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Section A.1: Three Ways of Using Percentages
Using percentages
We can use percentages in three different ways:
• To express a fraction of something. For example, ”A total of 10, 000 newspaper employees, 2.6% of
the newspaper work force, lost their jobs“ uses percentage to express a fraction of total newspaper
work force.
• To describe a change in something. For example, ”Cisco stock rose 5.7% last week, to $18“ uses percentage to describe a change in stock price.
• To compare two objects. For example, ”High definition television sets have 125% more resolution
than conventional TV sets, but cost 400% more“ uses percentage to compare the resolutions and the
costs of televisions.
Using Percentages as Fractions
Ex.1
If 10% of eighth-graders smoke and there are 50, 000 eighth-graders, how many eighth-graders smoke?
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Using Percentages to Describe Change
Absolute change and relative change
We can express the change of something in two ways:
• The absolute change describes the actual increase or decrease from a reference value to a new value:
absolute change = new value − reference value.
• The relative change is a fraction that describes the size of the absolute change in comparison to the
reference value:
new value − reference value
absolute change
=
.
relative change =
reference value
reference value
The relative change can be converted from a fraction to a percentage by multiplying by 100%. The relative
change formula leads to the following important rules:
• When a quantity doubles in value, its relative change is 1 = 100
100 = 100%.
• When a quantity triples in value, its relative change is 2 = 200%.
• When a quantity quadruples in value, its relative change is 3 = 300%. And so on.
Note that the absolute and relative change are positive if the new value is greater than the reference value
and the absolute and relative change are negative if the new value is less than the reference value.
Ex.2
Suppose the population of a town was 2, 000 in 1980 and 7, 000 in 2000. Find the absolute change and the
relative change.
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Ex.3 Depreciating a Computer.
You bought a computer three years ago for $1000. Today, it is worth only $300. Describe the absolute and
relative change in the computer’s value.
Using Percentages for Comparisons
Absolute change and relative difference
Percentages are commonly used to compare two numbers. There are two different ways to compare two
objects:
• The absolute change is the actual difference between the compared value and the reference value:
absolute difference = compared value − reference value.
• The relative difference describes the size of the absolute difference as a fraction of the reference value:
compared value − reference value
absolute difference
=
.
reference value
reference value
The relative difference formula gives a fraction. We can convert the answer to a percent difference by multiplying it by 100%.
The absolute and relative difference are positive if the compared value is greater than the reference value
and the absolute and relative change are negative if the compared value is less than the reference value.
relative difference =
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Ex.4
Suppose we want to compare the price of a $70, 000 Ferrari to the price of a $40, 000 Lexus. Describe the
absolute and relative difference.
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Section A.2: ”Of“ versus ”More Than“
”Of“ versus ”More Than“
There are two different ways to state a change with percentages: ”of“ and ”more than“.
In the case of ”more than“ we state the relative change. In the case we are using ”of“, we consider the ratio
of the new value and the old value.
• If the compared value is P % more than the reference value, it is (100 + P )% of the reference value.
• If the compared value is P % less than the reference value, it is (100 − P )% of the reference value.
Ex.5 Sale!
A store is having 50% off sale. How does an item’s sale price compare to its original price?
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Section A.3: Percentages of Percentages
Percentage Points versus %
When you see a change or difference expressed in percentage points, you can assume it is an absolute change or
difference. If it is expressed with the % sign or the word percent, it should be a relative change or difference.
Ex.6
Suppose your bank increases the interest rate on your savings account from 2% to 5%. What is the absolute
change? What is the relative change?
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Section A.4: Solving Percentage Problems
Solving Percentage Problems
If the compared value is P % more than the reference value, then
compared value = (100 + P )% × reference value
and
compared value
.
(100 + P )%
If the compared value is P % less than the reference value, then
reference value =
compared value = (100 − P )% × reference value
and
reference value =
compared value
.
(100 − P )%
Ex.7
Retail prices are 25% more than whole sale prices. If the whole sale price is $10, how much is the retail
price?
Ex.8
(1) You purchase a bicycle with a labeled (pre-tax) price of $760. The local sale tax rate is 7.6%. What is
your final cost (including tax)?
(2) Your receipt shows that you paid $107.69 for your new shoes, tax included. The local sales tax rate
is 6.2%. What was the labeled (pre-tax) price of the shoes?
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Ex.9
Consider the statement:
In the past four decades the percentage of bicycle in Italy decreased from 55 per cent, to 3 per cent.
What was the previous percentage of bicycle in Italy?
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Section A.5: Abuses of Percentages
Solving Percentage Problems
There are few common abuses of percentages: shifting reference values, less than nothing, average of percentages.
Beware of Shifting Reference Values
Ex.10
Because of losses by your employer, you agree to accept a temporary 10% pay cut. Your employer
promises to give you 10% pay raise after six months. Will the pay raise restore your original salary?
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Less Than Nothing
Ex.11 Impossible Sale.
A store advertises that it will take 150% off the price of all merchandise. What should happen when you
go to the counter to buy a $500 item?
Don’t Average Percentages
Ex.12
Suppose you got 70% of the questions correct on a midterm exam and 90% of the questions correct on the
final exam. Can you conclude that you answered 80% of all the questions correctly?
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Section B.1: Writing Large and Small Numbers
Large and small numbers
Working with large and small numbers is much easier when we write them in a special format called
scientific notation.
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.
Ex.13
one billion = 109 (ten to the ninth power)
6 billion
= 6 × 109
420
= 4.2 × 102
0.5
= 5 × 10−1
Ex.14 Numbers in Scientific Notation.
Rewrite each of the following statement using scientific notation.
(1) The U.S. federal debt is about $9, 100, 000, 000, 000.
(2) The diameter of a hydrogen nucleus is about 0.000000000000001 meter.
Approximations with Scientific Notation
Approximations with scientific notation
We can use scientific notation to approximate answers without a calculator.
Ex.15 Checking Answers with Approximations.
You and a friend are doing a rough calculation of how much garbage New York City residents produce
every day. You estimate that, on average, each of the 8 million residents produces 1.8 pounds or 0.0009
ton of garbage each day. Thus the total amount of garbage is
8, 000, 000 person × 0.0009 ton.
Your friend quickly presses the calculator buttons and tells you that the answer is 225 tons. Without using
your calculator, determine whether this answer is reasonable.
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Section B.2: Giving Meaning to Numbers
Giving meaning to numbers
Now that we have a method for writing large and small numbers, we can put numbers in perspective. We
will study three techniques to put the number in perspective: through estimation, through comparisons
and through scaling.
Perspective through Estimation
Definition of order of magnitude estimate
An order of magnitude estimate specifies only a broad range of values, such as ”in the ten thousands“ or ”in
the millions“.
Ex.16
We might say that the population of the United States is ”on the order of 300 million“, by which we mean
it is nearer to 300 million then to, say, 200 million or 400 million.
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Perspective through Comparisons
Ex.17
Consider $100 billion, which is more or less the wealth of the world’s richest individuals. It’s easy to say
a number like 100 billion, but how big is it?
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Perspective through Scaling
Ways of expressing scales
There are three ways of expressing scales:
• Verbally: A scale can be described in words such as ”One centimeter represents one kilometer“ or,
more simply, as ”1 cm = 1 km“.
• Geographically: A marked miniruler can show the scale visually.
• As a ratio: We can state the ratio of a scaled size to an actual size. For example, there are 100, 000 in
a kilometer. Thus, a scale where 1 centimeter represents 1 kilometer can be described as a scale ratio
of 1 to 100, 000.
Ex.18 Scale Ratio.
A city map states, ” One inch represents one mile“. What is the scale ratio for this map?
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Ex.19 Earth and Sun.
The distance from the Earth to the Sun is about 150 million kilometers. The diameter of the Sun is about
1.4 million kilometers and the diameter of the Earth is about 12, 760 kilometers. Put this numbers in
perspective by using a scale model of the scalar system with a 1 to a 10 billion scale.
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Section C.1: Significant Digits
Significant digits
The digits in a number that represents actual measurements and therefore have meaning are called significant digits.
Significant digits:
• Nonzero digits.
• Zeros that follow a nonzero digit and lie to the right of the decimal point.
• Zeros between nonzero digits or other significant zeros.
Not significant digits:
• Zeros to the left of the first nonzero digit.
• Zeros to the right of the last nonzero digit but before the decimal point.
Ex.20
• 132 pounds has 3 significant digits and implies a measurement to the nearest pound.
• 132.00 pounds has 5 significant digits and implies a measurement to the nearest hundredth of a
pound.
• 2 × 102 students has 1 significant digit and implies a measurement to the nearest hundred students.
• 2.00 × 102 students has 3 significant digits and implies exactly 200 students.
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Ex.21 Counting significant digits.
State the number of significant digits and the implied meaning of the following numbers:
(1) a time of 280 seconds;
(2) a length of 0.00675 meter;
(3) a population reported as 250, 000;
(4) a population reported as 2.50 × 105 .
Section C.2: Rounding
Significant digits
The basic process of rounding numbers take two steps:
• Step 1: Decide which decimal place (for example, tens, ones, tenths or hundredths) is the smallest
that should be kept.
• Step 2: Look at the number in the nearest place to the right (for example, if rounding the tenths,
look at hundredths). If the value in the next place is less than 5 round down, if it is 5 or greater than 5,
round up.
Ex.22
•
•
•
•
•
•
382.2593 rounded to the nearest thousandth is 382.259.
382.2593 rounded to the nearest hundredth is 382.26.
382.2593 rounded to the nearest tenth is 382.3.
382.2593 rounded to the nearest one is 382.
382.2593 rounded to the nearest ten is 380.
382.2593 rounded to the nearest hundred is 400.
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Ex.23 Rounding with significant digits.
For each of the following operations, give your answer with the specified number of significant digits:
(1) 7.7 mm × 9.92 mm; give your answer with 2 significant digits;
(2) 240, 000 × 72, 106; give your answer with 4 significant digits.
Section C.3: Understanding Errors
Types of Error: Random and Systematic
Types of error: random error and systematic error
There are two types of error:
Significant digits:
• Random errors occur because of random and inherently unpredictable events in the measurement
process. We can minimize the effect of random errors by making many measurements and averaging them.
• Systematic errors occur when there is a problem in the measurement system that affect all measurements in the same way, such as making them all too low or too high by the same amount. If we
discover a systematic error, we can go back and adjust the affected measurements.
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Ex.24
Suppose you work in a pediatric office and use a digital scale to weigh babies. If you have ever worked
with babies, you know that they usually aren’t very happy about being put on a scale. Their thrashing
and crying tends to shake the scale making the readout jump around. You could equally well record the
baby’s weight as anything between 14.5 and 15.0 pounds. The shaking of the scale introduces a random
error. If you measure the baby’s weight ten times, your measurements will probably be too high in some
case and to low in other cases. When you average the measurements you are likely to get a value that
better represents the true weight.
Now, suppose you have weighed babies all day. At the end of the day, you notice that the scale reads
1.2 pounds when there is nothing on it. In that case, every measurement you made was too high by 1.2
pounds. Therefore, we have a systematic error. Now that you know about this systematic error, you can
go back and adjust the affected measurements.
Ex.25 Errors in global warming data.
Scientists studying global warming need to know how the average temperature of the entire Earth, or
the global average temperature, has changed with time. Consider two difficulties (among many others) in
trying to interpret historical temperature data from the early 20th century:
(1) temperatures were measured with simple thermometers and the data were recorded by hand;
(2) most temperature measurements were recorded in or near urban areas, which tend to be warmer
than surrounding rural areas because heat released by human activity.
Discuss whether each of these two difficulties produces random or systematic errors, and consider the
implications of these errors.
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Size of Errors: Absolute versus Relative
Sizes of error: absolute error and relative error
There are two types of error:
• The absolute error describes how far a measured value lies from the true value:
absolute error = measured value − true value.
• The relative error compares the size of the error to the true value:
absolute error
measured value − true value
=
.
true value
true value
The absolute and the relative error are positive when the measured value is greater than the true value and
negative when the measured value is less than the true value.
Note that the above formula gives the relative error as a fraction which can be converted to a percentage.
relative error =
Ex.26
• Suppose you go to a store and ask 6 pounds of hamburger. However, because the store’s scale is
poorly calibrated, you actually get 4 pounds.
• Suppose you buy a car which the owner’s manual says weighs 3132 pounds, but you find that it
really weighs 3130 pounds.
Compute the absolute and relative error and discuss why you are disappointed in the first case but you
don’y care too much in the second case.
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Describing Results: Accuracy and Precision
Accuracy and precision
Once a measurement is reported, we should evaluate it to see whether it is believable in light of any potential errors. In particular, we should consider two key ideas about any reported value: its accuracy and its
precision. The term are often use interchangeably in English, but mathematically they are different.
• Accuracy describes how closely a measurement approximates a true value. An accurate measurement is very close to the true value.
• Precision describes the amount of detail in a measurement.
Ex.27 Accuracy.
If a census says that the population of your home town is 72, 453, but the true population is 96, 000, then
the census report is not very accurate. In contrast, if a company projects sales of $7.30 billion and true sales
turn out to be $7.32 billion, we would say that the projection is quite accurate.
Ex.28 Precision.
A distance given as 2.345 kilometers is more precise than a distance given as 2.3 kilometers because the first
number gives detail to the nearest 0.001 kilometer and the second number gives detail only to the nearest
0.1 kilometer.
Similarly, an income of $45, 678.90 has greater precision than an income of $46, 000 because the first income
is precise to the nearest penny and the second income is precise only to the nearest thousand dollars.
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Ex.29 Accuracy and Precision in your Weight.
Suppose your true weight is 102.4 pounds. The scale at the doctor’s office, which can be read only to the
nearest quarter pound, says that you weigh 102 14 puonds. The scale at the gym, which gives a digital
readout to the nearest 0.1 pound says that you weigh 100.7. Which scale is more precise? Which scale is
more accurate?
Summary: Dealing with Errors
Summary
• Errors can occur in many ways, but generally can be classified as one of two basic types: random
errors and systematic errors.
• Whatever the source of an error, its size can be described in two different ways: as an absolute error
or as a relative error.
• Once a measurement is reported, we can evaluate it in terms of its accuracy and its precision.
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Section C.3: Combining Measured Numbers
Combining measured numbers
In scientific or statistical work, researchers conduct careful analyses to determine how to combine numbers
properly. We can use two simple rounding rules:
• Rounding rule for addition and subtraction: Rounding your answer to the same precision as the
least precise number in the problem.
• Rounding rule for multiplication or division: Rounding your answer to the same number of significant digits as the measurement with the fewest significant digits.
Note: to avoid errors, you should do the rounding only after completing all the operations, not during
intermediate steps.
Ex.30
Suppose that you live in a city with a population of 300, 000. One day, your best friend move to your city
to share an apartment with you. What is the population of your city now?
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Section D.1: What Is an Index Number?
Definition of index number
An index number provides a simple way to compare measurements made at different times or in different
places. The value at one particular time (or place) must be chosen as the reference value. The index number
for any other time (or place) is
index number =
value
× 100.
reference value
Ex.31 Finding an index number.
The price of gasoline in 1965 was 31.2 cents. Suppose the current price of gasoline is $3.20 per gallon.
Using the 1975 price (= 56.7 cents) as the reference value, find the price index number for gasoline today
and in 1965.
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Making Comparisons with Index Numbers
Ex.32
(1) Suppose it cost $700 to fill a gas tank in 1975. How much would it have cost to fill the same tank in
2005? (price index for 2005 = 407.4, reference value = 1975 price).
(2) Suppose it cost $20.00 to fill the gas tank in 1995. How much would it have cost to fill the same tank
in 1955? (price index for 1995 = 212.5, price index for 1955 = 51.3, reference value = 1975 price).
Changing The Reference Value
Ex.33
Suppose the current price of gasoline is $3.20 per gallon. Using the 1985 price (= $1.196) as the reference
value, find its price index number. Compare this answer to the answer in Example 1, where 1975 was the
reference year.
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Section D.2: The Consumer Price Index
Inflation and Consumer Price Index
We have seen that the price of gas has risen substantially with time. Most other prices and wages have also
risen, a phenomenon we call inflation. Thus, changes in the actual price of gasoline are not very meaningful
unless we compare them to the overall rate of inflation, which is measured by the Consumer Price Index
(CPI).
The Consumer Price Index is based on an average of prices for a sample of more than 60, 000 goods, services,
and housing costs. It is computed and reported monthly.
The Consumer Price Index is just an index and the reference value is an average of prices during the period
1982 − 1984.
Ex.34
To find out how much higher typical prices were in 2005 than in 1995, compute the ratio of the CPIs for
the two years using the shorthand CPI2005 to represent the CPI for 2005 and the shorthand CPI1995 to
represent the CPI for 1995:
Ex.35 CPI changes.
Suppose you needed $30, 000 to maintain a particular standard of living in 2000. How much would you
have needed in 2005 to maintain the same standard of living? Assume that the average price of your
typical purchased has risen at the same rate as the Consumer Price Index (CPI).
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The Rate of Inflation
Rate of Inflation
The rate of inflation from one year to the next is usually defined as the relative change in the Consumer Price
Index.
Ex.36
Fing the inflation rate from 1998 to 1999 (CPI1998 = 163.0, CPI1999 = 166.6).
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Adjusting Prices for Inflation
Adjusting prices for inflation
Given a price in dollars per year X ($X ), the equivalent price in dollars for year Y ($Y ) is
CP IY
price in $Y = ( price in $X ) ×
CP IX
where X and Y represent the years, such as 1992 and 2005.
Ex.37
According to computer performance tests, a Macintosh computer that cost $1, 000 in 2005 had computing
power equivalent to that of a supercomputer that sold for $30 million in 1985. If computer prices had
risen with inflation, how much would the computing power of the 1985 supercomputer have cost in
2005? What does this tell us about the cost of computers?
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