Lesson 2-4 Warm-Up

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Lesson 2-4 Warm-Up
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(No Transcript)
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Ratios and Proportions (2-4)

• What is a ratio?
• What is a rate?
• What is a unit rate?

• ratio a comparison of two numbers by division
  There are three ways to compare two numbers
• a to b a b
• rate a ratio that compares two quantities
  measured in different units.
• Example is read as 210
  miles per or each 3 hours
• unit rate the rate in which the denominator is 1
  (rate per 1 unit of a given quantity) - Unit
  rate is used in everyday life to determine the
  best value. It can be found by simplifying the
  fraction to a denominator of 1 or simply
  dividing the numerator (top number) by the
  denominator (bottom number).
• Example
  or 210 ? 3 70 mph
• The above unit rate is read as 70 miles per
  hour (70 m/h or 70 mph)

a b
210 miles 3 hours
? 3 ? 3
70 miles 1 hours
210 miles 3 hours
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Ratio and Proportion
LESSON 2-4
Additional Examples
A brand of apple juice costs 1.56 for 48 oz.
Find the unit rate.
The unit rate is 3.25 / oz.
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Ratio and Proportion
LESSON 2-4
Additional Examples
In 2000, Lance Armstrong completed the 3630-km
Tour de France course in 92.5 hours. Traveling at
his average speed, how long would it take Lance
Armstrong to ride 295 km?
known
unknown
km.

hr.
3630 t 92.5 295 Write cross products.
3630t 3630
Traveling at his average speed, it would take
Lance approximately 7.5 hours to cycle 295 km.
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Ratios and Proportions (2-4)

• unit analysis (also called dimensional
  analysis) the process of changing units using
  conversion factors (unit rates in the form of
  fractions that include both the rate in the
  problem and the desired rate you want to
  convert, or change, to)
• To use dimensional analysis, start with the
  units. Multiply the given rate by conversion
  fractions in which the old units cancel out
  (one is on top of the fraction bar and the other
  is on the bottom) so that only the new units
  are left (Example To change the numerator unit,
  multiply by a conversion factor in which the new
  units are on top and the old units are on the
  bottom. To change the denominator unit, multiply
  by a conversion factor in which the new units are
  on the bottom and the old units are on the top ).
• Example To change hours to minutes or minutes to
  hours, you can multiply the number of hours by
  the conversion fraction (in red) as in
• 3 hours x
  180 minutes
• 300 minutes x
  5 hours

• What is a unit analysis?
• How do you use dimensional analysis to solve
  conversion problems in which you need to change
  the units in a rate?

3 hours 1
60 minutes 1 hour
180 minutes 1
300 minutes 1
1 hour 60 minutes
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Ratio and Proportion
LESSON 2-4
Additional Examples
The fastest recorded speed for an eastern gray
kangaroo is 40 mi per hour. What is the
kangaroos speed in feet per second?
5280 ft 1 mi
Use appropriate conversion factors

40 mi 1 h
5280 ft 1 mi.
1 h 60 min.


?
1 min 60 sec
58.6 ft 1 sec
3,600 3,600
211,200 ft 3,600 sec



?
make into a unit rate by making the denominator 1
The kangaroos speed is about 58.6 ft/s.
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Ratio and Proportion
LESSON 2-4
Additional Examples
y 3
3 4
Solve .
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Ratios and Proportions (2-4)

• proportion equal ratios (in other words, equal
  fractions)
• Example
• extremes of the proportion the first cross
  product of a proportion. In the above proportion,
  the extremes of the proportion are a and d.
• means of the proportion the second cross product
  of a proportion. In the above proportion, the
  extremes of the proportion are b and c.
• cross products the product of the means equals
  the product of the extremes (in the above
  example, ad bc).
• Example
• Proof That Cross Products Work For Equal
  Fractions Work

• What is a proportion?
• What is the extremes of the proportion?
• What is the means of the proportion?
• What are cross products?
• Why do cross products work?

a b
c d
for b ? 0 and d ? 0

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Ratio and Proportion
LESSON 2-4
Additional Examples
w 4.5
6 5
Use cross products to solve the proportion
.
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Ratio and Proportion
LESSON 2-4
Additional Examples
z 3 4
z 4 6
Solve the proportion .
2z 18 16
2z -34
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Ratio and Proportion
LESSON 2-4
Lesson Quiz
Solve. 1. Find the unit rate of a 12-oz bottle
of orange juice that sells for 1.29. 2. If you
are driving 65 mi/h, how many feet per second are
you driving? Solve each proportion. 3. 4.
5. 6.
10.75/oz.
about 95.3 ft/s
c 6
12 15
21 12
7 y
4.8
4


1 2
3 x 7
4 8
2 x x 4
25 35
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