one pace
• 24 palms make a man (72 inches or 6 feet)
• The length of a mans outspread arms is equal to
  his height.
• From the roots of his hair to the bottom of his
  chin is the tenth of a mans height
• From the bottom of the chin to the top of the
  head is one eighth of his height.
• The greatest width of the shoulders contains in
  itself the fourth part of a man.
• From the elbow to the tip of the hand will be the
  fifth part of a man.
• From the elbow to the angle of the armpit will be
  the eighth part of a man.
• The whole hand will be the tenth part of a man.
• The distance from the bottom of the chin to the
  nose and from the roots of the hair to the
  eyebrows is, in each case the same, and like the
  ear, a third part of the face.

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Are you Ideal?From AIMS

• Height vs. Arm Span Ratio Line Plot
• 0.92 0.93 0.94 0.95 0.96 0.97
  0.98 0.99 1.00 1.01 1.02 1.03
  1.04 1.05
• Ratio

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Are you Ideal?From AIMS

• Height vs. Arm Span Ration Box Plot
• 0.92 0.93 0.94 0.95 0.96 0.97
  0.98 0.99 1.00 1.01 1.02 1.03
  10.4 10.5
• Ratio

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• Respond to the following questions
• Explain the conclusion to the question - Is your
  arm span equal to your height?
• Generalize for all of the students in the class
  using the class data.
• Explain what your graphs showed.

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This is so Typicalfrom AIMS
Student Name Body ht/arm span Head ht./body ht Head with/head ht Eye-to-chin/head ht Leg length/body ht Shoulder ht/body ht Foot length/body ht









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• Discussion Questions
• Do men (boys) of the 1900's and 2000's have the
  same proportions as the Vitruvian Man?
• Do women (girls) of the 1900's and 2000's have
  the same proportions as the Vitruvian Man?
• Are taller men (boys) more likely to have
  Vitruvian Man proportions than shorter men
  (boys)?
• Are taller women (girls) more likely to have
  Vitruvian Man proportions than shorter women
  (girls)?
• Does the arm span increase linearly with height,
  i.e. is there a sort of best-fit line through the
  data points or does the taller you are mean the
  wider your arm span is?