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Contemporary Mathematics

9.1 The Metric System

Contemporary Mathematics9.1 The Metric System

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A weighing scale with a block of cheese on the pan.
Figure 9.2 A scale that measures weight in both metric and customary units. (credit: “Weighing the homemade cheddar” by Ruth Hartnup/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Identify units of measurement in the metric system and their uses.
  2. Order the six common prefixes of the metric system.
  3. Convert between like unit values.

Even if you don’t travel outside of the United States, many specialty grocery stores utilize the metric system. For example, if you want to make authentic tamales you might visit the nearest Hispanic grocery store. While shopping, you discover that there are two brands of masa for the same price, but one bag is marked 1,200 g and the other 1 kg. Which one is the better deal? Understanding the metric system allows you to understand that 1,200 grams is equivalent to 1.2 kg, so the 1,200 g bag is the better deal.

Units of Measurement in the Metric System

Units of measurement provide common standards so that regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. Understanding how metric units of measure relate to each other is essential to understanding the metric system:

  • The metric unit for distance is the meter (m). A person’s height might be written as 1.8 meters (1.8 m). A meter slightly longer than a yard (3 feet), while a centimeter is slightly less than half an inch.
  • The metric unit for area is the square meter (m2). The area of a professional soccer field is 7,140 square meters (7,140 m2).
  • The metric base unit for volume is the cubic meter (m3). However, the liter (L), which is a metric unit of capacity, is used to describe the volume of liquids. Soda is often sold in 2-liter (2 L) bottles.
  • The gram (g) is a metric unit of mass but is commonly used to express weight. The weight of a paper clip is approximately 1 gram (1 g).
  • The metric unit for temperature is degrees Celsius (°C). The temperature on a warm summer day might be 24 °C.

While the U.S. Customary System of Measurement uses ounces and pounds to distinguish between weight units of different sizes, in the metric system a base unit is combined with a prefix, such as kilo– in kilogram, to identify the relationship between smaller or larger units.

Checkpoint

When using abbreviations to represent metric measures, always separate the quantity and the units with a space, with no spaces between the letters or symbols in the units. For example, 7 millimeters is written as 7 mm, not 7mm.

Tech Check

It is important to be able to convert between the U.S. Customar System of Measurement and the metric system. However, in this chapter we’ll focus on converting units within the metric system. Why? Typing “200 centimeters in inches” into any browser search bar will instantly convert those measures for you (Figure 9.3). You’ll have an opportunity in the Projects to work between measurement systems.

A screenshot of the Google web browser shows 200 centimeters in inches entered in the search bar
Figure 9.3 (credit: Screenshot/Google)

Let’s be honest. Most of us use computers or smartphones to perform many of the calculations and conversions we were taught in math class. But there is value in understanding the metric system since it exists all around us, and most importantly, knowing how the different metric units relate to each other allows you to compare prices, find the right tool in a workshop, or acclimate when in another country. No matter the circumstance, you cannot avoid the metric system.

Example 9.1

Determining the Correct Base Unit

Which base unit would be used to express the following?

  1. amount of water in a swimming pool
  2. length of an electrical wire
  3. weight of one serving of peanuts

Your Turn 9.1

Determine the correct base measurement for each of the following.
1.
weight of a laptop
2.
width of a table
3.
amount of soda in a pitcher

While there are other base units in the metric system, our discussions in this chapter will be limited to units used to express length, area, volume, weight, and temperature.

Metric Prefixes

Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a centimeter is 102102, or 100 times smaller than a meter, while the kilometer is 103, or 1,000 times larger than a meter.

The metric system combines base units and unit prefixes reasonable to the size of a measured object or substance. The most used prefixes are shown in Table 9.1. An easy way to remember the order of the prefixes, from largest to smallest, is the mnemonic King Henry Died From Drinking Chocolate Milk.

Prefix kilo– hecto– deca– base unit deci– cent– milli–
Abbreviation kk hh dada dd cc mm
Magnitude 103103 102102 101101 100or1100or1 101101 102102 103103
Table 9.1 Metric Prefixes

Example 9.2

Ordering the Magnitude of Units

Order the measures from smallest unit to largest unit.

centimeter, millimeter, decimeter

Your Turn 9.2

1.
Order the measures from largest unit to smallest unit.
hectogram, decagram, kilogram

Example 9.3

Determining Reasonable Values for Length

What is a reasonable value for the length of a person’s thumb: 5 meters, 5 centimeters, or 5 millimeters?

Your Turn 9.3

1.
What is a reasonable estimate for the length of a hallway: 2.5 kilometers, 2.5 meters, or 2.5 centimeters?

Converting Metric Units of Measure

Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is 103103or 1,000 times greater than a gram. One kilogram is equivalent to 1,000 grams.

The tip of a highlighter measures approximately 1 cm. The letter “c” attached to the base unit of meter (m) means the unit used to express the measure is 102or1100102or1100 of a meter. One meter is equivalent to 100 centimeters.

A conversion factor is used to convert from smaller metric units to bigger metric units and vice versa. It is a number that when used with multiplication or division converts from one metric unit to another, both having the same base unit. In the metric system, these conversion factors are directly related to the powers of 10. The most common used conversion factors are shown in Figure 9.4.

Three illustrations show conversion factors for metric units.
Figure 9.4 Common Metric Conversion Factors for (a) Meters, (b) Liters, and (c) Grams

Example 9.4

Converting Metric Distances Using Multiplication

The firehouse is 13.45 km from the library. How many meters is it from the firehouse to the library?

Your Turn 9.4

1.
The record for the men’s high jump is 2.45 m. What is the record when expressed in centimeters?

Example 9.5

Converting Metric Capacity Using Division

How many liters is 3,565 milliliters?

Your Turn 9.5

1.
A bottle of cleaning solution measures 7.6 liters. How many decaliters is that?

Example 9.6

Converting Metric Units of Mass to Solve Problems

Caroline and Aiyana are working on a chemistry experiment together and must perform calculations using measurements taken during the experiment. Due to miscommunication, Caroline took measurements in centigrams and Aiyana used milligrams. Convert Caroline’s measurement of 125 centigrams to milligrams.

Your Turn 9.6

1.
Convert Aiyana’s measurement of 1,457 mg to centigrams.

Example 9.7

Converting Metric Units of Volume to Solve Problems

A bottle contains 500 mL of juice. If the juice is packaged in 24-bottle cases, how many liters of juice does the case contain?

Your Turn 9.7

1.
A hospital orders 250 doses of liquid amoxicillin. Each dose is 5 mL. How many liters of amoxicillin did the hospital order?

People in Mathematics

Valerie Antoine

In the 1970s, people were told that they must learn the metric system because the United States was soon going to convert to using metric measurements. Children and young adults probably watched educational cartoons about the metric system on Saturday mornings.

In 1975, President Gerald Ford signed the Metric Conversion Act and created a board of 17 people commissioned to coordinate the voluntary switch to the metric system in the United States. Among those 17 people was Valerie Antoine, an engineer who made it her life’s work to push for this change. Despite President Ronald Reagan dissolving the board in 1982, effectively killing the move to the metric system at the time, Antoine continued the movement out of her own home as the executive director of the U.S. Metric Association. Reagan’s decision followed intensive lobbying by American businesses whose factories used machinery designed to use customary measurements by workers trained in customary measurements. There was also intense public pressure from American citizens who didn’t want to go through the time consuming and expensive process of changing the country’s entire infrastructure. Fueled by a Congressional mandate in 1992 that required all federal agencies make the switch to the metric system, Antoine never gave up hope that the metric system would trickle down from the government and find its way into American schools, homes, and everyday life.

Example 9.8

Converting Grams to Solve Problems

The nutrition label on a jar of spaghetti sauce indicates that one serving contains 410 mg of sodium. You have poured two servings over your favorite pasta before recalling your doctor’s advice about keeping your sodium consumption below 1 g per meal. Have you followed your doctor’s recommendation?

Your Turn 9.8

1.
The FDA recommends that you consume less than 0.5 g of caffeine daily. A cup of coffee contains 95 mg of caffeine and a can of soda contains 54 mg. If you drink 2 cups of coffee and 3 cans of soda, have you kept your day’s caffeine consumption to the FDA recommendation? Explain.

Example 9.9

Comparing Different Units

A student carefully measured 0.52 cg of copper for a science experiment, but their lab partner said they need 6 mg of copper total. How many more centigrams of copper does the student need to add?

Your Turn 9.9

1.
Kyrie boasted he jumped out of an airplane at an altitude of 3,810 meters on his latest skydive trip. His friend said they beat Kyrie because their jump was at an altitude of 3.2 km. Whose skydive was at a greater altitude?

Who Knew?

The United States and the Metric System

Did you know that the metric system pervades daily life in the United States already? While Americans still may purchase gallons of milk and measure house sizes in square feet, there are many instances of the metric system. Photographers buy 35 mm film and use 50 mm lenses. When you have a headache, you might take 600 mg of ibuprofen. And if you are eating a low-carb diet you probably restrict your carb intake to fewer than 20 g of carbs daily. Did you know even the dollar is metric? In the video, Neil DeGrasse Tyson and comedian co-host Chuck Nice provide an amusing perspective on the metric system.

The International System of Units (SI) is the current international standard metric system and is the most widely used system around the world. In most English-speaking countries SI units such as meter, liter, and metric ton are spelled metre, litre, and tonne.

WORK IT OUT

Get to Know the Metric System

Just how much is the metric system a part of your life now? Probably more than you think. For the next 24 hours, take notice as you move through your daily activities. When you are shopping, are the package sizes provided in metric units? Change the weather app on your phone to display the temperature in degrees Celsius. Are you able to tell what kind of day it will be now? While the United States is not officially using the metric system, you will still find the metric system all around you.

Check Your Understanding

1.
Which metric base unit would be used to measure the height of a door?
2.
Which metric base unit would be used to measure your weight?
3.
Which is greater: 12 hectoliters or 12 centiliters?
4.
Convert 1,520 cm to meters (m).
5.
Convert 1.34 km to decameters (dam).
6.
Convert 12,700 cg to hectograms (hg).
7.
Convert 750 km to millimeters (mm).
8.
Which is the larger measurement: 0.04 dam or 40 cm?
For the following exercises, determine the base unit of the metric system described. Choose from liter, gram, or meter.
1 .
amount of soda in a bottle
2 .
weight of a book you are mailing
3 .
amount of gasoline needed to fill a car’s tank
4 .
height of a computer desk
5 .
weight of a dog at the veterinarian’s office
6 .
dimensions of the newest HD TV
7 .
distance a student athlete ran on the treadmill during their latest workout
8 .
amount of water to add to bleach for mopping floors
9 .
Write the order of the metric prefixes from greatest to least.
For the following exercises, choose the smaller of the two units.
10 .
decagrams or decigrams
11 .
centimeters or millimeters
12 .
liters or kiloliters
13 .
decigrams or centigrams
14 .
decameters or hectometers
15 .
milliliters or deciliters
16 .
Convert 158 hectometers to meters (m).
17 .
Convert 12.3 cg to grams (g).
18 .
Convert 160 dam to kilometer (km).
19 .
You purchase 10 kg of candy. You divide the candy into 25 bags. How many grams of candy are in each bag?
20 .
An outdoor track is 400 m long. If you run 10 laps around the track, how many kilometers have you run?
21 .
An aspirin tablet is 650 mg. If you take 2 aspirin twice in one day, how many grams of aspirin have you taken?
22 .
A juice box contains 450 mL of juice. If there are 6 juice boxes in a package, how many liters of juice are in the package?
23 .
Carlos consumed 4 cans of his favorite energy drink. If each can contains 111 mg of caffeine, how many grams of caffeine did he consume?
24 .
Celeste ran a total distance of 72,548 m in 1 week while training for a 5K fun run. How many kilometers did she run?
25 .
During week one of their diet, Dakota consumed 413 g of carbs. After speaking to their doctor, they only consumed 210 g of carbs the second week. How many fewer milligrams of carbs did they consume the second week?
26 .
Kaylea gained 2.3 kg of muscle weight in 9 months of working out. Cho gained 250 decagrams of muscle during that same time. Who gained more muscle weight? How many grams more?
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