Chapter 1 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

absolute value

additive identity

The additive identity is the number 0; adding 0 to any number does not change its value.

additive inverse

The opposite of a number is its additive inverse. A number and its additive inverse add to 0.

coefficient

The coefficient of a term is the constant that multiplies the variable in a term.

complex fraction

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

composite number

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

constant

A constant is a number whose value always stays the same.

counting numbers

The counting numbers are the numbers 1, 2, 3, …

decimal

A decimal is another way of writing a fraction whose denominator is a power of ten.

denominator

The denominator is the value on the bottom part of the fraction that indicates the number of equal parts into which the whole has been divided.

divisible by a number

equality symbol

equation

An equation is two expressions connected by an equal sign.

equivalent decimals

Two decimals are equivalent if they convert to equivalent fractions.

equivalent fractions

Equivalent fractions are fractions that have the same value.

evaluate an expression

To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

expression

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

factors

fraction

integers

The whole numbers and their opposites are called the integers: ...−3, −2, −1, 0, 1, 2, 3...

irrational number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

least common denominator

The least common denominator (LCD) of two fractions is the Least common multiple (LCM) of their denominators.

least common multiple

The least common multiple of two numbers is the smallest number that is a multiple of both numbers.

like terms

Terms that are either constants or have the same variables raised to the same powers are called like terms.

multiple of a number

n

n

multiplicative identity

The multiplicative identity is the number 1; multiplying 1 by any number does not change the value of the number.

multiplicative inverse

The reciprocal of a number is its multiplicative inverse. A number and its multiplicative inverse multiply to one.

number line

A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left.

numerator

The numerator is the value on the top part of the fraction that indicates how many parts of the whole are included.

opposite

origin

The origin is the point labeled 0 on a number line.

percent

A percent is a ratio whose denominator is 100.

prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

prime number

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

radical sign

rational number

p

q

real number

A real number is a number that is either rational or irrational.

reciprocal

repeating decimal

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.

simplified fraction

A fraction is considered simplified if there are no common factors in its numerator and denominator.

simplify an expression

To simplify an expression, do all operations in the expression.

square and square root

term

A term is a constant or the product of a constant and one or more variables.

variable

A variable is a letter that represents a number whose value may change.

whole numbers

The whole numbers are the numbers 0, 1, 2, 3, ....

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Key Concepts

• Place Value as in Figure 1.3.

• Name a Whole Number in Words Step 1. Start at the left and name the number in each period, followed by the period name. Step 2. Put commas in the number to separate the periods. Step 3. Do not name the ones period.

• Write a Whole Number Using Digits Step 1. Identify the words that indicate periods. (Remember the ones period is never named.) Step 2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas. Step 3. Name the number in each period and place the digits in the correct place value position.

• Round Whole Numbers Step 1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change. Step 2. Underline the digit to the right of the given place value. Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value. Step 4. Replace all digits to the right of the given place value with zeros.

• Divisibility Tests: A number is divisible by: 2 if the last digit is 0, 2, 4, 6, or 8. 3 if the sum of the digits is divisible by 3. 5 if the last digit is 5 or 0. 6 if it is divisible by both 2 and 3. 10 if it ends with 0.

• Find the Prime Factorization of a Composite Number Step 1. Find two factors whose product is the given number, and use these numbers to create two branches. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree. Step 3. If a factor is not prime, write it as the product of two factors and continue the process. Step 4. Write the composite number as the product of all the circled primes.

• Find the Least Common Multiple by Listing Multiples Step 1. List several multiples of each number. Step 2. Look for the smallest number that appears on both lists. Step 3. This number is the LCM.

• Find the Least Common Multiple Using the Prime Factors Method Step 1. Write each number as a product of primes. Step 2. List the primes of each number. Match primes vertically when possible. Step 3. Bring down the columns. Step 4. Multiply the factors.

• Notation The result is… $\begin{array}{lccl} \,\mathrm{∘}\,a+b & & & \,\mathrm{the sum of}\,a\,\mathrm{and}\,b \\ \,\mathrm{∘}\,a-b & & & \,\mathrm{the difference of}\,a\,\mathrm{and}\,b \\ \,\mathrm{∘}\,a\cdot{}b,ab,(a)(b)\,(a)b,a(b) & & & \,\mathrm{the product of}\,a\,\mathrm{and}\,b \\ \,\mathrm{∘}\,a\mathrm{÷}b,a/b,\frac{a}{b},ba & & & \,\mathrm{the quotient of}\,a\,\mathrm{and}\,b \\ \end{array}$

• Inequality $\begin{array}{lccl} \,\mathrm{∘}\,a<b\,\mathrm{is read}\,“a\,\mathrm{is less than}\,b” & & & a\,\mathrm{is to the left of}\,b\,\mathrm{on the number line} \\ \,\mathrm{∘}\,a>b\,\mathrm{is read}\,“a\,\mathrm{is greater than}\,b” & & & a\,\mathrm{is to the right of}\,b\,\mathrm{on the number line} \\ \end{array}$

• Inequality Symbols Words $\begin{array}{lccl} \,\mathrm{∘}\,a\ne{}b & & & \,a\,\mathrm{is}\,\mathrm{not equal to}\,b \\ \,\mathrm{∘}\,a<b & & & \,a\,\mathrm{is}\,\mathrm{less than}\,b \\ \,\mathrm{∘}\,a\le{}b & & & \,a\,\mathrm{is}\,\mathrm{less than or equal to}\,b \\ \,\mathrm{∘}\,a>b & & & \,a\,\mathrm{is}\,\mathrm{greater than}\,b \\ \,\mathrm{∘}\,a\ge{}b & & & \,a\,\mathrm{is}\,\mathrm{greater than or equal to}\,b \\ \end{array}$

• Grouping Symbols Parentheses $(\,)$ Brackets $[\,]$ Braces $\mathrm{\{}\,\mathrm{\}}$

• Exponential Notation ${a}^{n}$ means multiply $a$ by itself, $n$ times. The expression ${a}^{n}$ is read $a$ to the ${n}^{th}$ power.

• Order of Operations: When simplifying mathematical expressions perform the operations in the following order: Step 1. Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first. Step 2. Exponents: Simplify all expressions with exponents. Step 3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority. Step 4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

• Combine Like Terms Step 1. Identify like terms. Step 2. Rearrange the expression so like terms are together. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

• Addition of Positive and Negative Integers $\begin{array}{lccl} 5+3 & & & \,-5+(-3) \\ \,8 & & & \,-8 \\ \mathrm{both positive,} & & & \,\mathrm{both negative,} \\ \mathrm{sum positive} & & & \,\mathrm{sum negative} \\ -5+3 & & & \,5+(-3) \\ \,-2 & & & \,2 \\ \mathrm{different signs,} & & & \,\mathrm{different signs,} \\ \mathrm{more negatives} & & & \,\mathrm{more positives} \\ \mathrm{sum negative} & & & \,\mathrm{sum positive} \\ \end{array}$

• Property of Absolute Value: $|n|\ge{}0$ for all numbers. Absolute values are always greater than or equal to zero!

• Subtraction of Integers $\begin{array}{lccl} 5-3 & & & -5-(-3) \\ \,2 & & & \,-2 \\ 5\,\mathrm{positives} & & & 5\,\mathrm{negatives} \\ \mathrm{take away}\,3\,\mathrm{positives} & & & \mathrm{take away}\,3\,\mathrm{negatives} \\ \mathrm{2 positives} & & & \mathrm{2 negatives} \\ -5-3 & & & 5-(-3) \\ \,-8 & & & \,8 \\ 5\,\mathrm{negatives, want to} & & & 5\,\mathrm{positives, want to} \\ \mathrm{subtract}\,3\,\mathrm{positives} & & & \mathrm{subtract}\,3\,\mathrm{negatives} \\ \mathrm{need neutral pairs} & & & \mathrm{need neutral pairs} \\ \end{array}$

• Subtraction Property: Subtracting a number is the same as adding its opposite.

• Multiplication and Division of Two Signed Numbers Same signs—Product is positive Different signs—Product is negative

• Strategy for Applications Step 1. Identify what you are asked to find. Step 2. Write a phrase that gives the information to find it. Step 3. Translate the phrase to an expression. Step 4. Simplify the expression. Step 5. Answer the question with a complete sentence.

• Equivalent Fractions Property: If $a,b,c$ are numbers where $b\ne{}0,c\ne{}0,$ then $\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}$ and $\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}.$

• Fraction Division: If $a,b,c\,\mathrm{and}\,d$ are numbers where $b\ne{}0,c\ne{}0,\,\mathrm{and}\,d\ne{}0,$ then $\frac{a}{b}\mathrm{÷}\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}.$ To divide fractions, multiply the first fraction by the reciprocal of the second.

• Fraction Multiplication: If $a,b,c\,\mathrm{and}\,d$ are numbers where $b\ne{}0,\,\mathrm{and}\,d\ne{}0,$ then $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}.$ To multiply fractions, multiply the numerators and multiply the denominators.

• Placement of Negative Sign in a Fraction: For any positive numbers $a\,\mathrm{and}\,b,$$\frac{-a}{b}=\frac{a}{-b}=-\,\frac{a}{b}.$

• Property of One: $\frac{a}{a}=1;$ Any number, except zero, divided by itself is one.

• Simplify a Fraction Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first. Step 2. Simplify using the equivalent fractions property by dividing out common factors. Step 3. Multiply any remaining factors.

• Simplify an Expression with a Fraction Bar Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator. Step 2. Simplify the fraction.

• Fraction Addition and Subtraction: If $a,b,\,\mathrm{and}\,c$ are numbers where $c\ne{}0,$ then $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$ and $\frac{a}{c}-\,\frac{b}{c}=\frac{a-b}{c}.$ To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

• Strategy for Adding or Subtracting Fractions Step 1. Do they have a common denominator? Yes—go to step 2. No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator. Step 2. Add or subtract the fractions. Step 3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.

• Simplify Complex Fractions Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Divide the numerator by the denominator. Simplify if possible.

• Name a Decimal Step 1. Name the number to the left of the decimal point. Step 2. Write ”and” for the decimal point. Step 3. Name the “number” part to the right of the decimal point as if it were a whole number. Step 4. Name the decimal place of the last digit.

• Write a Decimal Step 1. Look for the word ‘and’—it locates the decimal point. Place a decimal point under the word ‘and.’ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right. Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word. Step 3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place. Step 4. Fill in zeros for place holders as needed.

• Round a Decimal Step 1. Locate the given place value and mark it with an arrow. Step 2. Underline the digit to the right of the place value. Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

• Add or Subtract Decimals Step 1. Write the numbers so the decimal points line up vertically. Step 2. Use zeros as place holders, as needed. Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

• Multiply Decimals Step 1. Determine the sign of the product. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors. Step 4. Write the product with the appropriate sign.

• Multiply a Decimal by a Power of Ten Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10. Step 2. Add zeros at the end of the number as needed.

• Divide Decimals Step 1. Determine the sign of the quotient. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places - adding zeros as needed. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Step 4. Write the quotient with the appropriate sign.

• Convert a Decimal to a Proper Fraction Step 1. Determine the place value of the final digit. Step 2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.

• Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

• Square Root Notation $\sqrt{m}$ is read ‘the square root of m.’ If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge{}0.$

• Order Decimals Step 1. Write the numbers one under the other, lining up the decimal points. Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match. Step 3. Compare the numbers as if they were whole numbers. Step 4. Order the numbers using the appropriate inequality sign.

• Commutative Property of Addition: If $a,b$ are real numbers, then $a+b=b+a.$ Multiplication: If $a,b$ are real numbers, then $a\cdot{}b=b\cdot{}a.$ When adding or multiplying, changing the order gives the same result.

• Associative Property of Addition: If $a,b,c$ are real numbers, then $(a+b)+c=a+(b+c).$ Multiplication: If $a,b,c$ are real numbers, then $(a\cdot{}b)\cdot{}c=a\cdot(b\cdot{}c).$ When adding or multiplying, changing the grouping gives the same result.

• Distributive Property: If $a,b,c$ are real numbers, then $a(b+c)=ab+ac$$(b+c)a=ba+ca$$a(b-c)=ab-ac$$(b-c)a=ba-ca$

• Identity Property of Addition: For any real number $a:\,a+0=a\,0+a=a$ 0 is the additive identity of Multiplication: For any real number $a:\,a\cdot{}1=a\,1\cdot{}a=a$$1$ is the multiplicative identity

• Inverse Property of Addition: For any real number $a,a+(-a)=0.$ A number and its opposite add to zero. $-a$ is the additive inverse of $a.$ of Multiplication: For any real number $a,(a\ne{}0)\,a\cdot\frac{1}{a}=1.$ A number and its reciprocal multiply to one. $\frac{1}{a}$ is the multiplicative inverse of $a.$

• Properties of Zero For any real number $a,$$a\cdot{}0=0\,0\cdot{}a=0$ – The product of any real number and 0 is 0. $\frac{0}{a}=0$ for $a\ne{}0$ – Zero divided by any real number except zero is zero. $\frac{a}{0}$ is undefined – Division by zero is undefined.

• Metric System of Measurement Length $\begin{array}{lll} \mathrm{1 kilometer (km)} & = & \mathrm{1,000 m} \\ \mathrm{1 hectometer (hm)} & = & \mathrm{100 m} \\ \mathrm{1 dekameter (dam)} & = & \mathrm{10 m} \\ \mathrm{1 meter (m)} & = & \mathrm{1 m} \\ \mathrm{1 decimeter (dm)} & = & \mathrm{0.1 m} \\ \mathrm{1 centimeter (cm)} & = & \mathrm{0.01 m} \\ \mathrm{1 millimeter (mm)} & = & \mathrm{0.001 m} \\ \mathrm{1 meter} & = & \mathrm{100 centimeters} \\ \mathrm{1 meter} & = & \mathrm{1,000 millimeters} \\ \end{array}$ Mass $\begin{array}{lll} \mathrm{1 kilogram (kg)} & = & \mathrm{1,000 g} \\ \mathrm{1 hectogram (hg)} & = & \mathrm{100 g} \\ \mathrm{1 dekagram (dag)} & = & \mathrm{10 g} \\ \mathrm{1 gram (g)} & = & \mathrm{1 g} \\ \mathrm{1 decigram (dg)} & = & \mathrm{0.1 g} \\ \mathrm{1 centigram (cg)} & = & \mathrm{0.01 g} \\ \mathrm{1 milligram (mg)} & = & \mathrm{0.001 g} \\ \mathrm{1 gram} & = & \mathrm{100 centigrams} \\ \mathrm{1 gram} & = & \mathrm{1,000 milligrams} \\ \end{array}$ Capacity $\begin{array}{lll} \mathrm{1 kiloliter (kL)} & = & \mathrm{1,000 L} \\ \mathrm{1 hectoliter (hL)} & = & \mathrm{100 L} \\ \mathrm{1 dekaliter (daL)} & = & \mathrm{10 L} \\ \mathrm{1 liter (L)} & = & \mathrm{1 L} \\ \mathrm{1 deciliter (dL)} & = & \mathrm{0.1 L} \\ \mathrm{1 centiliter (cL)} & = & \mathrm{0.01 L} \\ \mathrm{1 milliliter (mL)} & = & \mathrm{0.001 L} \\ \mathrm{1 liter} & = & \mathrm{100 centiliters} \\ \mathrm{1 liter} & = & \mathrm{1,000 milliliters} \\ \end{array}$

• Temperature Conversion To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula $C=\frac{5}{9}(F-32)$ To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula $F=\frac{9}{5}C+32$

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Review Exercises

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 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.

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.

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 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.

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.

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 >.

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.

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.

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In the following exercises, evaluate.

In the following exercises, simplify each expression.

Key Terms

absolute value

additive identity

The additive identity is the number 0; adding 0 to any number does not change its value.

additive inverse

The opposite of a number is its additive inverse. A number and its additive inverse add to 0.

coefficient

The coefficient of a term is the constant that multiplies the variable in a term.

complex fraction

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

composite number

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

constant

A constant is a number whose value always stays the same.

counting numbers

The counting numbers are the numbers 1, 2, 3, …

decimal

A decimal is another way of writing a fraction whose denominator is a power of ten.

denominator

The denominator is the value on the bottom part of the fraction that indicates the number of equal parts into which the whole has been divided.

divisible by a number

equality symbol

equation

An equation is two expressions connected by an equal sign.

equivalent decimals

Two decimals are equivalent if they convert to equivalent fractions.

equivalent fractions

Equivalent fractions are fractions that have the same value.

evaluate an expression

To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

expression

An expression is a number, a variable, or a combination of numbers and variables using operation symbols.

factors

fraction

integers

The whole numbers and their opposites are called the integers: ...−3, −2, −1, 0, 1, 2, 3...

irrational number

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.

least common denominator

The least common denominator (LCD) of two fractions is the Least common multiple (LCM) of their denominators.

least common multiple

The least common multiple of two numbers is the smallest number that is a multiple of both numbers.

like terms

Terms that are either constants or have the same variables raised to the same powers are called like terms.

multiple of a number

n

n

multiplicative identity

The multiplicative identity is the number 1; multiplying 1 by any number does not change the value of the number.

multiplicative inverse

The reciprocal of a number is its multiplicative inverse. A number and its multiplicative inverse multiply to one.

number line

A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left.

numerator

The numerator is the value on the top part of the fraction that indicates how many parts of the whole are included.

opposite

origin

The origin is the point labeled 0 on a number line.

percent

A percent is a ratio whose denominator is 100.

prime factorization

The prime factorization of a number is the product of prime numbers that equals the number.

prime number

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

radical sign

rational number

p

q

real number

A real number is a number that is either rational or irrational.

reciprocal

repeating decimal

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.

simplified fraction

A fraction is considered simplified if there are no common factors in its numerator and denominator.

simplify an expression

To simplify an expression, do all operations in the expression.

square and square root

term

A term is a constant or the product of a constant and one or more variables.

variable

A variable is a letter that represents a number whose value may change.

whole numbers

The whole numbers are the numbers 0, 1, 2, 3, ....

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Key Concepts

• Place Value as in Figure 1.3.

• Name a Whole Number in Words Step 1. Start at the left and name the number in each period, followed by the period name. Step 2. Put commas in the number to separate the periods. Step 3. Do not name the ones period.

• Write a Whole Number Using Digits Step 1. Identify the words that indicate periods. (Remember the ones period is never named.) Step 2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas. Step 3. Name the number in each period and place the digits in the correct place value position.

• Round Whole Numbers Step 1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change. Step 2. Underline the digit to the right of the given place value. Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value. Step 4. Replace all digits to the right of the given place value with zeros.

• Divisibility Tests: A number is divisible by: 2 if the last digit is 0, 2, 4, 6, or 8. 3 if the sum of the digits is divisible by 3. 5 if the last digit is 5 or 0. 6 if it is divisible by both 2 and 3. 10 if it ends with 0.

• Find the Prime Factorization of a Composite Number Step 1. Find two factors whose product is the given number, and use these numbers to create two branches. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree. Step 3. If a factor is not prime, write it as the product of two factors and continue the process. Step 4. Write the composite number as the product of all the circled primes.

• Find the Least Common Multiple by Listing Multiples Step 1. List several multiples of each number. Step 2. Look for the smallest number that appears on both lists. Step 3. This number is the LCM.

• Find the Least Common Multiple Using the Prime Factors Method Step 1. Write each number as a product of primes. Step 2. List the primes of each number. Match primes vertically when possible. Step 3. Bring down the columns. Step 4. Multiply the factors.

• Notation The result is… $\begin{array}{lccl} \,\mathrm{∘}\,a+b & & & \,\mathrm{the sum of}\,a\,\mathrm{and}\,b \\ \,\mathrm{∘}\,a-b & & & \,\mathrm{the difference of}\,a\,\mathrm{and}\,b \\ \,\mathrm{∘}\,a\cdot{}b,ab,(a)(b)\,(a)b,a(b) & & & \,\mathrm{the product of}\,a\,\mathrm{and}\,b \\ \,\mathrm{∘}\,a\mathrm{÷}b,a/b,\frac{a}{b},ba & & & \,\mathrm{the quotient of}\,a\,\mathrm{and}\,b \\ \end{array}$

• Inequality $\begin{array}{lccl} \,\mathrm{∘}\,a<b\,\mathrm{is read}\,“a\,\mathrm{is less than}\,b” & & & a\,\mathrm{is to the left of}\,b\,\mathrm{on the number line} \\ \,\mathrm{∘}\,a>b\,\mathrm{is read}\,“a\,\mathrm{is greater than}\,b” & & & a\,\mathrm{is to the right of}\,b\,\mathrm{on the number line} \\ \end{array}$

• Inequality Symbols Words $\begin{array}{lccl} \,\mathrm{∘}\,a\ne{}b & & & \,a\,\mathrm{is}\,\mathrm{not equal to}\,b \\ \,\mathrm{∘}\,a<b & & & \,a\,\mathrm{is}\,\mathrm{less than}\,b \\ \,\mathrm{∘}\,a\le{}b & & & \,a\,\mathrm{is}\,\mathrm{less than or equal to}\,b \\ \,\mathrm{∘}\,a>b & & & \,a\,\mathrm{is}\,\mathrm{greater than}\,b \\ \,\mathrm{∘}\,a\ge{}b & & & \,a\,\mathrm{is}\,\mathrm{greater than or equal to}\,b \\ \end{array}$

• Grouping Symbols Parentheses $(\,)$ Brackets $[\,]$ Braces $\mathrm{\{}\,\mathrm{\}}$

• Exponential Notation ${a}^{n}$ means multiply $a$ by itself, $n$ times. The expression ${a}^{n}$ is read $a$ to the ${n}^{th}$ power.

• Order of Operations: When simplifying mathematical expressions perform the operations in the following order: Step 1. Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first. Step 2. Exponents: Simplify all expressions with exponents. Step 3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority. Step 4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

• Combine Like Terms Step 1. Identify like terms. Step 2. Rearrange the expression so like terms are together. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.

• Addition of Positive and Negative Integers $\begin{array}{lccl} 5+3 & & & \,-5+(-3) \\ \,8 & & & \,-8 \\ \mathrm{both positive,} & & & \,\mathrm{both negative,} \\ \mathrm{sum positive} & & & \,\mathrm{sum negative} \\ -5+3 & & & \,5+(-3) \\ \,-2 & & & \,2 \\ \mathrm{different signs,} & & & \,\mathrm{different signs,} \\ \mathrm{more negatives} & & & \,\mathrm{more positives} \\ \mathrm{sum negative} & & & \,\mathrm{sum positive} \\ \end{array}$

• Property of Absolute Value: $|n|\ge{}0$ for all numbers. Absolute values are always greater than or equal to zero!

• Subtraction of Integers $\begin{array}{lccl} 5-3 & & & -5-(-3) \\ \,2 & & & \,-2 \\ 5\,\mathrm{positives} & & & 5\,\mathrm{negatives} \\ \mathrm{take away}\,3\,\mathrm{positives} & & & \mathrm{take away}\,3\,\mathrm{negatives} \\ \mathrm{2 positives} & & & \mathrm{2 negatives} \\ -5-3 & & & 5-(-3) \\ \,-8 & & & \,8 \\ 5\,\mathrm{negatives, want to} & & & 5\,\mathrm{positives, want to} \\ \mathrm{subtract}\,3\,\mathrm{positives} & & & \mathrm{subtract}\,3\,\mathrm{negatives} \\ \mathrm{need neutral pairs} & & & \mathrm{need neutral pairs} \\ \end{array}$

• Subtraction Property: Subtracting a number is the same as adding its opposite.

• Multiplication and Division of Two Signed Numbers Same signs—Product is positive Different signs—Product is negative

• Strategy for Applications Step 1. Identify what you are asked to find. Step 2. Write a phrase that gives the information to find it. Step 3. Translate the phrase to an expression. Step 4. Simplify the expression. Step 5. Answer the question with a complete sentence.

• Equivalent Fractions Property: If $a,b,c$ are numbers where $b\ne{}0,c\ne{}0,$ then $\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}$ and $\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}.$

• Fraction Division: If $a,b,c\,\mathrm{and}\,d$ are numbers where $b\ne{}0,c\ne{}0,\,\mathrm{and}\,d\ne{}0,$ then $\frac{a}{b}\mathrm{÷}\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}.$ To divide fractions, multiply the first fraction by the reciprocal of the second.

• Fraction Multiplication: If $a,b,c\,\mathrm{and}\,d$ are numbers where $b\ne{}0,\,\mathrm{and}\,d\ne{}0,$ then $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}.$ To multiply fractions, multiply the numerators and multiply the denominators.

• Placement of Negative Sign in a Fraction: For any positive numbers $a\,\mathrm{and}\,b,$$\frac{-a}{b}=\frac{a}{-b}=-\,\frac{a}{b}.$

• Property of One: $\frac{a}{a}=1;$ Any number, except zero, divided by itself is one.

• Simplify a Fraction Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first. Step 2. Simplify using the equivalent fractions property by dividing out common factors. Step 3. Multiply any remaining factors.

• Simplify an Expression with a Fraction Bar Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator. Step 2. Simplify the fraction.

• Fraction Addition and Subtraction: If $a,b,\,\mathrm{and}\,c$ are numbers where $c\ne{}0,$ then $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$ and $\frac{a}{c}-\,\frac{b}{c}=\frac{a-b}{c}.$ To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.

• Strategy for Adding or Subtracting Fractions Step 1. Do they have a common denominator? Yes—go to step 2. No—Rewrite each fraction with the LCD (Least Common Denominator). Find the LCD. Change each fraction into an equivalent fraction with the LCD as its denominator. Step 2. Add or subtract the fractions. Step 3. Simplify, if possible. To multiply or divide fractions, an LCD IS NOT needed. To add or subtract fractions, an LCD IS needed.

• Simplify Complex Fractions Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Divide the numerator by the denominator. Simplify if possible.

• Name a Decimal Step 1. Name the number to the left of the decimal point. Step 2. Write ”and” for the decimal point. Step 3. Name the “number” part to the right of the decimal point as if it were a whole number. Step 4. Name the decimal place of the last digit.

• Write a Decimal Step 1. Look for the word ‘and’—it locates the decimal point. Place a decimal point under the word ‘and.’ Translate the words before ‘and’ into the whole number and place it to the left of the decimal point. If there is no “and,” write a “0” with a decimal point to its right. Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word. Step 3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place. Step 4. Fill in zeros for place holders as needed.

• Round a Decimal Step 1. Locate the given place value and mark it with an arrow. Step 2. Underline the digit to the right of the place value. Step 3. Is this digit greater than or equal to 5? Yes—add 1 to the digit in the given place value. No—do not change the digit in the given place value. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

• Add or Subtract Decimals Step 1. Write the numbers so the decimal points line up vertically. Step 2. Use zeros as place holders, as needed. Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.

• Multiply Decimals Step 1. Determine the sign of the product. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors. Step 4. Write the product with the appropriate sign.

• Multiply a Decimal by a Power of Ten Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10. Step 2. Add zeros at the end of the number as needed.

• Divide Decimals Step 1. Determine the sign of the quotient. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places - adding zeros as needed. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Step 4. Write the quotient with the appropriate sign.

• Convert a Decimal to a Proper Fraction Step 1. Determine the place value of the final digit. Step 2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.

• Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.

• Square Root Notation $\sqrt{m}$ is read ‘the square root of m.’ If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge{}0.$

• Order Decimals Step 1. Write the numbers one under the other, lining up the decimal points. Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match. Step 3. Compare the numbers as if they were whole numbers. Step 4. Order the numbers using the appropriate inequality sign.

• Commutative Property of Addition: If $a,b$ are real numbers, then $a+b=b+a.$ Multiplication: If $a,b$ are real numbers, then $a\cdot{}b=b\cdot{}a.$ When adding or multiplying, changing the order gives the same result.

• Associative Property of Addition: If $a,b,c$ are real numbers, then $(a+b)+c=a+(b+c).$ Multiplication: If $a,b,c$ are real numbers, then $(a\cdot{}b)\cdot{}c=a\cdot(b\cdot{}c).$ When adding or multiplying, changing the grouping gives the same result.

• Distributive Property: If $a,b,c$ are real numbers, then $a(b+c)=ab+ac$$(b+c)a=ba+ca$$a(b-c)=ab-ac$$(b-c)a=ba-ca$

• Identity Property of Addition: For any real number $a:\,a+0=a\,0+a=a$ 0 is the additive identity of Multiplication: For any real number $a:\,a\cdot{}1=a\,1\cdot{}a=a$$1$ is the multiplicative identity

• Inverse Property of Addition: For any real number $a,a+(-a)=0.$ A number and its opposite add to zero. $-a$ is the additive inverse of $a.$ of Multiplication: For any real number $a,(a\ne{}0)\,a\cdot\frac{1}{a}=1.$ A number and its reciprocal multiply to one. $\frac{1}{a}$ is the multiplicative inverse of $a.$

• Properties of Zero For any real number $a,$$a\cdot{}0=0\,0\cdot{}a=0$ – The product of any real number and 0 is 0. $\frac{0}{a}=0$ for $a\ne{}0$ – Zero divided by any real number except zero is zero. $\frac{a}{0}$ is undefined – Division by zero is undefined.

• Metric System of Measurement Length $\begin{array}{lll} \mathrm{1 kilometer (km)} & = & \mathrm{1,000 m} \\ \mathrm{1 hectometer (hm)} & = & \mathrm{100 m} \\ \mathrm{1 dekameter (dam)} & = & \mathrm{10 m} \\ \mathrm{1 meter (m)} & = & \mathrm{1 m} \\ \mathrm{1 decimeter (dm)} & = & \mathrm{0.1 m} \\ \mathrm{1 centimeter (cm)} & = & \mathrm{0.01 m} \\ \mathrm{1 millimeter (mm)} & = & \mathrm{0.001 m} \\ \mathrm{1 meter} & = & \mathrm{100 centimeters} \\ \mathrm{1 meter} & = & \mathrm{1,000 millimeters} \\ \end{array}$ Mass $\begin{array}{lll} \mathrm{1 kilogram (kg)} & = & \mathrm{1,000 g} \\ \mathrm{1 hectogram (hg)} & = & \mathrm{100 g} \\ \mathrm{1 dekagram (dag)} & = & \mathrm{10 g} \\ \mathrm{1 gram (g)} & = & \mathrm{1 g} \\ \mathrm{1 decigram (dg)} & = & \mathrm{0.1 g} \\ \mathrm{1 centigram (cg)} & = & \mathrm{0.01 g} \\ \mathrm{1 milligram (mg)} & = & \mathrm{0.001 g} \\ \mathrm{1 gram} & = & \mathrm{100 centigrams} \\ \mathrm{1 gram} & = & \mathrm{1,000 milligrams} \\ \end{array}$ Capacity $\begin{array}{lll} \mathrm{1 kiloliter (kL)} & = & \mathrm{1,000 L} \\ \mathrm{1 hectoliter (hL)} & = & \mathrm{100 L} \\ \mathrm{1 dekaliter (daL)} & = & \mathrm{10 L} \\ \mathrm{1 liter (L)} & = & \mathrm{1 L} \\ \mathrm{1 deciliter (dL)} & = & \mathrm{0.1 L} \\ \mathrm{1 centiliter (cL)} & = & \mathrm{0.01 L} \\ \mathrm{1 milliliter (mL)} & = & \mathrm{0.001 L} \\ \mathrm{1 liter} & = & \mathrm{100 centiliters} \\ \mathrm{1 liter} & = & \mathrm{1,000 milliliters} \\ \end{array}$

• Temperature Conversion To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula $C=\frac{5}{9}(F-32)$ To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula $F=\frac{9}{5}C+32$

##

Review Exercises

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 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.

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.

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 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.

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.

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 >.

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.

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.

##

In the following exercises, evaluate.

In the following exercises, simplify each expression.

1. 26,915

ⓐ 1 ⓑ 2 ⓒ 9 ⓓ 5 ⓔ 6

2. 58,129,304

ⓐ 5 ⓑ 0 ⓒ 1 ⓓ 8 ⓔ 2

3. 6,104

4. 3,975,284

5. three hundred fifteen

6. ninety million, four hundred twenty-five thousand, sixteen

7. Round to the nearest ten.

ⓐ 407 ⓑ 8,564

8. 864,951

9. 6,15

10. $25-7$

1. $45\mathrm{÷}5$

2. $42\ge{}27$

3. $3\le{}20\mathrm{÷}4$

4. $6\cdot{}3+5$

5. $3n,{n}^{2},12,12{p}^{2},3,3{n}^{2}$

6. $11{x}^{2}+3x+6$

7. $17a+9a$

8. $9x+3x+8$

9. $7p+6+5p-4$

10. the sum of 8 and 12

1. the difference of $x$ and 4

2. the product of 6 and $y$

3. Adele bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let$b$ represent the cost of the blouse. Write an expression for the cost of the skirt.

4. ⓐ $6\mathrm{\_\_\_}2$ⓑ $-7\mathrm{\_\_\_}4$ⓒ $-9\mathrm{\_\_\_}-1$ⓓ $9\mathrm{\_\_\_}-3$

5. ⓐ $-8$ ⓑ 1

6. $-m$ whenⓐ $m=3$ⓑ $m=-3$

7. ⓐ $|7|$ ⓑ $|-25|$ ⓒ $|0|$

8. ⓐ $-8\mathrm{\_\_\_}|-8|$ⓑ $-|-2|\mathrm{\_\_\_}-2$

9. ⓐ $|x|$ when $x=-28$ ⓑ $|x|$ when $x=-15$

10. $-200+65$

1. ⓐ $15-6$ ⓑ $15+(-6)$

2. ⓐ $8-(-9)$ ⓑ $8+9$

3. $31-79$

4. $-31-11$

5. $-16-(-4+1)-7$

6. $-5(7)$

7. $-18(-2)$

8. $-28\mathrm{÷}7$

9. ${(-5)}^{3}$

10. $-4\cdot{}2\cdot{}11$

1. $-10(-4)\mathrm{÷}(-8)$

2. $x+8$ when ⓐ $x=-26$ⓑ $x=-95$

3. When $b=-11$, evaluate:ⓐ $b+6$ ⓑ $-b+6$

4. the sum of $-4$ and $-17$, increased by 32

5. the quotient of $-45$ and $-9$

6. Temperature The high temperature one day in Miami Beach, Florida, was $76^{\circ}$. That same day, the high temperature in Buffalo, New York was $-8^{\circ}$. What was the difference between the temperature in Miami Beach and the temperature in Buffalo?

7. $\frac{1}{4}$

8. $\frac{5}{6}$

9. $\frac{7}{21}$

10. $\frac{15}{20}$

1. $-\,\frac{168}{192}$

2. $\frac{11x}{11y}$

3. $\frac{2}{5}\cdot\frac{1}{3}$

4. $\frac{7}{12}(-\,\frac{8}{21})$

5. $-28p(-\,\frac{1}{4})$

6. $\frac{14}{5}(-15)$

7. $-\,\frac{4}{5}\mathrm{÷}\frac{4}{7}$

8. $\frac{5}{8}\mathrm{÷}\frac{a}{10}$

9. $\frac{7p}{12}\mathrm{÷}\frac{21p}{8}$

10. $\frac{2}{5}\mathrm{÷}(-10)$

1. $\frac{\frac{2}{3}}{\frac{8}{9}}$

2. $\frac{-\,\frac{9}{10}}{3}$

3. $\frac{\frac{r}{5}}{\frac{s}{3}}$

4. $\frac{4+11}{8}$

5. $\frac{30}{7-12}$

6. $\frac{22-14}{19-13}$

7. $\frac{5\cdot{}8}{-10}$

8. $\frac{15\cdot{}5-{5}^{2}}{2\cdot{}10}$

9. $\frac{2+4(3)}{-3-{2}^{2}}$

10. the quotient of c and the sum of d and 9.

1. $\frac{4}{9}+\frac{1}{9}$

2. $\frac{y}{3}+\frac{2}{3}$

3. $-\,\frac{1}{8}+(-\,\frac{3}{8})$

4. $\frac{4}{5}-\,\frac{1}{5}$

5. $\frac{y}{17}-\,\frac{9}{17}$

6. $-\,\frac{8}{d}-\,\frac{3}{d}$

7. $\frac{1}{3}+\frac{1}{5}$

8. $\frac{1}{5}-(-\,\frac{1}{10})$

9. $\frac{2}{3}+\frac{3}{4}$

10. $\frac{11}{12}-\,\frac{3}{8}$

1. $-\,\frac{9}{16}-(-\,\frac{4}{5})$

2. $1+\frac{5}{6}$

3. $\frac{{(\frac{1}{5})}^{2}}{2+{3}^{2}}$

4. $x+\frac{1}{2}$ whenⓐ $x=-\,\frac{1}{8}$ⓑ $x=-\,\frac{1}{2}$

5. $4{p}^{2}q$ when $p=-\,\frac{1}{2}$ and $q=\frac{5}{9}$