Write as a decimal.

Method 1 Use paper and pencil.

Answer: 0.0625

Method 2 Use a calculator.

Answer: 0.0625

ENTER1 16 0.0625

Write as a decimal.

Answer: 0.375

Write as a decimal.

Write as a decimal.

Write as the sum of an integer and a fraction.

Add.

Answer: 1.25

Write as a decimal.

Answer: 2.6

Write as a decimal.

The digits 12 repeat.

Answer:

Write as a decimal.

The digits 18 repeat.

Answer:

Answer: a. Write as a decimal.

b. Write as a decimal. Answer:

Answer: On a number line, 0.7 is to the right of 0.65, so

.

Write the sentence.

Write as a decimal.

In the tenths place, .

Replace with <, >, or = to make a true sentence.

Answer: <

Replace with <, >, or = to make a true sentence.

Grades Jeremy got a score of on his first quiz

and on his second quiz. Which grade was the

higher score?

Write the fractions as decimals and then compare the decimals.

Answer: The scores were the same, 0.80.

Quiz #1: Quiz #2:

Answer: the second recipe

Baking One recipe for cookies requires of a cup

of butter and a second recipe for cookies requires

of a cup of butter. Which recipe uses less butter?

Write as a fraction.

Write as an

improper fraction.

Answer:

Write 10 as a fraction.

Answer:

Answer:

a. Write as a fraction.

b. Write –6 as a fraction.

Answer:

Write 0.26 as a fraction or mixed number in simplest form.

0.26 is 26 hundredths.

Answer: Simplify. The GCF of 26 and100 is 2.

Write 2.875 as a fraction or mixed number in simplest form.

2.875 is 2 and 875 thousandths.

Answer: Simplify. The GCF of 875 and 1000 is 125.

Write each decimal as a fraction or mixed number in simplest form.

a. 0.84

b. 3.625

Answer:

Answer:

Write as a fraction in simplest form.

Let N represent the number.

Multiply each side by 100 because two digits repeat.

Subtract N from 100N to eliminate the repeating part, 0.3939...

Divide each side by 99.

Simplify.

Answer:

ENTER13 33 0.3939393939Check

Write as a fraction in simplest form.

Answer:

Identify all sets to which the number 15 belongs.

Answer: 15 is a whole number, an integer, a natural number, and a rational number.

Identify all sets to which the number belongs.

Answer: is a rational number.

Identify all sets to which the number 0.30303030... belongs.

Answer: 0.30303030... is a nonterminating, repeating decimal. So, it is a rational number.

Answer: integer, rational

Answer: rational

Answer: rational

Identify all sets to which each number belongs.

a. –7

b.

c. 0.24242424...

Find . Write the product in simplest form.

Multiply the numerators.

Multiply the denominators.

Answer: Simplify. The GCF of 10 and40 is 10.

Find .

Answer:

Find . Write the product in simplest form.

Multiply the numerators and multiply the denominators.

Answer: Simplify.

Divide 8 and 6 by their GCF, 2.

Find . Write the product in simplest form.

Answer:

Find . Write the product in simplest form.

Multiply the numerators and multiply the denominators.

Answer: Simplify.

Divide 2 and 4 by their GCF, 2.

Find .

Answer:

Find . Write the product in simplest form.

.

Rename and rename

Divide by the GCF, 3.

Multiply.

Simplify.Answer:

Find . Write the product in simplest form.

Answer:

Find . Write the product in simplest form.

The GCF of and q is q.

Answer: Simplify.

Find . Write the product in simplest form.

Answer:

Track The track at Cole’s school is mile around.

If Cole runs one lap in two minutes, how far (in miles)

does he run in 30 minutes?

Words Distance equals the rate multiplied by the time.

Variables Let d = distance, r = rate, and t = time.

Formula d = rt

Multiply.

Simplify.

Answer: Cole runs miles in 30 minutes.

Write the formula.mile per 2 minutes • 30 minutes

Divide by the common factors and units.

15

1

Check The problem asks for the distance. When you divide the common units, the answer is expressed in miles. So, the answer is reasonable.

Walking Bob walks mile in 12 minutes.

How far does he walk in 30 minutes?

Answer:

Find the multiplicative inverse of .

The product is 1.

Answer: The multiplicative inverse or reciprocal of

.

Find the multiplicative inverse of .

Write as an improper fraction.

Answer: The reciprocal of .

The product is 1.

Find the multiplicative inverse of each number.

a.

b.

Answer:

Answer:

Find . Write the quotient in simplest form.

Multiply by the multiplicative

inverse of , .

Divide 5 and 10 by their GCF, 5.

Answer: Simplify.

Find . Write the quotient in simplest form.

Answer:

Find . Write the quotient in simplest form.

Write 3 as .

Multiply by the multiplicative

inverse of , .

Answer: Multiply the numerators and multiply the denominators.

Find . Write the quotient in simplest form.

Answer:

Find . Write the quotient in simplest form.

Rename the mixed numbers as improper fractions.

Multiply by the multiplicative

inverse of , .

Divide out common factors.

Answer: Simplify.

Find . Write the quotient in simplest form.

Answer:

Find . Write the quotient in simplest form.

Multiply by the multiplicative

inverse of , .

Divide out common factors.

Answer: Simplify.

Find . Write the quotient in simplest form.

Answer:

Travel How many gallons of gas are needed to travel

miles if a car gets miles per gallon?

To find how many gallons, divide .

Write as improper fractions.

Multiply by the reciprocal of

.

Simplify.

Divide out commonfactors.

105 1

2 17

Answer: gallons of gas are needed.

Check Use dimensional analysis to examine the units.

Simplify.

Divide out the units.

The result is expressed as gallons. This agrees with your

answer of gallons of gas.

Sewing Emily has yards of fabric. She wants to

make pillows which each require yards of fabric

to complete. How many pillows can Emily make?

Answer: or 8 pillows

Example 1 Add Fractions

Example 2 Add Mixed Numbers

Example 3 Subtract Fractions

Example 4 Subtract Mixed Numbers

Example 5 Add Algebraic Fractions

Find . Write the sum in simplest form.

Estimate

The denominators are the same. Add the numerators.

Answer: Simplify and rename as a mixed number.

Find . Write the sum in simplest form.

Answer:

Find . Write the sum in simplest form.

Add the whole numbers and fractions separately.

Add the numerators.

Simplify.Answer:

Find . Write the sum in simplest form.

Answer:

Find . Write the difference in simplest form.

Estimate

The denominators are the same. Subtract the numerators.

Answer: Simplify.

Find . Write the difference in simplest form.

Answer:

Evaluate . Estimate

Write the mixed numbers as improper fractions.

Subtract the numerators.

Answer: Simplify.

Replace r with and q with .

Evaluate .

Answer:

Find . Write the sum in simplest form.

The denominators are the same.Add the numerators.

Add the numerators.

Answer: Simplify.

Find . Write the sum in simplest form.

Answer:

Find the LCM of 168 and 180.

Number Prime Factorization Exponential Form

168

180

The prime factors of both numbers are 2, 3, 5, and 7. Multiply the greatest powers of 2, 3, 5, and 7 appearing in either factorization.

Answer: The LCM of 168 and 180 is 2520.

Find the LCM of 144 and 96.

Answer: 288

Find the LCM of .

Answer: The LCM of .

Multiply the greatest power of each prime factor.

Find the LCM of .

Answer:

Find the LCD of .

Answer: The LCD of .

Multiply.

Write the prime factorization of 8 and 20.

Highlight the greatest power of each prime factor.

Find the LCD of .

Answer: 36

Find the LCD of .

Answer: The LCD of .

Answer:

Find the LCD of

Replace with <, >, or = to make a true statement.

The LCD of the fractions is . Rewrite the fractions using the LCD and then compare the numerators.

Multiply the fraction by to make

the denominator 105.

Multiply the fraction by to make

the denominator 105.

Answer: Since , then .

is to the left of on the number line.

Replace with <, >, or = to make a true statement.

Answer: <

Use as the common denominator.

Find .

Rename each fraction with the common denominator.

Add the numerators.Answer:

Find .

Answer:

Find . Estimate

Rename each fraction with the LCD.

Add the numerators.

Simplify.Answer:

The LCD is 30.

Find .

Answer:

Find . Write in simplest form.

Write the mixed numbers as improper fractions.

Simplify.

Rename fractions using the LCD, 24.

Simplify.Answer:

Add the numerators.

Find .

Answer:

Find .

The LCD is 16.

Rename using the LCD.

Answer: Subtract the numerators.

Find .

Answer:

Find .

Write the mixed numbers as improper fractions.

Rename the fractions using the LCD.

Simplify.

Answer: Subtract.

Find .

Answer:

Explore You know the distances Juyong jogged each day.

Plan Add the daily distances together to find the total distance. Estimate your answer.

Jogging Juyong jogged three days this week. She

jogged miles, miles, and miles. How far

did she jog altogether?

Rename the fractions with LCD, 20.

Examine Since is close to , the

answer is

reasonable.

Answer: Juyong jogged miles.

Solve

Add the like fractions.

Gardening Howard’s tomato plants grew inches

during the first week after sprouting, inches during

the second week, and inches the third week. Find

the total growth during the first three weeks after

sprouting.

Answer: inches

Movies The revenue of the 10 highest grossing movies as of June 2000 are given in the table. Find the mean, median, and mode of the revenues.

Answer: The mean revenue is $379.8 million.

Top 10 Movie Revenues

(millions of $)

601 330461 313431 309400 306357 290

To find the median, order the numbers from least to greatest.

Answer: The median revenue is $343.5 million.

290, 306, 309, 313, 330, 357, 400, 431, 461, 601

Answer: There is no mode because each number in the set occurs once.

There is an even number of items. Find the mean of the two middle numbers.

Test Scores The test scores for a class of nine students are 85, 93, 78, 99, 62, 83, 90, 75, 85. Find the mean, median, and mode of the test scores.

Answer: mean, 83.3; median, 85; mode, 85

Olympics The line plot below shows the number of gold medals earned by each country that participated in the 1998 Winter Olympic games in Nagano, Japan. Find the mean, median, and mode for the gold medals won.

Answer: The mean is 2.875.

There are 24 numbers. The median number is the average of the 12th and 13th numbers.

Answer: The median is 2.

The number 0 occurs most frequently in the set of data.

Answer: The mode is 0.

Families A survey of school-age children shows the family sizes displayed in the line plot below. Find the mean, median, and mode.

Answer: mean, 4.3; median, 5; mode, 5

Quiz scores The quiz scores for a math class are 8, 7, 6, 10, 8, 8, 9, 8, 7, 9, 8, 0, and 10. Identify an extreme value and describe how it affects the mean.

The data value 0 appears to be an extreme value. Calculate the mean with and without the extreme value to find how it affects the mean.

mean with extreme value mean without extreme value

Answer: The extreme value 0 decreases the mean by 8.2 – 7.5 or about 0.7.

Birth Weight The birth weights of ten newborn babies are given in pounds: 7.3, 8.4, 9.1, 7.9, 8.8, 6.5, 7.9, 4.1, 8.0, 7.5. Identify an extreme value and describe how it affects the mean.

Answer: 4.1; it decreases the mean by about 0.4.

Bob’s Books

The Reading

Place

1290 1400

1400 1450

1400 1550

1600 1600

3650 2000

The table shows the monthly salaries of the employees at two bookstores. Find the mean, median, and mode for each set of data. Based on the averages, which bookstore pays its employees better?

mean:

Bob’s Books

median: 1290, 1400, 1400, 1600, 3650median

mode: $1400

Bob’s Books

The Reading

Place

1290 1400

1400 1450

1400 1550

1600 1600

3650 2000

mean:

The Reading Place

mode: none

median: 1400, 1450, 1550, 1600, 2000median

Answer: The $3650 salary at Bob’s Books is an extreme value that increases the mean salary. The employees at The Reading Place are generally better paid as shown by the median.

Men Women

5 1

2 6

3 4

8 2

12 1

4 8

The number of hours spent exercising each week by men and women are given in the table. Find the mean, median, and mode for each set of data. Based on the averages, which gender exercises more?

Answer: Men: mean, 5.7; median, 4.5; mode, none

Women: mean, 3.7; median, 3; mode, 1

Men seem to exercise more.

Grid–In Test Item Jenny’s bowling average is 146. Today she bowled 138, 140, and 145. What does she need to score on her fourth game to maintain her average?

Read the Test Item

Find the sum of the first three games. Then write an equation to find the score needed on the fourth game.

Solve the Test Item

Step 1 Find the sum of the first three games x.

mean of the first three scores

Multiply each side by 3.

Simplify.

sum of first three scores

Step 2 Find the fourth score, x.

mean

Write an equation.

Multiply each side by 4 and simplify.

Subtract 423 from each side and simplify.

Substitution.

Answer: Jenny needs to score at least 161 to maintain her average of 146.

Emily scored 73, 82, and 85 on her first three math tests. What score does Emily need on the fourth test to give her an average of 82 for the four tests?

Answer: 88

Solve . Check your solution.

Write the equation.

Add to each side.

Simplify.

Rename the fractions using the LCD and add.

Answer: Simplify.

Check Write the original equation.

Simplify.

Replace y with .

Solve . Check your solution.

Answer:

Solve . Check your solution.

Write the equation.

Subtract 8.6 from each side.

Simplify.

Check: Write the original equation.

Simplify.Answer:

Replace m with 2.6.

Solve . Check your solution.

Answer: 6.1

Divide each side by 9.

Solve . Check your solution.

Write the equation.

Simplify.

Simplify.Answer:

Check: Write the original equation.

Replace a with 0.4.

Solve . Check your solution.

Answer: 0.8

Solve . Check your solution.

Write the equation.

Multiply each side by 6.

Simplify.Answer:

Check: Write the original equation.

Simplify.

Replace x with 48.

Solve . Check your solution.

Write the equation.

Simplify. Check the solution.Answer:

Multiply each side by .

Check: Write the original equation.

Simplify.

Replace t with –10.

a. Solve . Check your solution.

b. Solve . Check your solution.

Answer: 45

Answer: 16

State whether the sequence –5, –1, 3, 7, 11, ... is arithmetic. If it is, state the common difference and write the next three terms.

–5, –1, 3, 7, 11

Notice that , and so on.

Answer: The terms have a common difference of , so the sequence is arithmetic. Continue the pattern to find the next three terms.

11,

Answer: The next three terms of the sequence are 15, 19, and 23.

15, 19, 23

State whether the sequence 12, 7, 2, –3, –8, ... is arithmetic. If it is, state the common difference and write the next three terms.

Answer: arithmetic; –5; –13, –18, –23

State whether the sequence 0, 2, 6, 12, 20, ... is arithmetic. If it is, state the common difference and write the next three terms.

0, 2, 6, 12, 20

The terms do not have a common difference.

Answer: The sequence is not arithmetic. However, if the pattern continues, the next three differences will be 10, 12, and 14.

20, 30, 42, 56

The next three terms are 30, 42, and 56.

State whether the sequence 20, 17, 11, 2, –10, ... is arithmetic. If it is, state the common difference and write the next three terms.

Answer: not arithmetic

State whether the sequence 2, 4, 4, 8, 8, 16, 16 ... is geometric. If it is, state the common ratio and write the next three terms.

2, 4, 4, 8, 8, 16, 16

Answer: There is no common ratio. The sequence is not geometric.

27, –9, 3, –1,

State whether the sequence 27, –9, 3, –1, , ... is

geometric. If it is, state the common ratio and write

the next three terms.

Answer: The common ratio is , so the sequence is

geometric. Continue the pattern to find the next three terms.

, , ,

The next three terms are , , and .

a. State whether the sequence 2, 8, 32, 128, 512, ... is geometric. If it is, state the common ratio and write the next three terms.

b. State whether the sequence 100, 20, 4, , ... is

geometric. If it is, state the common ratio and write

the next three terms.

Answer: geometric; 4; 2048, 8192, 32,768

Answer: geometric; ; , ,

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