Mean, Median, and ModeThe mean is the average, or the sum of the datadivided by the number of data items. The mid-dle number in a set of ordered data is called themedian. The mode is the number that appearsmost often in a set of data.

A scientist counted the number of distinctsongs sung by seven different male birdsand collected the data shown below.

To determine the mean number ofsongs, add the total number of songs anddivide by the number of data items—inthis case, the number of male birds.

To find the median number of songs,arrange the data in numerical order andfind the number in the middle of the series.

27 28 29 35 36 36 40

The number in the middle is 35, so the median number of songs is 35.

The mode is the value that appearsmost frequently. In the data, 36 appearstwice, while each other item appears onlyonce. Therefore, 36 songs is the mode.

Find out how many minutes it takes eachstudent in your class to get to school.Then find the mean, median, and modefor the data.

ProbabilityProbability is the chance that an event willoccur. Probability can be expressed as a ratio, afraction, or a percentage. For example, whenyou flip a coin, the probability that the coin willland heads up is 1 in 2, or , or 50 percent.

The probability that an event will happencan be expressed in the following formula.

P(event) ≠

A paper bag contains 25 blue marbles,5 green marbles, 5 orange marbles, and 15 yellow marbles. If you close your eyesand pick a marble from the bag, what isthe probability that it will be yellow?

P(yellow marbles) ≠

P ≠ , or , or 30%

Each side of a cube has a letter on it. Twosides have A, three sides have B, and oneside has C. If you roll the cube, what isthe probability that A will land on top?

310

1550

15 yellow marbles50 marbles total

Number of times the event can occurTotal number of possible events

12

Mean 5 2317 5 33 songs

Skills Handbook

Math ReviewScientists use math to organize, analyze, and present data.This appendix will help you review some basic math skills.

Male Bird Songs

Bird A B C D E F G

Number36 29 40 35 28 36 27

of Songs

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AreaThe area of a surface is the number of squareunits that cover it. The front cover of your text-book has an area of about 600 cm2.

Area of a Rectangle and a Square Tofind the area of a rectangle, multiply its lengthtimes its width. The formula for the area of arectangle is

A ≠ � � w, or A ≠ �w

Since all four sides of a square have the samelength, the area of a square is the length of oneside multiplied by itself, or squared.

A ≠ s � s, or A ≠ s2

A scientist is studying the plants in a fieldthat measures 75 m � 45 m. What is thearea of the field?

A ≠ � � w

A ≠ 75 m � 45 m

A ≠ 3,375 m2

Area of a Circle The formula for the area ofa circle is

A ≠ π � r � r , or A ≠ πr2

The length of the radius is represented by r,and the value of p is approximately .

Find the area of a circle with a radius of14 cm.

A ≠ πr2

A ≠ 14 � 14 �

A ≠ 616 cm2

Find the area of a circle that has a radius of 21 m.

CircumferenceThe distance around a circle is called the circumference. The formula for finding the circumference of a circle is

C ≠ 2 � π � r , or C ≠ 2πr

The radius of a circle is 35 cm. Whatis its circumference?

C ≠ 2πr

C ≠ 2 � 35 �

C ≠ 220 cm

What is the circumference of a circle with a radius of 28 m?

VolumeThe volume of an object is the number of cubicunits it contains. The volume of a wastebasket,for example, might be about 26,000 cm3.

Volume of a Rectangular Object To findthe volume of a rectangular object, multiplythe object’s length times its width times itsheight.

V ≠ < � w � h, or V ≠ <wh

Find the volume of a box with length 24 cm, width 12 cm, and height 9 cm.

V ≠ <wh

V ≠ 24 cm � 12 cm � 9 cm

V ≠ 2,592 cm3

What is the volume of a rectangularobject with length 17 cm, width 11 cm,and height 6 cm?

227

227

227

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FractionsA fraction is a way to express a part of a whole.In the fraction , 4 is the numerator and 7 isthe denominator.

Adding and Subtracting Fractions Toadd or subtract two or more fractions that havea common denominator, first add or subtractthe numerators. Then write the sum or differ-ence over the common denominator.

To find the sum or difference of fractionswith different denominators, first find the leastcommon multiple of the denominators. This isknown as the least common denominator.Then convert each fraction to equivalent frac-tions with the least common denominator. Addor subtract the numerators. Then write the sumor difference over the common denominator.

Multiplying Fractions To multiply twofractions, first multiply the two numerators,then multiply the two denominators.

Dividing Fractions Dividing by a fraction is the same as multiplying by its reciprocal.Reciprocals are numbers whose numeratorsand denominators have been switched. To divide one fraction by another, first invert thefraction you are dividing by—in other words,turn it upside down. Then multiply the twofractions.

Solve the following: .

DecimalsFractions whose denominators are 10, 100, orsome other power of 10 are often expressed asdecimals. For example, the fraction can beexpressed as the decimal 0.9, and the fraction

can be written as 0.07.

Adding and Subtracting With DecimalsTo add or subtract decimals, line up the deci-mal points before you carry out the operation.

Multiplying With Decimals When youmultiply two numbers with decimals, the num-ber of decimal places in the product is equal tothe total number of decimal places in eachnumber being multiplied.

(one decimal place)(two decimal places)(three decimal places)

Dividing With Decimals To divide a decimalby a whole number, put the decimal point in thequotient above the decimal point in the dividend.

15.5 � 5

To divide a decimal by a decimal, you need torewrite the divisor as a whole number. Do thisby multiplying both the divisor and dividendby the same multiple of 10.

1.68 � 4.2 ≠ 16.8 � 42

Multiply 6.21 by 8.5.

0.442q16.8

3.15q15.5

46.23 2.37

109.494

278.6352 191.400

87.235

27.401 6.19

33.59

7100

910

37 4 4

5

25 4 7

8 5 25 3 8

7 5 2 3 85 3 7 5 16

35

56 3 2

3 5 5 3 26 3 3 5 10

18 5 59

56 2 3

4 5 1012 2 9

12 5 10 2 912 5 1

12

47

Skills Handbook

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Ratio and ProportionA ratio compares two numbers by division. Forexample, suppose a scientist counts 800 wolvesand 1,200 moose on an island. The ratio ofwolves to moose can be written as a fraction,

, which can be reduced to . The same ratiocan also be expressed as 2 to 3 or 2 � 3.

A proportion is a mathematical sentencesaying that two ratios are equivalent. Forexample, a proportion could state that

. You can sometimes set up a proportion to determine or estimate anunknown quantity. For example, suppose a sci-entist counts 25 beetles in an area of 10 squaremeters. The scientist wants to estimate thenumber of beetles in 100 square meters.

1. Express the relationship between beetlesand area as a ratio: , simplified to .

2. Set up a proportion, with x representingthe number of beetles. The proportioncan be stated as .

3. Begin by cross-multiplying. In otherwords, multiply each fraction’s numera-tor by the other fraction’s denominator.

5 � 100 ≠ 2 � x, or 500 ≠ 2x

4. To find the value of x, divide both sidesby 2. The result is 250, or 250 beetles in100 square meters.

Find the value of x in the following proportion: .

PercentageA percentage is a ratio that compares a num-ber to 100. For example, there are 37 graniterocks in a collection that consists of 100 rocks.The ratio can be written as 37%. Graniterocks make up 37% of the rock collection.

You can calculate percentages of numbersother than 100 by setting up a proportion.

Rain falls on 9 days out of 30 in June. Whatpercentage of the days in June were rainy?

To find the value of d, begin by cross-multiplying, as for any proportion:

There are 300 marbles in a jar, and 42 ofthose marbles are blue. What percentageof the marbles are blue?

d 5 30d 5 900309 3 100 5 30 3 d

9 days30 days 5 d%

100%

37100

67 5 x

49

52 5 x

100

52

2510

800 wolves1,200 moose 5 2 wolves

3 moose

23

8001,200

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Skills Handbook

Significant FiguresThe precision of a measurement depends on theinstrument you use to take the measurement.For example, if the smallest unit on the ruler ismillimeters, then the most precise measurementyou can make will be in millimeters.

The sum or difference of measurements canonly be as precise as the least precise measure-ment being added or subtracted. Round your answer so that it has the same number of digitsafter the decimal as the least precise measure-ment. Round up if the last digit is 5 or more,and round down if the last digit is 4 or less.

Subtract a temperature of 5.2�C from thetemperature 75.46�C.

75.46 – 5.2 ≠ 70.26

5.2 has the fewest digits after the decimal,so it is the least precise measurement.Since the last digit of the answer is 6,round up to 3. The most precise differencebetween the measurements is 70.3°C.

Add 26.4 m to 8.37 m. Round your answer according to the precision of themeasurements.

Significant figures are the number of nonzerodigits in a measurement. Zeroes betweennonzero digits are also significant. For example,the measurements 12,500 L, 0.125 cm, and 2.05kg all have three significant figures. When youmultiply and divide measurements, the onewith the fewest significant figures determines thenumber of significant figures in your answer.

Multiply 110 g by 5.75 g.

110 � 5.75 ≠ 632.5

Because 110 has only two significantfigures, round the answer to 630 g.

Scientific NotationA factor is a number that divides into anothernumber with no remainder. In the example, thenumber 3 is used as a factor four times.

An exponent tells how many times a numberis used as a factor. For example, 3 � 3 � 3 � 3can be written as 34. The exponent 4 indicatesthat the number 3 is used as a factor four times.Another way of expressing this is to say that 81is equal to 3 to the fourth power.

34≠ 3 � 3 � 3 � 3 ≠ 81

Scientific notation uses exponents and powers of ten to write very large or very smallnumbers in shorter form. When you write anumber in scientific notation, you write thenumber as two factors. The first factor is anynumber between 1 and 10. The second factor isa power of 10, such as 103 or 106.

The average distance between the planetMercury and the sun is 58,000,000 km. Towrite the first factor in scientific notation,insert a decimal point in the original num-ber so that you have a number between 1 and 10. In the case of 58,000,000, thenumber is 5.8.

To determine the power of 10, count thenumber of places that the decimal pointmoved. In this case, it moved 7 places.

58,000,000 km ≠ 5.8 � 107 km

Express 6,590,000 in scientific notation.

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