Numbers in the Real World

1
Numbers in the Real World

• Using percentages
• Putting numbers in perspective

2
Three ways of using Percentages

• A total of 2,000 people, 5 of the companys
  workforce, were laid off.
• Expresses a fraction of the total workforce
• Google stock fell 5.7 last week, to 519.27.
• Describes a change in the stock price
• High Definition TVs have 125 more resolution
  than conventional TVs, but cost 400 more.
• Compares resolution and costs of televisions

3
Using percentages as fractions

• Percent is just a fancy way of saying divided by
  100
• 9.8 means 9.8/100, or .098
• Example There are 436 cars on the parking lot,
  of which 32 are Fords. How many Fords are in
  the parking lot?
• 32 x 436 .32 x 436 139.52 or 139 Fords

4
Using percentages to describe change

• Percentages are often used to describe how a
  quantity changes with time.
• Absolute change
• New value - reference value
• Relative change
• (New value - reference value)/reference value
• Example you bought a computer 3 years ago for
  1000. Today it is worth 300. What is the
  absolute change and the relative change in the
  computers value?
• Absolute change 300. - 1000. -700.00
• Relative change (300.-1000.)/1000.
  -0.7-70

5
Using percentages for comparisons
Suppose you are comparing the prices of a 1989
Corvette And a 1989 Camaro. The Corvette costs
15,000 and the Camaro costs 12,000.
15,000 Corvette
12,000 Camaro
20 Less
25 More
6
Of versus More Than

• Consider a population that triples in size, from
  100 to 300.
• The new population is 200 more than the original
  population
• The new population is 300 of the original
  population
• If the compared value is P more than the
  reference value, it is (100P) of the reference
  value.
• If the compared value is P less than the
  reference value, it is (100-P) of the reference
  value.
• A store is having a 25 off sale. How does an
  items sale price compare to its original price?

7
Percentages of Percentages

• A Time Magazine story claims that the percentage
  of adults reading daily newspapers declined over
  a 25-year period from 78 to 64. Describe the
  change in readership.

8
Solving Percentage Problems

• If the compared value is P more than the
  reference value, then
• Compared value (100P) x reference value
• If the compared value is less than the reference
  value, then use (100-P) instead of (100P) in the
  equation.

9
Abuses of Percentages

• Beware of shifting reference values.
• Suppose you agree to take a temporary 10 pay cut
  because your company has a bad quarter. Your
  employer promises to give you a 10 raise in 6
  months, Will you be back to your original
  salary?
• Less than nothing
• An advertisement claims that replacing
  conventional light bulbs with CFL bulbs will use
  200 less energy.
• Dont average percentages
• Suppose that you got a 90 on your last quiz and
  a 70 on your exam. Will your class average be
  80?

10
Putting Numbers in Perspective

• What is a billion dollars, or a trillion dollars?
  These numbers are so large that to many people
  they are just words without real meaning. We
  cannot truly understand the issues of our time
  unless we have some sense of what these numbers
  mean.

11
Scientific Notation

• Scientific notation is a format in which a number
  is expressed as a number between 1 and 10
  multiplied by a power of 10.
• The U.S. federal debt is 9.1 x 1012
• The diameter of a hydrogen nucleus is about 1 x
  10-15 m
• Approximations are easy in scientific notation
• 5893 x 212 is around (6x103) x (2x102) 12 x 105

12
Giving Meaning to Numbers

• When we need to represent things as small as an
  atom and as large as the distances between solar
  systems, it is difficult to put it all in
  perspective. These techniques are useful for
  putting numbers in perspective
• Estimation
• Comparison
• Scaling

13
Perspective Through Estimation

• How high is 1000 ft.?
• (Think in terms of building height)
• We know that each story on a building is around
  10 ft. from floor to ceiling, so 1000 ft. is the
  height of a 100 story building.

14
Perspective Through Comparisons

• How much is 100 billion.
• (Think in terms of how long it would take to
  count if you counted 1 bill per second)

15
Perspective Through Scaling

• A city map states,One inch represents 1 mile.
  What is the scale ratio for this map?
• So the scale ratio is 63,360