# Foundations

## Systems of Measurement


### Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.
The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.
The equivalencies of measurements are shown in . The table also shows, in parentheses, the common abbreviations for each measurement.
In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.
To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction  When we multiply by this fraction we do not change the value, but just change the units.
But  also equals 1. How do we decide whether to multiply by  or  We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert  inches to feet, which multiplication will eliminate the inches?
The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.
When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.
Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

### Use Mixed Units of Measurement in the U.S. System

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.
Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!

### Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix kilo means thousand. One centimeter is  of a meter, just like one cent is  of one dollar.
The equivalencies of measurements in the metric system are shown in . The common abbreviations for each measurement are given in parentheses.
To make conversions in the metric system, we will use the same technique we did in the US system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a 5K or 10K race? The length of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.
As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.
To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.
We can apply this pattern when we make measurement conversions in the metric system. In , we changed 3,200 grams to kilograms by multiplying by  (or 0.001). This is the same as moving the decimal three places to the left.

### Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.

### Convert Between the U.S. and the Metric Systems of Measurement

Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.
 shows some of the most common conversions.
 shows how inches and centimeters are related on a ruler.
 shows the ounce and milliliter markings on a measuring cup.
 shows how pounds and kilograms marked on a bathroom scale.
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

### Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for  what does that mean?
The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written  The metric system uses degrees Celsius, written   shows the relationship between the two systems.

### Key Concepts

1. Metric System of Measurement


2. Temperature Conversion



### Section Exercises


### Practice Makes Perfect

Make Unit Conversions in the U.S. System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the U.S. System
In the following exercises, solve.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exercises, solve.
Convert Between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

### Chapter Review Exercises


### Introduction to Whole Numbers

Use Place Value with Whole Number
In the following exercises find the place value of each digit.
In the following exercises, name each number.
In the following exercises, write each number as a whole number using digits.
In the following exercises, round to the indicated place value.
In the following exercises, round each number to the nearest ⓐ hundred ⓑ thousand ⓒ ten thousand.
Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
In the following exercises, find the least common multiple of the following numbers using the multiples method.
In the following exercises, find the least common multiple of the following numbers using the prime factors method.

### Use the Language of Algebra

Use Variables and Algebraic Symbols
In the following exercises, translate the following from algebra to English.
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
In the following exercises, simplify
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
In the following exercises, identify the terms in each expression.
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the following phrases into algebraic expressions.

### Add and Subtract Integers

Use Negatives and Opposites of Integers
In the following exercises, order each of the following pairs of numbers, using < or >.
In the following exercises,, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, simplify.
Simplify Expressions with Absolute Value
In the following exercises,, simplify.
In the following exercises, fill in <, >, or = for each of the following pairs of numbers.
In the following exercises, simplify.
In the following exercises, evaluate.
Add Integers
In the following exercises, simplify each expression.
Subtract Integers
In the following exercises, simplify.
In the following exercises, simplify each expression.
Multiply Integers
In the following exercises, multiply.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercises, solve.

### Visualize Fractions

Find Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
Simplify Fractions
In the following exercises, simplify.
Multiply Fractions
In the following exercises, multiply.
Divide Fractions
In the following exercises, divide.
In the following exercises, simplify.
Simplify Expressions Written with a Fraction Bar
In the following exercises, simplify.
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.

### Add and Subtract Fractions

Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
In the following exercises, subtract.
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.

### Decimals

Name and Write Decimals
In the following exercises, write as a decimal.
In the following exercises, name each decimal.
Round Decimals
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide.
Convert Decimals, Fractions, and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.

### The Real Numbers

Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers.
In the following exercises, identify whether each number is rational or irrational.
In the following exercises, identify whether each number is a real number or not a real number.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
In the following exercises, order each of the following pairs of numbers, using < or >.
Locate Decimals on the Number Line
In the following exercises, locate on the number line.
In the following exercises, order each of the following pairs of numbers, using < or >.

### Properties of Real Numbers

Use the Commutative and Associative Properties
In the following exercises, use the Associative Property to simplify.
In the following exercises, simplify.
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
In the following exercises, find the multiplicative inverse of each number.
Use the Properties of Zero
In the following exercises, simplify.
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.

### Systems of Measurement

1.1 Define U.S. Units of Measurement and Convert from One Unit to Another
In the following exercises, convert the units. Round to the nearest tenth.
Use Mixed Units of Measurement in the U.S. System.
In the following exercises, solve.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exerices, solve.
Convert between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Chapter Practice Test

In the following exercises, evaluate.
In the following exercises, simplify each expression.