Math for Liberal Arts MAT 110: Chapter 3 Notes Uses and Abuses of - - PowerPoint PPT Presentation

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Math for Liberal Arts MAT 110: Chapter 3 Notes

Numbers in the Real World David J. Gisch

Uses and Abuses of Percentages

Three Ways of Using Percentages

• As fractions:

15% of the of the 850 students in a school were absent.

• To describe change:

The price of a stock increased 75% from $50 per share.

• For comparisons:

A Mercedes costs 25% more than a Lexus.

Absolute and R elative Change

The absolute change describes the actual increase or decrease from a reference value (starting number) to a new value: – The relative change is a fraction that describes the size

• f the absolute change in comparison to the reference

value: –

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Absolute Change

• Note that absolute change is the amount that
• ne entity changes.

▫ Population of U.S. ▫ Your weight ▫ The price of your car ▫ Google stock ▫ U.S. deficit

• Remember the reference value is the OLD value.

Absolute Change

Example: A diversified portfolio grows from $1500 to $2500. Calculate the absolute and relative change.

Absolute and R elative Change

The absolute difference is the actual difference between the compared value and the reference value:

–

The relative difference describes the size of the absolute difference as a fraction of the reference value:

–

• Absolute Difference
• Note that absolute difference is the amount of

difference between two different objects.

▫ Population of U.S and population of China. ▫ Your weight and the national norm. ▫ The price of a Toyota versus a Chevy. ▫ Google stock versus Apple stock.

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Absolute Difference

Example: Identify the reference value in each of the following.

• If the price of Cheetos is $5 and the price of Doritos is

$4, what is the price of Cheetos compared to Doritos?

• An Earth orbit is how much shorter than a Martian orbit

(in earth days)?

• How much more does a Corvette cost than a Prius?
• If you are 5’8” and I am 6’2”, how do our heights

compare?

Absolute Change and Difference

Example: The length of an Earth orbit (365.25 earth days) is ______ percent shorter than a Martian (686.971 earth days)

• rbit.

(a) Is it change or difference? (b) Which is the compared to amount and which is the reference amount? (c) Calculate the answer.

Absolute Change and Difference

Example: World population was 2.6 billion in 1950 and 7 billion in 2011. (a) Calculate the absolute and relative change. (b) Why is this not absolute and relative difference?

Change and Difference

Example: The average 3 bedroom house in New York is $485,000. In Iowa it is $135,000. Compare these amounts.

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Of versus More Than (or Less Than)

• If the compared value is P% more than the reference

value, it is % of the reference value.

• If the compared value is P% less than the reference

value, it is % of the reference value. Example: If Mr. Wiggin makes 35% more than Mr. Bean, how many times larger is Mr. Wiggin’s income?

S

• lving Problems with Percents

Turn it into a decimal

100 %

Percentages of Percentages

• When a change or difference is expressed

▫ in percentage points, assume it is an absolute change or difference. ▫ with the % sign or the word percent, it is a relative change or difference.

• For example, if a bank increases its CD rates

from 1.5% to 2%.

Percentages of Percentages

Example: Your grade was 80% but after a test it increases to 83%. You state that your grade went up 3%. Describe why this is correct or incorrect.

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S

• lving Percentage Problems

Example: The amount of violent acts of crime has decreased 8% over a decade to 1,240 per year. (a) Which is the reference value? (b) What was the amount 10 years ago?

S

• lving Percentage Problems

Example: An item is advertised on sale 20% off for $85. What was the original price?

S

• lving Percentage Problems

Example: A new car costs $22,500, which is a 4% increase from last year’s price. What was last year’s price (to the nearest dollar)?

S

• lving Percentage Problems

Example: You own a stock that decreased 22%. What increase would it take to return back to its original value?

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Putting Numbers in Perspective

Abuses of Percentages

• Percents don’t add up (shifting reference value)

▫ A 10% pay cut followed by a 10% pay raise.

 You do not get back to the same amount!!!

• Taking more than possible (less than nothing)

▫ You decrease your caloric intake by 150% to lose weight.

 So you went from 3000 calories to -1500 calories?

• Don’t average percents.

▫ If 70% of boys and 60% of the girls in a class voted to go to a water park, then 65% of the students in the class voted to go to the water park.

 Not necessarily the same amount of boys and girls.

Percents don’ t add up (shifting reference value)

Example: A politician promises, “If elected I will cut your taxes by 20% for each of the first three years of my term, for a total cut of 60%.” Suppose you currently pay $1000 in taxes.

Year Tax paid in Previous Year 20 % of Previous Year New Tax this Year

1 $1000 $200 $800 2 3

Don’ t Average Percents

Example: If 70% of boys and 60% of the girls in a class voted to go to a water park, then 65% of the students in the class voted to go to the water park. (a) Suppose there were 100 boys and 120 girls. (b) Suppose there were 50 boys and 75 girls.

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S cientific Notation

• Recall scientific notation

▫ 3 billion ▫ 2.41 10 ▫ 6.02 10

Giving Meaning to Numbers

An order of m agnitude estimate specifies a broad range

• f values.

Think of it as a level of precision and not an exact number.

• For example, whether it should be miles or inches.
• Millions or billions?
• Hours or minutes?

While order of magnitude is important I think it is better to be able to put numbers into perspective.

Numbers put into Perspetive

Example: If every person in America pitched in the same dollar amount each minute, every day for one year, what would it take to wipe out the 16.4 trillion dollar debt.

Numbers put into Perspetive

Example: The richest people in the world make about 100 billion dollars a year. Find a way to put that into perspective.

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• Chris Jordan, Math Art