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Contemporary Mathematics

9.4 Measuring Weight

Contemporary Mathematics9.4 Measuring Weight
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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.
Figure 9.13 Weight scale at the local Antigua market (credit: “Weight scale at the local Antigua market” by Lucía García González/Flickr, CC0 1.0 Public Domain Dedication)

Learning Objectives

After completing this section, you should be able to:

  1. Identify reasonable values for weight applications.
  2. Convert units of measures of weight.
  3. Solve application problems involving weight.

In the metric system, weight is expressed in terms of grams or kilograms, with a kilogram being equal to 1,000 grams. A paper clip weighs about 1 gram. A liter of water weighs about 1 kilogram. In fact, in the same way that 1 liter is equal in volume to 1 cubic decimeter, the kilogram was originally defined as the mass of 1 liter of water. In some cases, particularly in scientific or medical settings where small amounts of materials are used, the milligram is used to express weight. At the other end of the scale is the metric ton (mt), which is equivalent to 1,000 kilograms. The average car weighs about 2 metric tons.

Any discussion about metric weight must also include a conversation about mass. Scientifically, mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The amount of mass of an object remains constant no matter where the object is. Identical objects located on Earth and on the moon will have the same mass, but the weight of the objects will differ because the moon has a weaker gravitational force than Earth. So, objects with the same mass will weigh less on the moon than on Earth.

Since there is no easy way to measure mass, and since gravity is just about the same no matter where on Earth you go, people in countries that use the metric system often use the words mass and weight interchangeably. While scientifically the kilogram is only a unit of mass, in everyday life it is often used as a unit of weight as well.

Reasonable Values for Weight

To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given weight is reasonable for the situation.

Example 9.31

Identifying Reasonable Units for Weight

Which is the more reasonable value for the weight of a newborn baby:

  • 3.5 kg or
  • 3.5 g?

Your Turn 9.31

1.
Which represents a reasonable value for the weight of a penny:
2.5 g or 2.5 kg

Example 9.32

Determining Reasonable Values for Weight

Which of the following represents a reasonable value for the weight of three lemons?

  • 250 g,
  • 2,500 g, or
  • 250 kg?

Your Turn 9.32

1.
Which represents a reasonable value for the weight of a car:
1,300 g, 130 kg, or 1,300 kg?

Who Knew?

How Do You Measure the Weight of a Whale?

It is impossible to weigh a living whale. Fredrik Christiansen from the Aarhus Institute of Advanced Studies in Denmark developed an innovative way to measure the weight of whales. Using images taken from a drone and computer modeling, the weight of a whale can be estimated with great accuracy.

Example 9.33

Explaining Reasonable Values for Weight

The blue whale is the largest living mammal on Earth. Which of the following is a reasonable value for the weight of a blue whale: 149 g, 149 kg, or 149 mt? Explain your answer.

Your Turn 9.33

1.
The Etruscan shrew is one of the world’s smallest mammals. It has a huge appetite, eating almost twice its weight in food each day. Its heart beats at a rate of 25 beats per second! Which of the following is a reasonable value for the weight of an Etruscan shrew: 2 g, 2 kg, or 2 mt? Explain your answer.

Converting Like Units of Measures for Weight

Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used conversion factors for metric weight are illustrated in Figure 9.14.

An illustration shows three units: milligram, gram, and kilogram.
Figure 9.14 Common Conversion Factors for Metric Weight Units

Example 9.34

Converting Metric Units of Weight Using Multistep Division

How many kilograms are in 24,300,000 milligrams?

Your Turn 9.34

1.
How many kilograms are in 175,000 milligrams?

Example 9.35

Converting Metric Units of Weight Using Multiplication

The average ostrich weighs approximately 127 kilograms. How many grams does an ostrich weigh?

Your Turn 9.35

1.
The world’s heaviest tomato weighed 4.869 kg when measured on July 15, 2020. How much did the tomato weigh in grams?

Example 9.36

Converting Metric Units of Weight Using Multistep Multiplication

How many milligrams are there in 0.025 kilograms?

Your Turn 9.36

1.
How many milligrams are there in 1.23 kilograms?

Solving Application Problems Involving Weight

From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious eating.

Example 9.37

Comparing Weights to Solve Problems

The maximum weight for a child to safely use a car seat is 29 kilograms. If a child weighs 23,700 grams, can the child safely use the car seat?

Your Turn 9.37

1.
The dosage recommendations for a popular brand of acetaminophen are listed in table below. What is the recommended dosage for a child who weighs 17,683 grams?
Weight Dosage
11 kg to 15 kg 5 mL
16 kg to 21 kg 7.5 mL
22 kg to 27 kg 10 mL

Example 9.38

Solving Multistep Weight Problems

A recipe for scones calls for 350 grams of flour. How many kilograms of flour are required to make 4 batches of scones?

Your Turn 9.38

1.
A croissant recipe calls for 500 g of flour. How many kilograms of flour are required to make 10 batches of croissants?

Example 9.39

Solving Complex Weight Problems

The average tomato weighs 140 grams. A farmer needs to order boxes to pack and ship their tomatoes to local grocery stores. They estimate that this year’s harvest will yield 125,000 tomatoes. A box can hold 12 kilograms of tomatoes. How many boxes does the farmer need?

Your Turn 9.39

1.
The average potato weighs 225 grams. A grocery chain orders 5,000 bags of potatoes. Each bag weighs 5 kg. Approximately how many potatoes did they order?

Check Your Understanding

For the following exercises, determine the most reasonable value for each weight.
25.
Candy bar:
50 kg, 50 g, or 50 mg
26.
Lion:
180 kg, 180 g, or 180 mg
27.
Basketball:
624 kg, 624 g, or 624 mg
For the following exercises, convert the given weight to the units shown.
28.
8,900 g = __________ kg
29.
17 g = __________ mg
30.
0.07 kg = __________ g
For the following exercises, determine the total weight in the units shown.
31.
three 48 g granola bars ________ kg
32.
seven 28 g cheese slices ________ mg
33.
six 15 mg tea bags ________ g

Section 9.4 Exercises

For the following exercises, determine the most reasonable value for each weight.
1.
Aspirin tablet:
300 kg, 300 g, or 300 mg
2.
Elephant:
5,000 kg, 5,000 g, or 5,000 mg
3.
Baseball:
145 kg, 145 g, or 145 mg
4.
Orange:
115 kg, 115 g, or 115 mg
5.
Pencil:
6 kg, 6 g, or 6 mg
6.
Automobile:
1,300 kg, 1,300 g, or 1,300 mg
For the following exercises, convert the given weight to the units shown.
7.
3,500 g = __________ kg
8.
53 g = __________ mg
9.
0.02 kg = __________ g
10.
200 mg = __________ g
11.
2.3 g = _________ mg
12.
20 kg = _______ mg
13.
2,300 kg = __________ g
14.
8,700 mg = _________ g
15.
9,730 mg = _______ kg
16.
0.0078 kg = __________ g
17.
2.34 g = _________ mg
18.
234.5 mg = _______ g
For the following exercises, determine the total weight in the units shown.
19.
Three 350 mg tablets ________ g
20.
Seven 115 g soap bars ________ kg
21.
Six 24 g batteries ________ mg
22.
Fifty 3.56 g pennies ________ mg
23.
Eight 2.25 kg bags of potatoes ________ g
24.
Four 23 kg sacks of flour ________ g
25.
Ten 2.5 kg laptops ________ g
26.
Seven 1,150 g chickens ________ kg
27.
Ninety 4,500 mg marbles ________ kg
28.
There are 26 bags of flour. Each bag weighs 5 kg. What is the total weight of the flour?
29.
The average female hippopotamus weighs 1,496 kg. The average male hippopotamus weighs 1,814 kg. How much heavier, in grams, is the male hippopotamus than the female hippopotamus?
30.
Twelve pieces of cardboard weigh 72 grams. What is the weight of one piece of cardboard?
31.
Miguel’s backpack weighs 2.4 kg and Shanayl’s backpack weighs 2,535 grams. Whose backpack is heavier and by how much?
32.
A souvenir chocolate bar weighs 1.815 kg. If you share the candy bar equally with two friends, how many grams of chocolate does each person get?
33.
You purchase 10 bananas that weigh 50 grams each. If bananas cost $5.50 per kilogram, how much did you pay?
34.
A family-size package of ground meat costs $15.75. The package weighs 4.5 kg. What is the cost per gram of the meat?
35.
A box containing 6 identical books weighs 7.2 kg. The box weighs 600 g. What is the weight of each book in grams?
36.
A 2.316 kg bag of candy is equally divided into 12 party bags. What is the weight of the candy, in grams, in each party bag?
37.
A store has 450 kg of flour at the beginning of the day. At the end of the day the store has 341 kg of flour. If flour costs $0.35 per kilogram, how much flour, in dollars, did the store sell that day?
38.
A local restaurant offers lobster for $110 per kilogram. What is the price for a lobster that weighs 450 grams?
39.
A student’s backpack weighs 575 grams. Their books weigh 3.5 kg. If the student’s weight while wearing their backpack is 58.25 kg, how much does the student weigh in kilograms?
40.
The weight of a lamb is 41 kg 340 g. What is the total weight, in kilograms, of four lambs of the same weight?
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