CHAPTER 4 RATIONAL NUMBERSLesson 1: Terminating and Repeating DecimalsWrite Fractions as Decimals

Our decimal system is based on powers of 10 such as 10, 100, and 1,000. If the denominator of a fraction has a power of 10, you can use place value to write the fraction as a decimal.

Words Fraction DecimalSeven tenths 710 0.7

nineteen hundredthsone-hundred five thousandthsExample 1: Write the fraction of mixed number as a decimal. 74100

74100

=0.74

So, 74100 = 0.74Example 2: Write the fraction of mixed number as a decimal.a. 720

So, 720=0.35

b. 5 355 35=5+3

5= 5 + 0.75

So, 5 35 = 5.75

Got it? 1 & 2a. 310 b. 325 c. −6 12

Example 3:Write 38 as a decimal. Example 4:

So, 38=0.375 .

Write 79 as a decimal.Example 5:Write −140 as a decimal.

Got it? 3, 4, & 5a. −78 b. 2 18 c. −311 d. 8 13

Example 6:We can convert decimals back to fractions. Find the fraction of the fish in the aquarium that are gold fish. Write in simplest form. 0.15 = 15100

= 320So, 320 of fish are goldfish.

Got it? 6Find the fraction of the fish in the aquarium made up by each fish. Write in simplest form. a. molly b. guppy c. angelfish

Guided Practice:Write each fraction or mixed number as a decimal. Use bar notation if needed.1. 25 2. −910 3. 59

So, −140 =0.025 .

So, 79=0.777…¿

4. During a hockey game, an ice resurfacer travels 0.75 mile. What fraction represents this distance?

Journal: How can you write a fraction as a decimal?

Lesson 2: Compare and Order Rational NumbersA _______________________________ ____________________________________ is common multiple of the denominators of two or more fractions. The _________________ ________________________ _________________________________ or __________ is the LCM or the denominators. You can use the LCD to compare fractions. You can also use a number line. Example 1: Fill in the with <, >, or = to make ⍜ −1 5

6 ⍜ −1 16 a true sentence. Graph each rational number on a number line.Mark off equal-size increments of 16 between -2 and -1.

The number line shows that −1 56 < −1 16 .Got it? 1: Fill in the with <, >, or = to make ⍜ −5 5

9 ⍜ −5 19 a true sentence.

Example 2: Fill in the with <, >, or = to make ⍜ 712 ⍜ 8

18 a true sentence.The LCD is 36.

712

= 7 x312 x3

¿ 2136

818

= 8 x 218 x 2

¿ 1636

2136

> 1636So, 712> 818

Got it? 2a. 56 ⍜ 79 b. 15 ⍜ 750 c. −916 ⍜− 710

Example 3: In Mr. Huang’s class, 20% of students own roller shoes. In Mrs. Tevino’s class, 5 out of 28 students own roller shoes. In which class does a greater fraction of student own roller shoes? Express each number as a decimal and then compare.

20% = 0.2 528 = 5 28 ≈ 0.1724

Since 0.2 > 0.1724, 20% > 528 .Mr. Huang’s class has more student who own roller shoes.Got it? 3 In a second class period, 37.5% of students like to bowl. In a fifth period class, 12 out of 29 student like to bowl. In which class does a greater fraction of students like to bowl?

You can order rational numbers using place value.Example 4: Order the set {3.44, п, 3.14, 3.4} from least to greatest. Line up the decimal points and compare using place value.3.1403.14159…3.4403.4444…..

3.140 < 3.141 < 3.440 < 3.444…So the order is {3.44, п, 3.14, 3.4}Got it? 4 Order the set {23%, 0.21, 14 , 15} from least to greatest.

Example 5: Nolan is the quarterback on the football team. He completed 67% of his passes in the first game. He completed 0.64, 35 , and 68% of his passes in the next three games. List Nolan’s completed passing numbers from least to greatest. 67% = 0.67 3

5=¿0.6 68% = 0.680.6 < 0.64 < 0.67 < 0.68

35 , 0.64, 67%, 68%

Guided Practice:Fill in the with >, < or = to make the sentence true. ⍜1. −45 ⍜−15 2. 1 34 ⍜1 58

3. Elliot and Shanna are both soccer goalies. Elliot saves 3 goals out of 4. Shanna saves 7 out of 11. What has the better average, Elliot or Shanna? Explain. Journal: How can you compare two fractions?

Lesson 3: Add and Subtract Like FractionsKey Concept:Words: To add or subtract like fractions, add or subtract the ________________________ and write the result over the denominator.Examples: Algebra:

ac+bc=a+b

c,where c≠0

ac−bc=a−b

c,wherec ≠0

Example 1: Add. Write in simplest form. 59+ 29

59+ 29=5+29

¿ 79

Example 2: Add. Write in simplest form. −35

+(−15 )

−35

+(−15 )=−3+(−1)5

¿−45

Got it? 1 & 2a. 13+ 23 c. −37 +( 17 )

b. −25 +(−25 ) d. −14 + 14

Example 3: Sofia ate 35 of a cheese pizza. Jack ate 15 of a cheese pizza and 25 of a pepperoni pizza. How much pizza did Sofia and Jack eat altogether?

Got it? 3 Eduardo used fabric to make three costumes. He used 14 yard for the first, 24 yard for the second, and 34 yard for the third costume. How much fabric did Eduardo use altogether?

Example 4: Find −58 −38 .

−58

−38=−58

+(−38 )¿−5+(−3)

8

¿ −88

∨−1

Example 5: Find 58−78 .58−78=5−7

8

¿−28

∨−14

Got it? 4 & 5a. 59−29 c. −1112 −(−512 )

b. −59 −29

Example 6: About 6100 of the population of the United States lives in Florida. Another 4100 lives in Ohio. About what fraction more of the U.S. population lives in Florida than in Ohio?

6100

− 4100

= 2100= 2100∨ 150

About 150 more of the U.S. population lives in Florida than in Ohio.Guided Practice:Add or subtract. Write in simplest form.1. 35+ 15 = ____________ 2. 27+ 17=¿¿ 3. ( 58 + 1

8 )+ 38=¿¿

4. −45 −(−15 )=¿¿ 5. 515−(−114 )=¿¿ 6. 27−67=¿¿

5. Of the 50 states in the United States, 14 have an Atlantic Ocean coastline and 5 have a pacific Ocean coastline. What fraction of U.S. states have either an Atlantic Ocean or Pacific Ocean? Journal Write: What is a simple rule for adding and subtracting like fractions?

Lesson 4: Add and Subtract Unlike FractionsTo add or subtract fractions with different denominators:

Rename the fractions using the ________________________ __________________________________ ___________ or ______________________ as with like fractions If necessary, _______________________________ the sum or difference.

Example 1: Find 12+ 14 .The least common denominator is 4. Use the LCD.

12+ 14=1 x22 x2

+ 1 x14 x 1

¿ 24+ 14= 34

Got it? 1 a. 16+ 23 b. 910+(−12 )

c. 14 + 38 d. −13 +(−14 )

Example 2: Find (−34 +59 )+ 74 .

(−34 +59 )+ 74=( 59 +(−34 ))+ 74¿ 59+(−34 + 7

4 )= 59+1∨1 59Got it? 2a. 25+( 47 + 3

5 ) b. (−310 +58 )+ 2310

Example 3: Find −23 −12.

Use the LCD.−23

−12=−2 x23 x2

−1x 32x 3

¿−46−36

¿ −4−36

∨−76

∨−1 16

Got it? 3a. 58−14 b. 34−13 c. 12−(−25 )

Example 4:Use the table to find the fraction of the total population that has type A or type B blood. To find the fraction of the total population, add 2150 and 110.2150

+ 110

=21x 150x 1

+ 1 x510 x5= 2150+ 550= 2650∨1325

So, 1325 of the population has type A or type B blood.Guided Practice:Add or subtract. Write in simplest form.1. 35+ 110 = __________ 2. −56 +(49 )= __________ 3. ( 78 + 3

11 )+ 18=¿¿_________

4. 45 – 310 = __________ 5. 38 – (−14 ) = __________ 6. 34−13 = __________

7. Cassandra cuts 516 inch off the top of a photo and 38 inch off the bottom. How much shorter is the total height of the photo row?

Journal: Compare adding unlike fractions to adding like fractions.

Lesson 5: Add and Subtract Unlike FractionsTo add or subtract mixed numbers, first add or subtract the _____________________. If necessary, rename them using the _________. Then add or subtract the _____________ ____________________ and ________________________ if necessary. Sometimes when you subtract mixed numbers, the fraction in the ______________ mixed number is less than the fraction in the second mixed number. In this case, rename one or both fractions in order to subtract. Example 1: Find 7 49 +10 2

9 . Write in simplest form. Example 2: Find 856−2 13 . Write in simplest form.

Got it? 1 & 2a. 618+2 58 b. 515+2 310 c. 1 59 +4 16

d. 545−1 310 e. 13 78−9 34 f. 8 23−2 12

Example 3: Find 213−1 23 . Write in simplest form. Rename Mixed Numbers Write an Improper Fractions

Got it? 3a. 7−1 12 b. 538−4 1112 c. 11 25−2 35

d. 8−3 34 e. 3 14−1 34 f. 16−5 56

Example 4: An urban planner is designing a skateboard park. The length of the skateboard park is 12012 feet. The length of the parking lot is 40 13 feet. What will be the length of the park and the parking lot combined?120 1

2+40 1

3=120 3

6+40 2

6

¿160+ 56

¿160 56The total length is 160 56 feet.

Guided Practice:Add or subtract. Write in simplest form.1. 812+3 45 2. 7 56−3 16 3. 11−6 38

4. A hybrid car’s gas tank can hold 11 910 gallons of gasoline. It contains 834 gallons of gasoline. How much more gasoline is needed to fill the tank?

Journal: How can you subtract mixed numbers when the fraction in the first mixed number is less than the fraction in the second mixed number?

Lesson 6: Multiply FractionsWords: To multiply fractions, multiply the numerators and multiply the denominators. Examples: 12 x 23=1x 22x 3

∨26 Algebra: ab x cd= a•cb • d∨ acbd ,where b ,d ≠0

Example 1: Multiply. Write in simplest form. 12 x 1312x 13=16= 16

Example 2: Multiply. Write in simplest form. 2 x(−34 )2 x(−34 )=21 x−34 =

2(−3)1(4)= −64 =−1 1

2

Example 3: Multiply. Write in simplest form. 27 x (−38 )

1x (−3)7 x4

∨−328Got it? 1, 2, & 3a. 35 x 12 b. 23 x (−4) c. −13 x (−37 )

Example 4: Find 12 x 4 25 .Write in simplest form.

Got it? 4a. 14 x8 49 b. 5 13 x3 c. −1 78 x (−2 25 )

Example 5: Humans sleep about 13 of each day. Let each year equal 365 14 days. Determine the number of days in a year the average human sleeps.

Guided Practice:Multiply. Write in simplest form.1. 23 x 13=¿¿ 2. −14 x (−89 )=¿¿ 3. 2 14 x 23=¿¿

4. The weight of an object on Mars is about 25 its weight on Earth. How much would an 8012-pound dog weigh on Mars?

Journal: How is the process of multiplying fractions different from the process of adding fractions?

Lesson 7: Convert Between SystemsYou can multiply by fractions to convert between __________________________ and _______________________ units. The table below lists common customary and metric units.

What do you notice about customary and metric units?

Example 1: Convert 17.22 inches to centimeters. Round to the nearest hundredth if necessary.

Example 2: Convert 5 kilometers to miles. Round to the nearest hundredth if necessary.

Got it? 1 & 2Complete. Round to the nearest hundredth if necessary. a. 6 yards ≈ _____________ meters b. 1.6 centimeters ≈ _______________ inches

c. 17 meters ≈ __________________ yards

Example 3: Convert 828.5 milliliters to cups. Round to the nearest hundredth if necessary.

Example 4: Convert 3.4 quarts to milliliters. Round to the nearest hundredth if necessary.

Example 5: Convert 4.25 kilograms to pounds. Round to the nearest hundredth if necessary.

Got it? 3, 4, & 5Complete. Round to the nearest hundredth if necessary. a. 7.44 cups ≈ ________________ mL b. 22.09 lb ≈ _______________ kilogramsc. 35.85 liters ≈ __________________ gallonsExample 6: An Olympic-size swimming pool is 50 meters long. About how many feet long is the pool?

Guided Practice:Complete. Round to the nearest hundredth if necessary. 1. 3.7 yards ≈ _____________ meters 2. 11.07 pt ≈ _____________ mL 3. 650 lb ≈ _____________kg

4. About how many feet does a team of athletes run in a 1,600-meter relay race?

5. Raheem bought 3 pounds of bananas. About how many kilograms did he buy?

Journal: How can you use dimensional analysis to convert between measurement systems?

Lesson 8: Divide FractionsWords: To divide by a fraction, multiply by its multiplicative inverse, or ________________________.Examples: 78 ÷ 34=78 ∙ 43 Algebra: ab ÷ cd=a

b∙ dc,whereb , c ,∧d≠0

Dividing 3 by 14 is the same as multiplying 3 by the reciprocal of 14 , which is 4.

Example 1: Find 13 ÷5.

13÷5=1

3÷ 51

¿ 13x 15

¿ 115

Example 2: Find 34 ÷(−12 ). 34÷(−12 )= 34 x (−21 )

¿−64

¿−1 12Got it? 1 & 2a. Find 34 ÷ 14 b. Find −45 ÷ 89 c. Find −34 ÷(−23 )

Example 3: Find 23 ÷3 13 . Write in simplest form.23÷3 13=23÷ 103

¿ 23x 310

¿ 210

=15

Got it? 3a. Find 5÷1 13 b. Find −34 ÷1 12 c. Find 2 13 ÷ 5

Example 4: The side pieces of a butterfly house are 8 14 inches long. How many side pieces can be cut from a board measuring 49 12 inches long?

Guided Practice:Divide. Write in simplest form.1. 18 ÷ 13=¿¿ 2. −3÷(−67 )=¿¿ 3. −78 ÷ 34=¿¿

4. On Saturday, Lindsay walked 312 miles in 125 hours. What was her walking pace in miles per hour? Write in simplest form.

Journal: How is dividing fractions related to multiplying?