Halfway Between Two Numbers To find a rational number

In Exercises 1–12, fill in the blanks with an appropriate word, phrase, or symbol(s). The set of rational numbers is the set of numbers of the form \(\frac{p}{q}\) where p and q are ___________ and \(q \neq 0\).

For the rational number p q , p is called the .

For the rational number p q , q is called the .

Rational numbers such as 23 4 and -11 2 are examples of numbers.

Rational numbers such as 11 4 and -3 2 are examples of fractions.

The number 1 2 can be represented as a(n) decimal number.

The number 1 3 can be represented as a(n) decimal number.

In the decimal number 0.285714, the 2 is in the position ____.

In the decimal number 0.285714, the 8 is in the position.

The product of a number and its must equal 1.

When adding or subtracting two fractions with unlike denominators, first rewrite each fraction with the ______ common denominator.

The rational numbers 1 2 and 5 10 are examples of fractions.

In Exercises 1322, reduce each fraction to lowest terms. 3 6

In Exercises 1322, reduce each fraction to lowest terms. 15 20

In Exercises 1322, reduce each fraction to lowest terms. 28 63

In Exercises 1322, reduce each fraction to lowest terms. 36 56

In Exercises 1322, reduce each fraction to lowest terms.95 125

In Exercises 13–22, reduce each fraction to lowest terms. \(\frac{13}{221}\)

In Exercises 13–22, reduce each fraction to lowest terms. \(\frac{112}{176}\)

In Exercises 1322, reduce each fraction to lowest terms.120 135

In Exercises 21-26, convert each mixed number to an improper fraction. 2 \(\frac{3}{7}\)

In Exercises 2126, convert each mixed number to an improper fraction.3 5 6

In Exercises 2126, convert each mixed number to an improper fraction.1 15 16

In Exercises 2126, convert each mixed number to an improper fraction.7 1 5

In Exercises 2126, convert each mixed number to an improper fraction.4 15 16

In Exercises 2126, convert each mixed number to an improper fraction.11 9 16

In Exercises 2730, write the number of inches indicated by the arrows as an improper fraction.

In Exercises 2730, write the number of inches indicated by the arrows as an improper fraction.

In Exercises 2730, write the number of inches indicated by the arrows as an improper fraction.

In Exercises 2730, write the number of inches indicated by the arrows as an improper fraction.

In Exercises 3136, convert each improper fraction to a mixed number. 83

In Exercises 31–36, convert each improper fraction to a mixed number. \(\frac{19}{7}\)

In Exercises 3136, convert each improper fraction to a mixed number.73 6

In Exercises 3136, convert each improper fraction to a mixed number.157 12

In Exercises 3136, convert each improper fraction to a mixed number.878 15

In Exercises 3136, convert each improper fraction to a mixed number. 1028 21

In Exercises 3744, express each rational number as terminating or repeating decimal number. 310

In Exercises 3744, express each rational number as terminating or repeating decimal number.7 8

In Exercises 3744, express each rational number as terminating or repeating decimal number.2 3

In Exercises 3744, express each rational number as terminating or repeating decimal number. 1 6

In Exercises 37–44, express each rational number as terminating or repeating decimal number. \(\frac{3}{8}\)

In Exercises 3744, express each rational number as terminating or repeating decimal number. 23 7

In Exercises 3744, express each rational number as terminating or repeating decimal number.13 6

In Exercises 3744, express each rational number as terminating or repeating decimal number.115 15

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 0.25

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.0.13

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 0.175

In Exercises 45–52, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 0.375

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.0.2

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 0.251

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.. 0.0131

In Exercises 4552, express each terminating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.0.2345

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.0.2

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.0.7

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 0.9

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.0.23

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 1.36

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. 0.135

In Exercises 53–60, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms. \(2.0 \overline{5}\)

In Exercises 5360, express each repeating decimal number as a quotient of two integers. If possible, reduce the quotient to lowest terms.2.49

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.1 3 # 6 7

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.1 4 , 5 8

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.a -3 8 b a -16 15 b

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.a -3 5 b , 10 21

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 6170, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.2 5 + 1 3

In Exercises 71-80, perform the indicated operation and reduce your answer to lowest terms. \(\frac{5}{6}\) - \(\frac{3}{4}\)

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.5 12 + 7 36

In Exercises 71–80, perform the indicated operation and reduce your answer to lowest terms. \(\frac{5}{9}-\frac{7}{54}\)

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.17 25 - 43 100

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.12 + 1 48 + 1 72

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.3 5 + 7 15 + 9 75

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.30 - 3 40 - 7 50

In Exercises 7180, perform the indicated operation and reduce your answer to lowest terms.4 25 - 9 100 - 7 40

In Exercises 8186, use the appropriate formula to evaluate the expression.5 8 + 1 3

In Exercises 8186, use the appropriate formula to evaluate the expression.4 7 + 2 5

In Exercises 81-86, use the appropriate formula to evaluate the expression. \(\frac{5}{6}\) - \(\frac{7}{8}\)

In Exercises 8186, use the appropriate formula to evaluate the expression.7 3 - 5 12

In Exercises 8186, use the appropriate formula to evaluate the expression.3 8 + 5 12

In Exercises 81–86, use the appropriate formula to evaluate the expression. \(\left(\frac{2}{3}+\frac{1}{4}\right)-\frac{3}{5}\)

In Exercises 8792, evaluate each expression.a 1 3 # 6 7 b + 1 4

In Exercises 8792, evaluate each expression.a 1 6 , 2 3 b - 1 7

In Exercises 87–92, evaluate each expression. \(\left(\frac{3}{4}+\frac{1}{6}\right) \div\left(2-\frac{7}{6}\right)\)

In Exercises 87–92, evaluate each expression. \(\left(\frac{1}{3} \cdot \frac{3}{7}\right)+\left(\frac{3}{5} \cdot \frac{10}{11}\right)\)

In Exercises 8792, evaluate each expression.a3 - 4 9 b , a4 + 2 3 b

In Exercises 8792, evaluate each expression.a 2 5 , 4 9 b a 3 5 # 6b

In Exercises 93-104, write an expression that will solve the problem and then evaluate the expression.Height Increase When David Conway finished his first year of high school, his height was 69 \(\frac{7}{8}\) inches. When he returned to school after the summer, Davids height was 71 \(\frac{5}{8}\) inches. How much did Davids height increase over the summer?

In Exercises 93-104, write an expression that will solve the problem and then evaluate the expression.Thistles Diane Helbing has four different varieties of thistles invading her pasture. She estimates that of these thistles, \(\frac{1}{2}\) are Canada thistles, \(\frac{1}{4}\) are bull thistles, \(\frac{1}{6}\) are plumeless thistles, and the rest are musk thistles. What fraction of the thistles are musk thistles?

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Stairway Height A stairway consists of 14 stairs, each 8 5 8 inches high. What is the vertical height of the stairway?

In Exercises 93–104, write an expression that will solve the problem and then evaluate the expression. Alphabet Soup Margaret Cannata’s recipe for alphabet soup calls for (among other items) \(\frac{1}{4}\) cup snipped parsley, \(\frac{1}{8}\) teaspoon pepper, and \(\frac{1}{2}\) cup sliced carrots. Margaret is expecting company and needs to multiply the amounts of the ingredients by \(1 \frac{1}{2}\) times. Determine the amount of (a) snipped parsley, (b) pepper, and (c) sliced carrots she needs for the soup.

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Sprinkler System To repair his sprinkler system, Tony Gambino needs a total of 20 5 16 inches of PVC pipe. He has on hand pieces that measure 2 1 4 inches, 3 7 8 inches, and 4 1 4 inches in length. If he can combine these pieces and use them in the repair, how long a piece of PVC pipe will Tony need to purchase to repair his sprinkler system?

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Crop Storage Todd Schroeder has a silo on his farm in which he can store silage made from his various crops. He currently has a silo that is 1 4 full of corn silage, 2 5 full of hay silage, and 1 3 full of oats silage. What fraction of Todds silo is currently in use?

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Department Budget Jaime Bailey is chair of the humanities department at Santa Fe Community College. Jaime has a budget in which 1 2 of the money is for photocopying, 2 5 of the money is for computer-related expenses, and the rest of the money is for student tutors in the foreign languages lab. What fraction of Jaimes budget is for student tutors?

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Art Supplies Denise Viale teaches kindergarten and is buying supplies for her class to make papier-mch piggy banks. Each piggy bank to be made requires 1 1 4 cups of flour. If Denise has 15 students who are going to make piggy banks, how much flour does Denise need to purchase?

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Height of a Computer Stand The instructions for assembling a computer stand include a diagram illustrating its dimensions. Find the total height of the stand. 30 1 4 " 4 1 2 " 24 1 8

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Width of a Picture The width of a picture is 24 7 8 in., as shown in the diagram. Find x, the distance from the edge of the frame to the center.

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Traveling Interstate 5 While on vacation, Janet Mazerella traveled the full length of Interstate Highway 5 from the U.S. Canadian border to the U.S. Mexican border. She recorded the following travel times between cities: Blaine, Washington, to Seattle: 1 hour 49 minutes; Seattle to Portland: 2 hours 48 minutes; Portland to Sacramento: 9 hours 6 minutes; Sacramento to Los Angeles: 6 hours 3 minutes; Los Angeles to San Diego: 2 hours 9 minutes; San Diego to San Ysidro, California: 22 minutes. a) Write each of these times as a fraction or as a mixed number with minutes represented as a fraction with a denominator of 60. Do not reduce the fractional part of the mixed number. b) What is Janets total driving time from Blaine, Washington, to San Ysidro, California? Give your answer as a mixed number and in terms of hours and minutes. California Oregon Washington Blaine Seattle Portland Sacramento Los Angeles San Diego San Ysidro

In Exercises 93104, write an expression that will solve the problem and then evaluate the expression.Traveling Across Texas While on vacation, Omar Rodriguez traveled the entire length of Interstate Highway 10 in Texas. Omar recorded the following travel times between cities on his trip. Anthony to El Paso: 25 minutes; El Paso to San Antonio: 7 hours 54 minutes; San Antonio to Houston: 3 hours 1 minute; Houston to Beaumont: 1 hour 23 minutes; Beaumont to Orange: 28 minutes. a) Write each of these times as a fraction or as a mixed number with minutes represented as a fraction with a denominator of 60. Do not reduce the fractional part of the mixed number. b) What is Omars total driving time from Anthony to Orange? Give your answer as a mixed number and in terms of hours and minutes.

Cutting Lumber If a piece of wood 8 3 4 ft long is to be cut into four equal pieces, find the length of each piece. (Allow 1 8 in. for each saw cut.)

Dimensions of a Room A rectangular room measures 8 ft 3 in. by 10 ft 8 in. by 9 ft 2 in. high. a) Determine the perimeter of the room in feet. Write your answer as a mixed number. b) Calculate the area of the floor of the room in square feet. c) Calculate the volume of the room in cubic feet. Use volume = length X width X height. Write your answer as a mixed number.

Hanging a Picture The back of a framed picture that is to be hung is shown above on the right. A nail is to be hammered into the wall, and the picture will be hung by the wire on the nail. a) If the center of the wire is to rest on the nail and a side of the picture is to be 20 in. from the window, how far from the window should the nail be placed? b) If the top of the frame is to be 26 1 4 in. from the ceiling, how far from the ceiling should the nail be placed? (Assume the wire will not stretch.) c) Repeat part (b) if the wire will stretch 1 4 in. when the picture is hung.

Increasing a Book Size The dimensions of the cover of a book have been increased from 8 1 2 in. by 9 1 4 in. to 8 1 2 in. by 10 1 4 in. By how many square inches has the surface area increased? Use area = length * width.

Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can find a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 109114, find a rational number between the two numbers in each pair. Many answers are possible. 0.21 and 0.22

Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can find a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 109114, find a rational number between the two numbers in each pair. Many answers are possible.4.005 and 4.05

Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can find a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 109114, find a rational number between the two numbers in each pair. Many answers are possible. -2.176 and -2.175

Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can find a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 109114, find a rational number between the two numbers in each pair. Many answers are possible. 1.3457 and 1.34571

Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can find a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 109–114, find a rational number between the two numbers in each pair. Many answers are possible. 4.872 and 4.873

Dense Set of Numbers A set of numbers is said to be a dense set if between any two distinct members of the set there exists a third distinct member of the set. The set of integers is not dense since between any two consecutive integers there is not another integer. For example, between 1 and 2 there are no other integers. The set of rational numbers is dense because between any two distinct rational numbers there exists a third distinct rational number. For example, we can find a rational number between 0.243 and 0.244. The number 0.243 can be written as 0.2430, and 0.244 can be written as 0.2440. There are many numbers between these two. Some of them are 0.2431, 0.2435, and 0.243912. In Exercises 109114, find a rational number between the two numbers in each pair. Many answers are possible.3.7896 and -3.7895

Halfway Between Two Numbers To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 115120, find a rational number halfway between the two fractions in each pair.1 4 and 3 4

Halfway Between Two Numbers To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 115120, find a rational number halfway between the two fractions in each pair.1 5 and 2 5

Halfway Between Two Numbers To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 115–120, find a rational number halfway between the two fractions in each pair. \(\frac{1}{100} \text { and } \frac{1}{10}\)

Halfway Between Two Numbers To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 115–120, find a rational number halfway between the two fractions in each pair. \(\frac{7}{13}\) and \(\frac{8}{13}\)

Halfway Between Two Numbers To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 115–120, find a rational number halfway between the two fractions in each pair. \(\frac{1}{99}\) and \(\frac{1}{9}\)

Halfway Between Two Numbers To find a rational number halfway between any two rational numbers given in fraction form, add the two numbers together and divide their sum by 2. In Exercises 115–120, find a rational number halfway between the two fractions in each pair. \(\frac{1}{2}\) and \(\frac{2}{3}\)

Cooking Oatmeal Following are the instructions given on a box of oatmeal. Determine the amount of water (or milk) and oats needed to make 1 1 2 servings by: a) Adding the amount of each ingredient needed for 1 serving to the amount needed for 2 servings and dividing by 2. b) Adding the amount of each ingredient needed for 1 serving to half the amount needed for 1 serving.

Consider the rational number 0.9. a) Use the method from Example 8 on page 234 to convert 0.9 to a quotient of integers. b) Find a number halfway between 0.9 and 1 by adding the two numbers and dividing by 2. c) Find 1 3 + 2 3 by adding the fractions. Now express 1 3 and 2 3 as repeating decimals and find the sum using the decimal representation of 1 3 and 2 3d) What conclusion can you draw from parts (a), (b), and (c)?

Paper Folding Read the Recreational Mathematics box on page 230. Next, take a clean sheet of white paper and fold it in half, fold it in half again, and fold it in half a third time. Now unfold the paper. You should have eight equal regions of the paper. a) What fraction does each region represent? b) Next, color three of the regions red and one of the regions blue, as shown in the diagram below. Use the paper to determine the following sum; 3 8 + 1 8 Write your answer in reduced form. c) Next, return the paper to its folded state and then fold the paper in half again. Use the blue region of the paper to determine the following product: 1 16 * 2. d) Use the blue region of the paper to determine the following quotient: 1 8 , 2.

The ancient Greeks are often considered the first true mathematicians. Write a report summarizing the ancient Greeks contributions to rational numbers. Include in your report what they learned and believed about the rational numbers. References include encyclopedias, history of mathematics books, and Internet Web sites.