1 7 What Is an Average? Suppose that your class is doing an experiment to determine the boiling point of a particular liquid. Working in groups, your ...

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7

MATH SKILLS

What Is an Average? Suppose that your class is doing an experiment to determine the boiling point of a particular liquid. Working in groups, your classmates come up with several answers that are all slightly different. Your teacher asks you to determine which temperature best represents all of the varying results from the class. A mathematical tool called an average, or mean, will help you solve the problem. An average allows you to simplify a list of numbers into a single number that approximates the value of all of them. Check it out! PROCEDURE: To calculate the average of any set of numbers, first add all of the

numbers to find the sum. Then divide the sum by the amount of numbers in your set. The result is the average of your numbers. SAMPLE PROBLEM: Find the average of the following set of numbers:

5, 4, 7, 8 Step 1: Find the sum.

5 4 7 8 24 Step 2: Divide the sum by the amount of numbers in your set. Because there are

four numbers in your set, divide the sum by 4. 24 24 4 6 or 4 6 MATH SKILLS

The average of the numbers is 6.

Practice Your Skills! 1. Find the average of each of the following sets of numbers. a. 19 m, 11 m, 29 m, 62 m, 14 m

▼ ▼ ▼

Copyright © by Holt, Rinehart and Winston. All rights reserved.

Be sure to show your work for the following problems:

19 m 11 m 29 m 62 m 14 m 135 m; 135 m 5 27 m

b. 12 cm, 16 cm, 25 cm, 15 cm 12 cm 16 cm 25 cm 15 cm 68 cm; 68 cm 4 17 cm

c. 31˚C, 42˚C, 35˚C, 38˚C, 59˚C 31˚C 42˚C 35˚C 38˚C 59˚C 205˚C; 205˚C 5 41˚C

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Name

Date

Class

What Is an Average? continued

Use the data in the tables to complete the following problems. Be sure to show your work. Height of Students (cm) Students

Grade 6

Grade 7

Grade 8

Grade 9

Gretchen

152

156

159

163

Dylan

151

152

157

162

Sergio

144

147

150

152

2. Calculate the average of Gretchen’s and Dylan’s heights in the 8th grade. 159 cm 157 cm 316 cm; 316 cm 2 158 cm

3. What is the average height of all three students in Grade 6? 152 cm 151 cm 144 cm 447 cm; 447 cm 3 149 cm

Year

Arizona

New Mexico

Oklahoma

Texas

1993

10

7

17

85

1994

16

11

24

84

1995

12

5

7

72

1996

13

5

37

91

4. What was the average number of wildfires to occur annually in New Mexico for the years 1993–1996? 7 11 5 5 28; 28 4 7

5. What was the average number of wildfires for all four states in 1995? 12 5 7 72 96; 96 4 24

6. What was the average number of wildfires to occur annually in Texas for the years 1993–1996? 85 84 72 91 332; 332 4 83

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Number of Wildfires in 1993–1996

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MATH SKILLS

What Is SI? To make sharing information easier, most of the world uses the SI of measurement. SI, which stands for Système Internationale d’Unités, is a standard for measuring mass, length, volume, and other quantities. It is used by all scientists to avoid the confusion of comparing data that is based on Quantity Unit Symbol different measuring systems. Three common SI units are length meter m in the chart at right. Obviously, these three units are not volume liter L suitable for all measuring needs. But most quantities can be measured using one of these units with one of the premass gram g fixes in the chart below.

kilohectodeca-

Powers of 10

Symbol

Example

1000

(103)

k

kilogram (kg)

100

(102)

h

hectoliter (hL)

10

(10 )

da

decameter (dam)

—

meter (m), gram (g), liter (L)

d

decigram (dg)

—

1

deci-

0.1

1

(101) 2

centi-

0.01 (10 )

c

centimeter (cm)

milli-

0.001 (10–3)

m

milliliter (mL) MATH SKILLS

Prefix

in the chart above. If you are converting from a smaller prefix to a larger prefix (moving up the chart), divide your number by a power of 10. If you are converting from a larger prefix to a smaller prefix (moving down the chart), multiply your number by a power of 10. SAMPLE PROBLEM A: Convert 500 decimeters (dm) to kilometers (km).

▼ ▼ ▼

Step 1: Find the prefixes of the numbers.

decimeters to kilometers Step 2: Notice that you will move up the chart four places when converting

from deci- to kilo-. Therefore, you will divide your number by 10 10 10 10, or 10,000. 500 10,000 5 0 0 → 0.05 500 decimeters (dm) 0.05 kilometers (km)

SAMPLE PROBLEM B: 2.5 centiliters (cL) is how many milliliters (mL)? Step 1: Find the prefixes.

centiliters to milliliters Step 2: Because you move down the chart one place when converting from

centi- to milli-, multiply your number by 10. 2.5 10 2. 5 → 25 2.5 centiliters (cL) 25 milliliters (mL)

Copyright © by Holt, Rinehart and Winston. All rights reserved.

PROCEDURE: To convert between SI units, first find the prefixes of your numbers

MATH SKILLS FOR SCIENCE

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Date

Class

What Is SI? continued

Work with the System! 1. Write True or False next to each statement. a. 12 hg 1.2 kg

True ______________

False b. 54 cm 5.4 mm ______________

c. 0.5 dL 0.005 cL

False ______________

d. 4.5 g 0.45 dag

e. 111 cm 1.11 m

True ______________

False f. 7 cL 70,000 kL ______________

True ______________

2. Fill in the missing numbers and units in the equations below. a. 25 mm

2.5

cm

b. 27 kg

270,000

dg

c. 50 cm

0.005

hm

d. 1.2 dL

0.12

L

900

mL

f. 41 hm 4,100,000

e. 0.9 L

mm

3. 1 m

10

dm

100

cm

1000

4. 5 kg

50

hg

50,000

dg

500,000

mm cg

5. Special balances can weigh to the 0.00000001 g. How many kilograms is this? 0.00000001 g 0.00000000001 kg

6. A chemistry experiment calls for 5 g of baking soda. Your measuring spoon holds 5000 mg of powder. How many scoops will you need for the experiment?

Challenge Yourself! Some SI prefixes are almost never used because they are so small or large. A micrometer (m) is 10–6 m, while a nanometer is 10–9 m. A gigameter is 109 m. 7. a. How many nanometers are in 1 gigameter? 1 gigameter 1,000,000,000,000,000,000 nanometers

b. How many gigameters are in 1,000,000,000,000 micrometers? 1,000,000,000,000 micrometers 0.001 gigameters

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HOLT SCIENCE AND TECHNOLOGY

Copyright © by Holt, Rinehart and Winston. All rights reserved.

5 g 5000 mg; You will need 1 scoop.

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A Formula for SI Catch-up Scientists use SI all the time. But most people in the United States still use non-SI units. So what do you do if you have data in non-SI units and you want to convert the data into SI units, or vice versa? Have no fear! Conversion charts, like the one shown below, can help you accomplish the task with ease. SI Conversion Chart Multiply by

inches (in.) feet (ft)

2.54

centimeters (cm)

30.50

centimeters (cm)

yards (yd)

0.91

meters (m)

miles (mi)

1.61

kilometers (km)

ounces (oz)

28.35

pounds (lb)

0.45

kilograms (kg)

29.57

milliliters (mL)

fluid ounces (fl oz)

grams (g)

cups (c)

0.24

liters (L)

pints (pt)

0.47

liters (L)

quarts (qt)

0.94

liters (L)

gallons (gal)

3.79

liters (L)

PROCEDURE: To convert from non-SI units to SI units, find the non-SI unit in the

left column and multiply it by the number in the center column. The resulting number will be in the SI unit in the right column. To convert a SI unit into a non-SI unit, find the SI unit in the right column and divide by the number in the center column to get the non-SI unit on the left.

▼ ▼ ▼

Copyright © by Holt, Rinehart and Winston. All rights reserved.

To find

MATH SKILLS

If you know

SAMPLE PROBLEM: Convert 15 gal into liters (L).

15 3.79 56.85 L

Complete the Conversions! 1. Use the SI conversion chart to do the following conversions (round to the nearest hundredths): a. 15 oz

(15 28.35) 425.25g __________________

b. 40 cm

(40 2.54) 15.75 in. __________________

c. 2 c

(2 0.24) 0.48 __________________ L

d. 27 m

(27 0.91) 29.67 yd __________________

e. 5.5 gal

(5.5 3.79) 20.85 L __________________

f. 115 lb

(115 0.45) 51.75 kg __________________

MATH SKILLS FOR SCIENCE

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A Formula for SI Catch-up, continued

2. A chemistry experiment calls for 6 mL of HCl (hydrochloric acid). How many fluid ounces is this? 6 mL 29.57 fl oz/mL 0.2 fl oz

3. Simone wants to compete in a 15 km run. The farthest she can run is 10 mi. Can she finish the race? 10 mi 1.61 km/mi 16.1 km; Yes, she can finish the race.

4. A cake recipe calls for 1 cup of milk. How many milliliters is this? 1 c 0.24 L/c 0.24 L; 0.24 L 240 mL

5. Julie is 162 cm tall. How tall is she in feet? 162 cm 30.5 cm/ft 5.31 ft

6. George ran 1000 yd in gym class. How many kilometers did he run? 1000 yd 0.91 m/yd 900 m; 900 m 1000 km/m 0.9 km

7. Alejandro weighed 8 lb, 4 oz when he was born. How many grams did he weigh?

3600 g 113.4 g 3713.4 g. He weighed 3713.4 g.

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HOLT SCIENCE AND TECHNOLOGY

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8 lb 0.45 kg/lb 3.6 kg; 3.6 kg 1000 g/kg 3600 g; 4 oz 28.35 g/oz 113.4 g;

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MATH SKILLS

Counting the Zeros A power of 10 is a number that can have 10 as its only factors. For instance, (10 10) 100 and (10 10 10 10) 10,000 are both powers of 10. Multiplying and dividing by powers of 10 is as easy as counting the zeros and moving your decimal point the same number of places.

Part 1: Multiplying by Powers of 10 PROCEDURE: To multiply a number by a power of 10, move the decimal point to

the right the same number of places as there are zeros in the power of 10. If there are not enough places in your number to do this, you will need to add zeros to the number as place holders. SAMPLE PROBLEM: Multiply 8.25 by 10, 100, and 1000.

10 8.25 8. 2 5 → 82.5 100 8.25 8. 2 5 → 825 1000 8.25 8. 2 5 → 8250

It’s Your Turn! 1. Write your answers on the lines, and remember to place commas in the appropriate places.

e. 10 11.9

__________________________

71

→ 7100

11. 9 → 119

d. 1000 41

____________________________

f. 67 10,000

____________________________

41.

67

→ 41,000

→ 670,000

Part 2: Dividing by Powers of 10 PROCEDURE: To divide a number by a power of 10, move the decimal point to

the left as many places as there are zeros in the power of 10. SAMPLE PROBLEM: Divide 763 by 10, 1000, and 100,000.

763 10 76 3 → 76.3 763 1000 7 6 3 → 0.763 763 100,000 7 6 3 → 0.00763 2. Divide by powers of 10. 5 5 → 0.055 a. 55 1000 __________________ b. 9907 100 → 62 620 __________________ c. 620 10 d. 4.01 100 04 → 0.00004 e. 0.04 1000 __________________ f. 996 10,000

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HOLT SCIENCE AND TECHNOLOGY

99 0 7 → 99.07 ____________________ 4.01 → 0.0401 ____________________ 9 9 6 → 0.0996 ____________________

Copyright © by Holt, Rinehart and Winston. All rights reserved.

__________________________

____________________________

c. 71 100

9. 3 8 1 → 938.1

b. 9.381 100

→ 60

__________________________

6

a. 10 6

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