Chapter 5 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

circumference of a circle

circumference

diameter of a circle

A diameter of a circle is a line segment that passes through a circle’s center connecting two points on the circle.

equivalent decimals

Two decimals are equivalent decimals if they convert to equivalent fractions.

mean

median

• Half the data values are less than or equal to the median.

• Half the data values are greater than or equal to the median.

mode

The mode of a set of numbers is the number with the highest frequency.

radius of a circle

A radius of a circle is a line segment from the center to any point on the circle.

rate

A rate compares two quantities of different units. A rate is usually written as a fraction.

ratio

repeating decimal

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

unit price

unit price

unit rate

A unit rate is a rate with denominator of 1 unit.

##

Key Concepts

• Name a decimal number. 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 number from its name. 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.

• Convert a decimal number to a fraction or mixed number. Step 1. Look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If it is not zero, the decimal converts to a mixed number. Write the whole number. Step 2. Determine the place value of the final digit. Step 3. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit Step 4. Simplify the fraction, if possible.

• Order decimals. Step 1. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match. Step 2. Compare the numbers to the right of the decimal point as if they were whole numbers. Step 3. Order the numbers using the appropriate inequality sign.

• 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 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. Rewrite the number, removing all digits to the right of the given place value.

• Add or subtract decimals. Step 1. Write the numbers vertically so the decimal points line up. 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 decimal numbers. Step 1. Determine the sign of the product. Step 2. Write the numbers in vertical format, lining up the numbers on the right. Step 3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Step 4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders. Step 5. Write the product with the appropriate sign.

• Multiply a decimal by a power of 10. 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. Write zeros at the end of the number as placeholders if needed.

• Divide a decimal by a whole number. Step 1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. Step 2. Divide as usual.

• Divide decimal numbers. 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 to the right, writing 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.

• 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.

• Convert a Fraction to a Decimal To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

• Properties of Circles $r$ is the length of the radius $d$ is the length of the diameter The circumference is $2\pi{}r$. $\,C=2\pi{}r$ The area is $\pi{r}^{2}$. $\,A=\pi{r}^{2}$

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

• Properties of Equality

Table

5.8

• Calculate the mean of a set of numbers. Step 1. Write the formula for the mean $\mathrm{mean}=\frac{\mathrm{sum of values in data set}}{n}$ Step 2. Find the sum of all the values in the set. Write the sum in the numerator. Step 3. Count the number, n, of values in the set. Write this number in the denominator. Step 4. Simplify the fraction. Step 5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.

• Find the median of a set of numbers. Step 1. List the numbers from least to greatest. Step 2. Count how many numbers are in the set. Call this $n$. Step 3. Is $n$ odd or even? If $n$ is an odd number, the median is the middle value. If $n$ is an even number, the median is the mean of the two middle values

• Identify the mode of a set of numbers. Step 1. List the data values in numerical order. Step 2. Count the number of times each value appears. Step 3. The mode is the value with the highest frequency.

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

• Use a strategy for applications with square roots. Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Write a complete sentence that answers the question.

##

Review Exercises

Name Decimals

In the following exercises, name each decimal.

Write Decimals

In the following exercises, write as a decimal.

Convert Decimals to Fractions or Mixed Numbers

In the following exercises, convert each decimal to a fraction. Simplify the answer if possible.

Locate Decimals on the Number Line

Order Decimals

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

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 Decimals

In the following exercises, multiply.

Divide Decimals

In the following exercises, divide.

Use Decimals in Money Applications

In the following exercises, use the strategy for applications to solve.

Convert Fractions to Decimals

In the following exercises, convert each fraction to a decimal.

Order Decimals and Fractions

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

In the following exercises, write each set of numbers in order from least to greatest.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify

Find the Circumference and Area of Circles

In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle.

Determine Whether a Decimal is a Solution of an Equation

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

Solve Equations with Decimals

In the following exercises, solve.

Translate to an Equation and Solve

In the following exercises, translate and solve.

Find the Mean of a Set of Numbers

In the following exercises, find the mean of the numbers.

Find the Median of a Set of Numbers

In the following exercises, find the median.

Find the Mode of a Set of Numbers

In the following exercises, identify the mode of the numbers.

Use the Basic Definition of Probability

In the following exercises, solve. (Round decimals to three places.)

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction. Simplify the answer if possible.

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction. Simplify the answer if possible.

Find Unit Rates

In the following exercises, find the unit rate.

Find Unit Price

In the following exercises, find the unit price.

In the following exercises, find each unit price and then determine the better buy.

Translate Phrases to Expressions with Fractions

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

Simplify Expressions with Square Roots

In the following exercises, simplify.

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

##

In the following exercises, simplify each expression.

In the following exercises, solve.

In the following exercises, simplify.

Key Terms

circumference of a circle

circumference

diameter of a circle

A diameter of a circle is a line segment that passes through a circle’s center connecting two points on the circle.

equivalent decimals

Two decimals are equivalent decimals if they convert to equivalent fractions.

mean

median

• Half the data values are less than or equal to the median.

• Half the data values are greater than or equal to the median.

mode

The mode of a set of numbers is the number with the highest frequency.

radius of a circle

A radius of a circle is a line segment from the center to any point on the circle.

rate

A rate compares two quantities of different units. A rate is usually written as a fraction.

ratio

repeating decimal

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

unit price

unit price

unit rate

A unit rate is a rate with denominator of 1 unit.

##

Key Concepts

• Name a decimal number. 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 number from its name. 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.

• Convert a decimal number to a fraction or mixed number. Step 1. Look at the number to the left of the decimal. If it is zero, the decimal converts to a proper fraction. If it is not zero, the decimal converts to a mixed number. Write the whole number. Step 2. Determine the place value of the final digit. Step 3. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit Step 4. Simplify the fraction, if possible.

• Order decimals. Step 1. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match. Step 2. Compare the numbers to the right of the decimal point as if they were whole numbers. Step 3. Order the numbers using the appropriate inequality sign.

• 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 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. Rewrite the number, removing all digits to the right of the given place value.

• Add or subtract decimals. Step 1. Write the numbers vertically so the decimal points line up. 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 decimal numbers. Step 1. Determine the sign of the product. Step 2. Write the numbers in vertical format, lining up the numbers on the right. Step 3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Step 4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders. Step 5. Write the product with the appropriate sign.

• Multiply a decimal by a power of 10. 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. Write zeros at the end of the number as placeholders if needed.

• Divide a decimal by a whole number. Step 1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. Step 2. Divide as usual.

• Divide decimal numbers. 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 to the right, writing 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.

• 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.

• Convert a Fraction to a Decimal To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

• Properties of Circles $r$ is the length of the radius $d$ is the length of the diameter The circumference is $2\pi{}r$. $\,C=2\pi{}r$ The area is $\pi{r}^{2}$. $\,A=\pi{r}^{2}$

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

• Properties of Equality

Table

5.8

• Calculate the mean of a set of numbers. Step 1. Write the formula for the mean $\mathrm{mean}=\frac{\mathrm{sum of values in data set}}{n}$ Step 2. Find the sum of all the values in the set. Write the sum in the numerator. Step 3. Count the number, n, of values in the set. Write this number in the denominator. Step 4. Simplify the fraction. Step 5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.

• Find the median of a set of numbers. Step 1. List the numbers from least to greatest. Step 2. Count how many numbers are in the set. Call this $n$. Step 3. Is $n$ odd or even? If $n$ is an odd number, the median is the middle value. If $n$ is an even number, the median is the mean of the two middle values

• Identify the mode of a set of numbers. Step 1. List the data values in numerical order. Step 2. Count the number of times each value appears. Step 3. The mode is the value with the highest frequency.

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

• Use a strategy for applications with square roots. Identify what you are asked to find. Write a phrase that gives the information to find it. Translate the phrase to an expression. Simplify the expression. Write a complete sentence that answers the question.

##

Review Exercises

Name Decimals

In the following exercises, name each decimal.

Write Decimals

In the following exercises, write as a decimal.

Convert Decimals to Fractions or Mixed Numbers

In the following exercises, convert each decimal to a fraction. Simplify the answer if possible.

Locate Decimals on the Number Line

Order Decimals

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

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 Decimals

In the following exercises, multiply.

Divide Decimals

In the following exercises, divide.

Use Decimals in Money Applications

In the following exercises, use the strategy for applications to solve.

Convert Fractions to Decimals

In the following exercises, convert each fraction to a decimal.

Order Decimals and Fractions

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

In the following exercises, write each set of numbers in order from least to greatest.

Simplify Expressions Using the Order of Operations

In the following exercises, simplify

Find the Circumference and Area of Circles

In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle.

Determine Whether a Decimal is a Solution of an Equation

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

Solve Equations with Decimals

In the following exercises, solve.

Translate to an Equation and Solve

In the following exercises, translate and solve.

Find the Mean of a Set of Numbers

In the following exercises, find the mean of the numbers.

Find the Median of a Set of Numbers

In the following exercises, find the median.

Find the Mode of a Set of Numbers

In the following exercises, identify the mode of the numbers.

Use the Basic Definition of Probability

In the following exercises, solve. (Round decimals to three places.)

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction. Simplify the answer if possible.

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction. Simplify the answer if possible.

Find Unit Rates

In the following exercises, find the unit rate.

Find Unit Price

In the following exercises, find the unit price.

In the following exercises, find each unit price and then determine the better buy.

Translate Phrases to Expressions with Fractions

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

Simplify Expressions with Square Roots

In the following exercises, simplify.

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

##

In the following exercises, simplify each expression.

In the following exercises, solve.

In the following exercises, simplify.

1. $0.375$

2. $5.24$

3. $-4.09$

4. negative five hundredths

5. $0.825$

1. $3.64$

2. $0.6\mathrm{\_\_\_}0.8$

3. $0.803\mathrm{\_\_\_\_}0.83$

4. $12.529$

5. $5.897$

1. $1.83-4.2$

2. $(-6.4)(0.25)$

3. $(15.4)(1000)$

4. Jessie put $8$ gallons of gas in her car. One gallon of gas costs $\text{\textdollar 3.528.}$ How much did Jessie owe for all the gas?

5. Alice bought a roll of paper towels that cost $\text{\textdollar 2.49.}$ She had a coupon for $\text{\textdollar 0.35}$ off, and the store doubled the coupon. How much did Alice pay for the paper towels?

1. $-\frac{21}{4}$

2. $\frac{6}{11}$

3. $\frac{3}{5}\mathrm{\_\_\_}0.$

4. $-\frac{5}{12}\mathrm{\_\_\_}-0.42$

5. $0.33\mathrm{\_\_\_}\frac{5}{16}$

1. $\frac{7}{9},0.75,\frac{11}{15}$

2. $-\frac{2}{5}\cdot\frac{9}{10}+0.14$

3. $\mathrm{radius}=\mathrm{3.5 ft.}$

4. $\mathrm{diameter}=\mathrm{11 cm}$

5. $y+3.2=-1.5\,$ⓐ $\,y=1.7\,$ⓑ $\,y=-4.7$

1. $0.45v=-40.5\,$ⓐ $\,v=-18.225\,$ⓑ $\,v=-90$

2. $h+5.91=2.4$

3. $p-4.3=-1.65$

4. $j-7.42=-3.7$

5. $-8.6x=34.4$

1. $\frac{a}{0.3}=-24$

2. $\frac{s}{-2.5}=-10$

3. The product of $-5.9$ and $x$ is $-3.54.$

4. The sum of $m$ and $(-4.03)$ is $6.8.$

5. $\text{\textdollar 270}$, $\text{\textdollar 310.50}$, $\text{\textdollar 243.75}$, and$\text{\textdollar 252.15}$

1. In the last three months, Raul’s water bills were $\text{\textdollar 31.45},\,\text{\textdollar 48.76},\,\mathrm{and}\,\text{\textdollar 42.60}.$ Find the mean.

2. The number of clients at Miranda’s beauty salon each weekday last week were $18,7,12,16,\,\mathrm{and}\,20.$ Find the median number of clients.

3. Luc has to read $3$ novels and $12$ short stories for his literature class. The professor will choose one reading at random for the final exam. Find the probability that the professor will choose a novel for the final exam. Express your answer as a fraction and as a decimal.

4. $56$ to $32$

5. $1.2$ to $1.8$

1. $2\frac{1}{3}\,\mathrm{to}\,5\frac{1}{4}$

2. $28$ inches to $3$ feet

3. $90$ pounds per $7.5$ square inches

4. $\text{\textdollar 612.50}$ for $35$ hours

5. $90$ pounds per $7.5$ square inches

1. $\text{\textdollar 612.50}$ for $35$ hours

2. Highlighters: $6$ for $\text{\textdollar 2.52}$

3. Anna bought a pack of $8$ kitchen towels for $\text{\textdollar 13.20}.$ How much did each towel cost? Round to the nearest cent if necessary.

4. Vitamins: $60$ tablets for $\text{\textdollar 6.49}$ or $100$ for $\text{\textdollar 11.99}?$

5. $a$ adults to $45$ children

1. the ratio of $19$ and the sum of $3$ and $n$

2. $-\sqrt{81}$

3. $\sqrt{-36}$

4. $\sqrt{155}$

5. $\sqrt{64{b}^{2}}$

1. $\sqrt{225{m}^{2}{n}^{2}}$

2. $\sqrt{49{y}^{2}}$

3. $\sqrt{121{c}^{2}{d}^{2}}$

4. Landscaping Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of $30$ square feet. How long can a side of the flower garden be?

5. Accident investigation The skid marks of a car involved in an accident were $216$ feet. How fast had the car been going before applying the brakes?

1. Write $1.73$ as a fraction.

2. Round $16.749$ to the nearest ⓐ tenth ⓑ hundredth ⓒ whole number

3. $0.96\mathrm{÷}(-12)$

4. $m+3.7=2.5$

5. Three friends went out to dinner and agreed to split the bill evenly. The bill was $\text{\textdollar 79.35}.$ How much should each person pay?

1. The ages, in months, of $10$ children in a preschool class are: $55$, $55$, $50$, $51$, $52$, $50$, $53$, $51$, $55$, $49$ Find the ⓐ mean ⓑ median ⓒ mode

2. Find each unit price and then the better buy. Laundry detergent: $64$ ounces for $\text{\textdollar 10.99}$ or $48$ ounces for $\text{\textdollar 8.49}$

3. $\sqrt{144{n}^{2}}$

4. Yanet wants a square patio in her backyard. She has $225$ square feet of tile. How long can a side of the patio be?