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$: To express a fraction To describe a change To compare more ...meucci/Meucci_files/Review3.pdf · To express a fraction of something. For example, ”A total of 10;000 newspaper employees,$
Chapter 3 Review Math 1030

Section A.1: Three Ways of Using Percentages

Using percentagesWe can use percentages in three different ways:

• To express a fraction of something. For example, ”A total of 10, 000 newspaper employees, 2.6% ofthe newspaper work force, lost their jobs“ uses percentage to express a fraction of total newspaperwork force.

• To describe a change in something. For example, ”Cisco stock rose 5.7% last week, to $18“ uses per-centage to describe a change in stock price.

• To compare two objects. For example, ”High definition television sets have 125% more resolutionthan conventional TV sets, but cost 400% more“ uses percentage to compare the resolutions and thecosts of televisions.

Using Percentages as Fractions

Ex.1If 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 changeWe 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 thereference value:

relative change =absolute changereference value

=new value − reference value

reference value.

The relative change can be converted from a fraction to a percentage by multiplying by 100%. The relativechange formula leads to the following important rules:

• When a quantity doubles in value, its relative change is 1 = 100100 = 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 valueand the absolute and relative change are negative if the new value is less than the reference value.

Ex.2Suppose the population of a town was 2, 000 in 1980 and 7, 000 in 2000. Find the absolute change and therelative 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 andrelative change in the computer’s value.

Using Percentages for Comparisons

Absolute change and relative differencePercentages are commonly used to compare two numbers. There are two different ways to compare twoobjects:

• 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:

relative difference =absolute difference

reference value=

compared value − reference valuereference value

.

The relative difference formula gives a fraction. We can convert the answer to a percent difference by multi-plying it by 100%.The absolute and relative difference are positive if the compared value is greater than the reference valueand the absolute and relative change are negative if the compared value is less than the reference value.

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Ex.4Suppose we want to compare the price of a $70, 000 Ferrari to the price of a $40, 000 Lexus. Describe theabsolute 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 ratioof 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 ordifference. If it is expressed with the % sign or the word percent, it should be a relative change or difference.

Ex.6Suppose your bank increases the interest rate on your savings account from 2% to 5%. What is the absolutechange? What is the relative change?

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Section A.4: Solving Percentage Problems

Solving Percentage ProblemsIf the compared value is P% more than the reference value, then

compared value = (100 + P )%× reference value

andreference value =

compared value(100 + P )%

.

If the compared value is P% less than the reference value, then

compared value = (100− P )%× reference value

andreference value =

compared value(100− P )%

.

Ex.7Retail prices are 25% more than whole sale prices. If the whole sale price is $10, how much is the retailprice?

Ex.8

(1) You purchase a bicycle with a labeled (pre-tax) price of $760. The local sale tax rate is 7.6%. What isyour final cost (including tax)?

(2) Your receipt shows that you paid $107.69 for your new shoes, tax included. The local sales tax rateis 6.2%. What was the labeled (pre-tax) price of the shoes?

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Ex.9Consider 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 ProblemsThere are few common abuses of percentages: shifting reference values, less than nothing, average of per-centages.

Beware of Shifting Reference Values

Ex.10Because of losses by your employer, you agree to accept a temporary 10% pay cut. Your employerpromises 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 yougo to the counter to buy a $500 item?

Don’t Average Percentages

Ex.12Suppose you got 70% of the questions correct on a midterm exam and 90% of the questions correct on thefinal 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 numbersWorking with large and small numbers is much easier when we write them in a special format calledscientific notation.

Scientific notationScientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by apower 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 notationWe 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 produceevery day. You estimate that, on average, each of the 8 million residents produces 1.8 pounds or 0.0009ton 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 usingyour calculator, determine whether this answer is reasonable.

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Section B.2: Giving Meaning to Numbers

Giving meaning to numbersNow that we have a method for writing large and small numbers, we can put numbers in perspective. Wewill study three techniques to put the number in perspective: through estimation, through comparisonsand through scaling.

Perspective through Estimation

Definition of order of magnitude estimateAn order of magnitude estimate specifies only a broad range of values, such as ”in the ten thousands“ or ”inthe millions“.

Ex.16We might say that the population of the United States is ”on the order of 300 million“, by which we meanit is nearer to 300 million then to, say, 200 million or 400 million.

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Perspective through Comparisons

Ex.17Consider $100 billion, which is more or less the wealth of the world’s richest individuals. It’s easy to saya number like 100 billion, but how big is it?

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Perspective through Scaling

Ways of expressing scalesThere 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 ratioof 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 about1.4 million kilometers and the diameter of the Earth is about 12, 760 kilometers. Put this numbers inperspective 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 digitsThe digits in a number that represents actual measurements and therefore have meaning are called signifi-cant 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 digitsThe basic process of rounding numbers take two steps:

• Step 1: Decide which decimal place (for example, tens, ones, tenths or hundredths) is the smallestthat 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 dig-its:

(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 errorThere are two types of error:Significant digits:

• Random errors occur because of random and inherently unpredictable events in the measurementprocess. We can minimize the effect of random errors by making many measurements and averag-ing them.

• Systematic errors occur when there is a problem in the measurement system that affect all measure-ments in the same way, such as making them all too low or too high by the same amount. If wediscover a systematic error, we can go back and adjust the affected measurements.

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Ex.24Suppose you work in a pediatric office and use a digital scale to weigh babies. If you have ever workedwith babies, you know that they usually aren’t very happy about being put on a scale. Their thrashingand crying tends to shake the scale making the readout jump around. You could equally well record thebaby’s weight as anything between 14.5 and 15.0 pounds. The shaking of the scale introduces a randomerror. If you measure the baby’s weight ten times, your measurements will probably be too high in somecase and to low in other cases. When you average the measurements you are likely to get a value thatbetter represents the true weight.Now, suppose you have weighed babies all day. At the end of the day, you notice that the scale reads1.2 pounds when there is nothing on it. In that case, every measurement you made was too high by 1.2pounds. Therefore, we have a systematic error. Now that you know about this systematic error, you cango 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, orthe global average temperature, has changed with time. Consider two difficulties (among many others) intrying 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 theimplications of these errors.

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Size of Errors: Absolute versus Relative

Sizes of error: absolute error and relative errorThere 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:

relative error =absolute error

true value=

measured value − true valuetrue value

.

The absolute and the relative error are positive when the measured value is greater than the true value andnegative 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.

Ex.26

• Suppose you go to a store and ask 6 pounds of hamburger. However, because the store’s scale ispoorly calibrated, you actually get 4 pounds.

• Suppose you buy a car which the owner’s manual says weighs 3132 pounds, but you find that itreally weighs 3130 pounds.

Compute the absolute and relative error and discuss why you are disappointed in the first case but youdon’y care too much in the second case.

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Describing Results: Accuracy and Precision

Accuracy and precisionOnce a measurement is reported, we should evaluate it to see whether it is believable in light of any poten-tial errors. In particular, we should consider two key ideas about any reported value: its accuracy and itsprecision. 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 measure-ment 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, thenthe census report is not very accurate. In contrast, if a company projects sales of $7.30 billion and true salesturn 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 firstnumber gives detail to the nearest 0.001 kilometer and the second number gives detail only to the nearest0.1 kilometer.Similarly, an income of $45, 678.90 has greater precision than an income of $46, 000 because the first incomeis 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 thenearest quarter pound, says that you weigh 102 1

4 puonds. The scale at the gym, which gives a digitalreadout to the nearest 0.1 pound says that you weigh 100.7. Which scale is more precise? Which scale ismore accurate?

Summary: Dealing with Errors

Summary

• Errors can occur in many ways, but generally can be classified as one of two basic types: randomerrors and systematic errors.

• Whatever the source of an error, its size can be described in two different ways: as an absolute erroror 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 numbersIn scientific or statistical work, researchers conduct careful analyses to determine how to combine numbersproperly. We can use two simple rounding rules:

• Rounding rule for addition and subtraction: Rounding your answer to the same precision as theleast precise number in the problem.

• Rounding rule for multiplication or division: Rounding your answer to the same number of sig-nificant 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 duringintermediate steps.

Ex.30Suppose that you live in a city with a population of 300, 000. One day, your best friend move to your cityto 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 numberAn index number provides a simple way to compare measurements made at different times or in differentplaces. The value at one particular time (or place) must be chosen as the reference value. The index numberfor any other time (or place) is

index number =value

reference value× 100.

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 todayand 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 in2005? (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 tankin 1955? (price index for 1995 = 212.5, price index for 1955 = 51.3, reference value = 1975 price).

Changing The Reference Value

Ex.33Suppose the current price of gasoline is $3.20 per gallon. Using the 1985 price (= $1.196) as the referencevalue, find its price index number. Compare this answer to the answer in Example 1, where 1975 was thereference year.

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Section D.2: The Consumer Price Index

Inflation and Consumer Price IndexWe have seen that the price of gas has risen substantially with time. Most other prices and wages have alsorisen, a phenomenon we call inflation. Thus, changes in the actual price of gasoline are not very meaningfulunless 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 period1982− 1984.

Ex.34To find out how much higher typical prices were in 2005 than in 1995, compute the ratio of the CPIs forthe two years using the shorthand CPI2005 to represent the CPI for 2005 and the shorthand CPI1995 torepresent 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 youhave needed in 2005 to maintain the same standard of living? Assume that the average price of yourtypical purchased has risen at the same rate as the Consumer Price Index (CPI).

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The Rate of Inflation

Rate of InflationThe rate of inflation from one year to the next is usually defined as the relative change in the Consumer PriceIndex.

Ex.36Fing the inflation rate from 1998 to 1999 (CPI1998 = 163.0, CPI1999 = 166.6).

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Adjusting Prices for Inflation

Adjusting prices for inflationGiven a price in dollars per year X ($X ), the equivalent price in dollars for year Y ($Y ) is

price in $Y = ( price in $X)× CPIYCPIX

where X and Y represent the years, such as 1992 and 2005.

Ex.37According to computer performance tests, a Macintosh computer that cost $1, 000 in 2005 had computingpower equivalent to that of a supercomputer that sold for $30 million in 1985. If computer prices hadrisen with inflation, how much would the computing power of the 1985 supercomputer have cost in2005? What does this tell us about the cost of computers?

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