8/30/2013 Math for Liberal Arts MAT 110: Chapter 3 Notes Uses and Abuses of Percentages Numbers in the Real World David J. Gisch Three Ways of Using Percentages Absolute and R elative Change The absolute change describes the actual increase or • As fractions: decrease from a reference value (starting number) to a new value: 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. The relative change is a fraction that describes the size of the absolute change in comparison to the reference • For comparisons: value: �������� ������ � ��� ����� – ��������� ����� A Mercedes costs 25% more than a Lexus. ��������� ����� �������� ������ � ��� � ��� ��� 1
8/30/2013 Absolute Change Absolute Change Example: A diversified portfolio grows from $1500 to • Note that absolute change is the amount that $2500. Calculate the absolute and relative change. one 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 and R elative Change Absolute Difference The absolute difference is the actual difference between • Note that absolute difference is the amount of the compared value and the reference value: 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. The relative difference describes the size of the absolute difference as a fraction of the reference value: �������� ���������� � �������� ����� – ��������� ����� ��������� ����� �������� ���������� � ��� � ��� ��� 2
8/30/2013 Absolute Difference Absolute Change and Difference Example: Identify the reference value in each of the Example: The length of an Earth orbit (365.25 earth days) is following. ______ percent shorter than a Martian (686.971 earth days) orbit. • If the price of Cheetos is $5 and the price of Doritos is $4, what is the price of Cheetos compared to Doritos? (a) Is it change or difference? (b) Which is the compared to amount and which is the • An Earth orbit is how much shorter than a Martian orbit reference amount? (in earth days)? • How much more does a Corvette cost than a Prius? (c) Calculate the answer. • If you are 5’8” and I am 6’2”, how do our heights compare? Absolute Change and Difference Change and Difference Example: World population was 2.6 billion in 1950 and 7 Example: The average 3 bedroom house in New York is billion in 2011. $485,000. In Iowa it is $135,000. Compare these amounts. (a) Calculate the absolute and relative change. (b) Why is this not absolute and relative difference? 3
8/30/2013 Of versus More Than (or Less Than ) S olving Problems with Percents • If the compared value is P% more than the reference Turn it value, it is ���� � ��% of the reference value. into a • If the compared value is P% less than the reference decimal 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? �������� ����� 100 � � % � ��� ����� Percentages of Percentages Percentages of Percentages Example: Your grade was 80% but after a test it increases • When a change or difference is expressed to 83%. You state that your grade went up 3%. Describe ▫ in percentage points, assume it is an absolute why this is correct or incorrect. 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%. 4
8/30/2013 S olving Percentage Problems S olving Percentage Problems Example: The amount of violent acts of crime has Example: An item is advertised on sale 20% off for $85. decreased 8% over a decade to 1,240 per year. What was the original price? (a) Which is the reference value? (b) What was the amount 10 years ago? S olving Percentage Problems S olving Percentage Problems Example: A new car costs $22,500, which is a 4% increase Example: You own a stock that decreased 22%. What from last year’s price. What was last year’s price (to the increase would it take to return back to its original value? nearest dollar)? 5
8/30/2013 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? Putting Numbers in Perspective • 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. Don’ t Average Percents Percents don’ t add up (shifting reference value) Example: A politician promises, “If elected I will cut your Example: If 70% of boys and 60% of the girls in a class taxes by 20% for each of the first three years of my term, voted to go to a water park, then 65% of the students in the for a total cut of 60%.” class voted to go to the water park. Suppose you currently pay $1000 in taxes. (a) Suppose there were 100 boys and 120 girls. Year Tax paid in 20 % of Previous New Tax this Previous Year Year Year 1 $1000 $200 $800 2 (b) Suppose there were 50 boys and 75 girls. 3 6
8/30/2013 S cientific Notation Giving Meaning to Numbers • Recall scientific notation An order of m agnitude estimate specifies a broad range of values. ▫ 3 billion 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? ▫ 2.41 � 10 � • Hours or minutes? While order of magnitude is important I think it is better ▫ 6.02 � 10 �� to be able to put numbers into perspective. Numbers put into Perspetive Numbers put into Perspetive Example: If every person in America pitched in the same Example: The richest people in the world make about 100 dollar amount each minute, every day for one year, what billion dollars a year. Find a way to put that into would it take to wipe out the 16.4 trillion dollar debt. perspective. 7
8/30/2013 • Chris Jordan, Math Art 8
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