Chapter 4 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.

##

Worked example

complex fraction

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

equivalent fractions

Equivalent fractions are two or more fractions that have the same value.

fraction

least common denominator (LCD)

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

mixed number

proper and improper fractions

proper

improper

reciprocal

simplified fraction

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

##

Key Concepts

• Property of OneAny number, except zero, divided by itself is one.$\frac{a}{a}=1$, where $a\ne{}0$.

• Mixed NumbersA mixed number consists of a whole number $a$ and a fraction $\frac{b}{c}$ where $c\ne{}0$. It is written as follows: $a\frac{b}{c}\,c\ne{}0$

• Proper and Improper FractionsThe fraction $\frac{a}{b}$ is a proper fraction if $a<b$ and an improper fraction if $a\ge{}b$.

• Convert an improper fraction to a mixed number.Step 1. Divide the denominator into the numerator. Step 2. Identify the quotient, remainder, and divisor. Step 3. Write the mixed number as quotient$\frac{\mathrm{remainder}}{\mathrm{divisor}}$.

• Convert a mixed number to an improper fraction.Step 1. Multiply the whole number by the denominator.Step 2. Add the numerator to the product found in Step 1.Step 3. Write the final sum over the original denominator.

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

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

• 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. Step 2. Simplify, using the equivalent fractions property, by removing common factors. Step 3. Multiply any remaining factors.

• Fraction MultiplicationIf $a,b,c,$ and $d$ are numbers where $b\ne{}0$and $d\ne{}0$, then $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$.

• ReciprocalA number and its reciprocal have a product of $1$. $\frac{a}{b}\cdot\frac{b}{a}=1$ Opposite Absolute Value Reciprocal has opposite sign is never negative has same sign, fraction invertsTable 4.4

• Fraction DivisionIf $a,b,c,$ and $d$ are numbers where $b\ne{}0$, $c\ne{}0$ and $d\ne{}0$ , then

4.1 To divide fractions, multiply the first fraction by the reciprocal of the second.

• Multiply or divide mixed numbers.Step 1. Convert the mixed numbers to improper fractions. Step 2. Follow the rules for fraction multiplication or division. Step 3. Simplify if possible.

• Simplify a complex fraction.Step 1. Rewrite the complex fraction as a division problem. Step 2. Follow the rules for dividing fractions. Step 3. Simplify if possible.

• Placement of negative sign in a fraction.For any positive numbers $a$ and $b$, $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$.

• Simplify an expression with a fraction bar.Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Simplify the fraction.

• Fraction AdditionIf $a,b,$ and $c$ are numbers where $c\ne{}0$, then $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$.To add fractions, add the numerators and place the sum over the common denominator.

• Fraction SubtractionIf $a,b,$ and $c$ are numbers where $c\ne{}0$, then $\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}$.To subtract fractions, subtract the numerators and place the difference over the common denominator.

• Find the least common denominator (LCD) of two fractions.Step 1. Factor each denominator into its primes. Step 2. List the primes, matching primes in columns when possible. Step 3. Bring down the columns. Step 4. Multiply the factors. The product is the LCM of the denominators. Step 5. The LCM of the denominators is the LCD of the fractions.

• Equivalent Fractions PropertyIf $a,b$, and $c$ are whole numbers where $b\ne{}0$, $c\ne{}0$ then $\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}$ and $\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}$

• Convert two fractions to equivalent fractions with their LCD as the common denominator.Step 1. Find the LCD. Step 2. For each fraction, determine the number needed to multiply the denominator to get the LCD. Step 3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2. Step 4. Simplify the numerator and denominator.

• Add or subtract fractions with different denominators.Step 1. Find the LCD. Step 2. Convert each fraction to an equivalent form with the LCD as the denominator. Step 3. Add or subtract the fractions. Step 4. Write the result in simplified form.

• Summary of Fraction OperationsFraction multiplication: Multiply the numerators and multiply the denominators.

4.2 Fraction division: Multiply the first fraction by the reciprocal of the second.

4.3 Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

4.4 Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

4.5

• Simplify complex fractions.Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Divide the numerator by the denominator. Step 4. Simplify if possible.

• Add mixed numbers with a common denominator.Step 1. Add the whole numbers. Step 2. Add the fractions. Step 3. Simplify, if possible.

• Subtract mixed numbers with common denominators.Step 1. Rewrite the problem in vertical form. Step 2. Compare the two fractions. If the top fraction is larger than the bottom fraction, go to Step 3. If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction. Step 3. Subtract the fractions. Step 4. Subtract the whole numbers. Step 5. Simplify, if possible.

• Subtract mixed numbers with common denominators as improper fractions.Step 1. Rewrite the mixed numbers as improper fractions. Step 2. Subtract the numerators. Step 3. Write the answer as a mixed number, simplifying the fraction part, if possible.

• Determine whether a number is a solution to an equation.Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Addition, Subtraction, and Division Properties of EqualityFor any numbers a, b, and c, if $a=b$, then $a+c=b+c$. Addition Property of Equality if $a=b$, then $a-c=b-c$. Subtraction Property of Equality if $a=b$, then $\frac{a}{c}=\frac{b}{c}$, $c\ne{}0$. Division Property of Equality

• The Multiplication Property of EqualityFor any numbers $ab$ and $c,a=b$, then $ac=bc$. If you multiply both sides of an equation by the same quantity, you still have equality.

##

Review Exercises

In the following exercises, name the fraction of each figure that is shaded.

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

In the following exercises, convert the improper fraction to a mixed number.

In the following exercises, convert the mixed number to an improper fraction.

In the following exercises, locate the numbers on a number line.

In the following exercises, order each pair of numbers, using $<$ or $\mathrm{>.}$

In the following exercises, simplify.

In the following exercises, multiply.

In the following exercises, find the reciprocal.

In the following exercises, divide.

In the following exercises, perform the indicated operation.

In the following exercises, translate the English phrase into an algebraic expression.

In the following exercises, simplify the complex fraction

In the following exercises, simplify.

In the following exercises, add.

In the following exercises, subtract.

In the following exercises, find the least common denominator.

In the following exercises, change to equivalent fractions using the given LCD.

In the following exercises, perform the indicated operations and simplify.

In the following exercises, evaluate.

In the following exercises, perform the indicated operation.

In the following exercises, determine whether the each number is a solution of the given equation.

In the following exercises, solve the equation.

In the following exercises, translate and solve.

##

Convert the improper fraction to a mixed number.

Convert the mixed number to an improper fraction.

Locate the numbers on a number line.

In the following exercises, simplify.

Evaluate.

In the following exercises, solve the equation.

Key Terms

complex fraction

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

equivalent fractions

Equivalent fractions are two or more fractions that have the same value.

fraction

least common denominator (LCD)

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

mixed number

proper and improper fractions

proper

improper

reciprocal

simplified fraction

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

##

Key Concepts

• Property of OneAny number, except zero, divided by itself is one.$\frac{a}{a}=1$, where $a\ne{}0$.

• Mixed NumbersA mixed number consists of a whole number $a$ and a fraction $\frac{b}{c}$ where $c\ne{}0$. It is written as follows: $a\frac{b}{c}\,c\ne{}0$

• Proper and Improper FractionsThe fraction $\frac{a}{b}$ is a proper fraction if $a<b$ and an improper fraction if $a\ge{}b$.

• Convert an improper fraction to a mixed number.Step 1. Divide the denominator into the numerator. Step 2. Identify the quotient, remainder, and divisor. Step 3. Write the mixed number as quotient$\frac{\mathrm{remainder}}{\mathrm{divisor}}$.

• Convert a mixed number to an improper fraction.Step 1. Multiply the whole number by the denominator.Step 2. Add the numerator to the product found in Step 1.Step 3. Write the final sum over the original denominator.

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

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

• 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. Step 2. Simplify, using the equivalent fractions property, by removing common factors. Step 3. Multiply any remaining factors.

• Fraction MultiplicationIf $a,b,c,$ and $d$ are numbers where $b\ne{}0$and $d\ne{}0$, then $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$.

• ReciprocalA number and its reciprocal have a product of $1$. $\frac{a}{b}\cdot\frac{b}{a}=1$ Opposite Absolute Value Reciprocal has opposite sign is never negative has same sign, fraction invertsTable 4.4

• Fraction DivisionIf $a,b,c,$ and $d$ are numbers where $b\ne{}0$, $c\ne{}0$ and $d\ne{}0$ , then

4.1 To divide fractions, multiply the first fraction by the reciprocal of the second.

• Multiply or divide mixed numbers.Step 1. Convert the mixed numbers to improper fractions. Step 2. Follow the rules for fraction multiplication or division. Step 3. Simplify if possible.

• Simplify a complex fraction.Step 1. Rewrite the complex fraction as a division problem. Step 2. Follow the rules for dividing fractions. Step 3. Simplify if possible.

• Placement of negative sign in a fraction.For any positive numbers $a$ and $b$, $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$.

• Simplify an expression with a fraction bar.Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Simplify the fraction.

• Fraction AdditionIf $a,b,$ and $c$ are numbers where $c\ne{}0$, then $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$.To add fractions, add the numerators and place the sum over the common denominator.

• Fraction SubtractionIf $a,b,$ and $c$ are numbers where $c\ne{}0$, then $\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}$.To subtract fractions, subtract the numerators and place the difference over the common denominator.

• Find the least common denominator (LCD) of two fractions.Step 1. Factor each denominator into its primes. Step 2. List the primes, matching primes in columns when possible. Step 3. Bring down the columns. Step 4. Multiply the factors. The product is the LCM of the denominators. Step 5. The LCM of the denominators is the LCD of the fractions.

• Equivalent Fractions PropertyIf $a,b$, and $c$ are whole numbers where $b\ne{}0$, $c\ne{}0$ then $\frac{a}{b}=\frac{a\cdot{}c}{b\cdot{}c}$ and $\frac{a\cdot{}c}{b\cdot{}c}=\frac{a}{b}$

• Convert two fractions to equivalent fractions with their LCD as the common denominator.Step 1. Find the LCD. Step 2. For each fraction, determine the number needed to multiply the denominator to get the LCD. Step 3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2. Step 4. Simplify the numerator and denominator.

• Add or subtract fractions with different denominators.Step 1. Find the LCD. Step 2. Convert each fraction to an equivalent form with the LCD as the denominator. Step 3. Add or subtract the fractions. Step 4. Write the result in simplified form.

• Summary of Fraction OperationsFraction multiplication: Multiply the numerators and multiply the denominators.

4.2 Fraction division: Multiply the first fraction by the reciprocal of the second.

4.3 Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

4.4 Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

4.5

• Simplify complex fractions.Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Divide the numerator by the denominator. Step 4. Simplify if possible.

• Add mixed numbers with a common denominator.Step 1. Add the whole numbers. Step 2. Add the fractions. Step 3. Simplify, if possible.

• Subtract mixed numbers with common denominators.Step 1. Rewrite the problem in vertical form. Step 2. Compare the two fractions. If the top fraction is larger than the bottom fraction, go to Step 3. If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction. Step 3. Subtract the fractions. Step 4. Subtract the whole numbers. Step 5. Simplify, if possible.

• Subtract mixed numbers with common denominators as improper fractions.Step 1. Rewrite the mixed numbers as improper fractions. Step 2. Subtract the numerators. Step 3. Write the answer as a mixed number, simplifying the fraction part, if possible.

• Determine whether a number is a solution to an equation.Step 1. Substitute the number for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

• Addition, Subtraction, and Division Properties of EqualityFor any numbers a, b, and c, if $a=b$, then $a+c=b+c$. Addition Property of Equality if $a=b$, then $a-c=b-c$. Subtraction Property of Equality if $a=b$, then $\frac{a}{c}=\frac{b}{c}$, $c\ne{}0$. Division Property of Equality

• The Multiplication Property of EqualityFor any numbers $ab$ and $c,a=b$, then $ac=bc$. If you multiply both sides of an equation by the same quantity, you still have equality.

##

Review Exercises

In the following exercises, name the fraction of each figure that is shaded.

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

In the following exercises, convert the improper fraction to a mixed number.

In the following exercises, convert the mixed number to an improper fraction.

In the following exercises, locate the numbers on a number line.

In the following exercises, order each pair of numbers, using $<$ or $\mathrm{>.}$

In the following exercises, simplify.

In the following exercises, multiply.

In the following exercises, find the reciprocal.

In the following exercises, divide.

In the following exercises, perform the indicated operation.

In the following exercises, translate the English phrase into an algebraic expression.

In the following exercises, simplify the complex fraction

In the following exercises, simplify.

In the following exercises, add.

In the following exercises, subtract.

In the following exercises, find the least common denominator.

In the following exercises, change to equivalent fractions using the given LCD.

In the following exercises, perform the indicated operations and simplify.

In the following exercises, evaluate.

In the following exercises, perform the indicated operation.

In the following exercises, determine whether the each number is a solution of the given equation.

In the following exercises, solve the equation.

In the following exercises, translate and solve.

##

Convert the improper fraction to a mixed number.

Convert the mixed number to an improper fraction.

Locate the numbers on a number line.

In the following exercises, simplify.

Evaluate.

In the following exercises, solve the equation.

1. $\frac{63}{11}$

2. $9\frac{4}{5}$

3. Find three fractions equivalent to $-\frac{4}{3}.$ Show your work, using figures or algebra.

4. $-2\frac{1}{2}\mathrm{\_\_\_}-3$

1. $-\frac{90}{120}$

2. $-\frac{8x}{8y}$

3. $-\frac{1}{3}\cdot\frac{12}{7}$

4. $6m\cdot\frac{4}{11}$

1. $\frac{15}{4}$

2. $-\frac{1}{4}$

3. $\frac{4}{5}\mathrm{÷}3$

4. $\frac{5}{18}\mathrm{÷}(-\frac{b}{9})$

1. $-5\frac{7}{12}\cdot{}4\frac{4}{11}$

2. the quotient of $V$ and the difference of $h$ and $6$

3. $\frac{\frac{8}{9}}{-4}$

4. $\frac{8\cdot{}4-{5}^{2}}{3\cdot{}12}$

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

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

3. $\frac{x}{10}+\frac{7}{10}$

4. $\frac{11}{12}-\frac{5}{12}$

1. $-\frac{31}{30}-\frac{7}{30}$

2. $\frac{11}{15}-\frac{5}{15}-(-\frac{2}{15})$

3. $\frac{3}{8}$ and $\frac{5}{6},$ LCD $=24$

4. $\frac{1}{3},\frac{3}{4}$ and $\frac{4}{5},$ LCD $=60$

1. $\frac{11}{12}-\frac{2}{3}$

2. $-\frac{11}{36}-\frac{11}{20}$

3. $\frac{y}{10}-\frac{1}{3}$

4. $\frac{4}{11}\mathrm{÷}\frac{2}{7d}$

1. $\frac{{(\frac{2}{3})}^{2}}{{(\frac{5}{8})}^{2}}$

2. $y-\frac{4}{5}$ when

1. ⓐ $y=-\frac{4}{5}$

2. ⓑ $y=\frac{1}{4}$

3. $4\frac{1}{3}+9\frac{1}{3}$

4. $5\frac{8}{11}+2\frac{4}{11}$

1. $9\frac{13}{20}-4\frac{11}{20}$

2. $2\frac{11}{12}-1\frac{7}{12}$

3. $x-\frac{1}{2}=\frac{1}{6}$:

1. ⓐ $x=1$

2. ⓑ $x=\frac{2}{3}$

3. ⓒ $x=-\frac{1}{3}$

4. $n+\frac{9}{11}=\frac{4}{11}$

1. $x-\frac{1}{6}=\frac{7}{6}$

2. $h-(-\frac{7}{8})=-\frac{2}{5}$

3. $-z=23$

4. The difference of $q$ and one-tenth is $\frac{1}{2}.$

1. Three-eighths of $y$ is $24.$

2. $3\frac{2}{7}$

3. $\frac{5}{20}$

4. $\frac{1}{3}\cdot\frac{3}{4}$

1. $-36u(-\frac{4}{9})$

2. $-\frac{5}{6}\mathrm{÷}\frac{5}{12}$

3. $\frac{9a}{10}\mathrm{÷}\frac{15a}{8}$

4. $\frac{\,\frac{p}{2}\,}{\,\frac{q}{5}\,}$

1. $-\frac{3}{13}+(-\frac{4}{13})$

2. $\frac{2}{5}+(-\frac{7}{5})$

3. $-\frac{3}{4}\mathrm{÷}\frac{x}{3}$

4. $y+\frac{3}{5}=\frac{7}{5}$

1. $f+(-\frac{2}{3})=\frac{5}{12}$

2. $-\frac{2}{3}c=18$