Algebra 1A Unit 04 - Woodland Hills School District 04...Algebra 1A Unit 04 Chapter 3… sections...

Algebra 1A

Unit 04 Chapter 3… sections 1-9 Chapter 4… section 7-8

GUIDED NOTES

NAME _________________________

Teacher _______________

Period ___________

1

Date: ______________________

Notes Section 3-1: Writing Equations

Writing Equations: Some verbal expressions that suggest the equals sign: Example #1: Translate each sentence into an equation. a.) Nine times y subtracted from 95 equals 37. b.) A number b divided by three is equal to six less than c. c.) Five times the number a is equal to three times the sum of b and c. Four-Step Problem-Solving Plan: 1.) 2.) 3.) 4.) Example #2: You know that 2,000,000 gallons of ice cream are produced in the United States each day. You want to know how many days it will take to produce 40,000,000 gallons of ice cream. Use the Four-Step Plan.

1

2

Example #3: A popular jellybean manufacturer produces 1,250,000 jellybeans per hour. How many hours does it take them to produce 10,000,000 jellybeans? Use the Four-Step Plan. Example #4: Translate the sentence into a formula. The perimeter of a square equals four times the length of the side. Example #5: Translate the sentence into a formula. The perimeter of a rectangle equals two times the length plus two times the width. Writing Verbal Sentences: Example #4: Translate each equation into a verbal sentence. a.) 5212 −=− x

b.) 6

32 cba =+

c.) 1453 =+m d.) 2yvw =+

2

1

Date: ______________________

Notes – Part 1 Section 3-2: Solving Equations by Using Addition and Subtraction

Solve Using Addition: ADDITION PROPERTY OF EQUALITY

:

Example #1: Solve the following equations by using the Addition Property of Equality. a.) 2948 =−m b.) 2712 −=−h c.) 2314 =+− s d.) 36.504.2 =−k Solve Using Subtraction: SUBTRACTION PROPERTY OF EQUALITY

:

Example #2: Solve the following equations by using the Subtraction Property of Equality. a.) 14297 =+d b.) 9263 =+k

c.) 8647 =+ m d.) 32

54=+y

3

2

e.) 17)4( =−−g f.) 91)(18 =−− f Solve by Adding or Subtracting: Example #3: Solve the following. a.) 36102 =+c b.) 3417 =+− t

c.) 31

52=−y d.) 32.3345.65 += a

e.) 32

71=+b f.) 39)8( −=−−k

4

1

Date: ______________________

Notes – Part 2 Section 3-2: Solving Equations by Using Addition and Subtraction

Write an equation for the following problems. Then solve the equations and check your solutions. 1.) A number increased by 5 is equal to 42. Find the number. 2.) Fourteen more than a number is equal to twenty seven. Find the number. 3.) Twenty one subtracted from a number is –8. Find the number. 4.) A number increased by –37 is –91. Find the number. 5.) The Washington Monument in Washington, D.C., was built in two phases. During the first phase, from 1848-1854, the monument was built to a height of 152 feet. From 1854 until 1878, no work was done. Then from 1878 to 1888, the additional construction resulted in its final height of 555 feet. How much of the monument was added during the second construction phase? Write an equation to solve the problem.

5

1

Date: ______________________

Notes Section 3-3: Solving Equations by Using Multiplication and Division

Solve Using Multiplication: MULTIPLICATION PROPERTY OF EQUALITY

:

Example #1: Solve the following equations by using the Multiplication Property of Equality.

a.) 78=

x b.) 53

15=

z

c.) 4

9−

=k d.)

211

412 =

g

Solve Using Division: DIVISION PROPERTY OF EQUALITY

:

Example #2: Solve the following equations by using the Division Property of Equality. a.) 213 =y b.) 842 −=− g

7

2

c.) 777 =− t d.) 3243 =f

Solve Using Multiplication or Division: Example #3: Solve the following.

a.) 43

11=

s b.) 541

833 =− k

c.) b1575 −=− d.) 14311 =w e.) 968 =− x Write and Solve an Equation: Example #4: Write an equation for the problems below. Then solve the equations.

a.) Negative fourteen times a number equals 224.

b.) One sixth times the weight on Earth equals the weight on the moon.

8

1

Date: ______________________

Notes Section 3-4: Solving Multi-Step Equations

Multi-Step Equations

:

Example #1: Solve each equation. Then check your solution. a.) 60177 =−m b.) 37135 =−q

c.) 14218

=+t d.) 119

12−=−

s

e.) 69

15−=

−p f.) 238

−=−+r

9

2

Example #2: Write an equation and solve each problem. a.) Twelve is eight plus two times a number. b.) Eight more than five times a number is negative 42. c.) Find three consecutive integers whose sum is 96. Example #3: Solve each equation. Then check your solution. a.) 2725 =+x b.) 2796 =+x c.) 51165 =+x

d.) 34814 =−n e.) 8.15.16.0 =−x f.) 10487

=−p

g.) 14

1216 −=

d h.) 1335

−=+−g i.)

8)1(74

−−−

=−x

10

1

Date: ______________________

Notes Section 3-5: Solving Equations with the Variable on Each Side

Variables on Each Side

:

Example #1: Solve each equation. Then check your solution. a.) 18102 −=+− xx b.) 2758 −=+ pp

c.) )7049(71)82(4 +=− rr d.) )72(6)1218(

31

−=+ yy

e.) mmm 3)7(552 −−=+ f.) )432(10)25(8 cc +=−

g.) 235)1(3 −=−+ rr h.) )40020(51)20(4 +=+ tt

11

2

Example #2: Write and equation and solve. The sum of one half of a number and 6 equals one third of the number. What is the number? Example #3: When exercising, a person’s pulse rate should not exceed a certain limit, which depends on his or her age. This maximum rate is represented by the expression )220(8.0 a− , where a is age in years. Find the age of a person whose maximum pulse is 152. Show all work!

12

1

Date: ______________________

Notes – Part 1 Section 3-6: Ratios and Proportions

Ratios and Proportions:

Ratio

– a comparison of ________ numbers by ___________________

The ratio of x to y can be expressed in the following ways: 1.) 2.) 3.) What is the ratio of girls to boys in this classroom? What is the ratio of teachers to students in this classroom? What is the ratio of boys to all students in this classroom? *** Remember – Simplify your ratios!!! Proportion

– an equation stating that two _________ are __________

Example #1: Determine whether the following pairs of ratios are proportional. a.)

32 and

96 b.)

54 and

3024 c.)

61 and

245

13

2

Example #2: Use cross products to determine whether each pair of ratios form a proportion. a.)

4.17.0,

8.04.0 b.)

2824,

86

Means and Extremes

If dc

ba= , then bcad = …

Example #3: Solve each proportion. a.)

1624

15=

n b.) x6

43= c.)

47.81.16.0 n=

14

1

Date: ______________________

Notes – Part 2 Section 3-6: Ratios and Proportions

Create a proportion for each situation and solve. Show all of your work! 1.) Tom earns $152 in 4 days. At that rate, how many days will it take him to earn $532? 2.) Ashely drove 248 miles in 4 hours. At that rate, how long will it take her drive an additional 93 miles? 3.) A blueprint for a house states that 2.5 inches equals 10 feet. If the length of a wall is 12 feet, how long is the wall in the blueprint? 4.) A research study shows that three out of every twenty pet owners got their pet from a breeder. Of the 122 animals cared for by a veterinarian, how many would you expect to have been bought from a breeder?

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1

Date: ______________________

Notes Section 3-7: Percent of Change

Percent of Change:

Percent of Change

– when an ________________ or ________________ is expressed as a percent

Example #1: State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. a.) original: 25 b.) original : 20 c.) original: 32 new: 28 new: 4 new: 40 Example #2: Find amount after sales tax. a.) A meal for two at a restaurant costs $32.75. If the sales tax is 7%, what is the total price of the meal? b.) A concert ticket costs $45. If the sales tax is 6.25%, what is the total price of the ticket?

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2

Example #3: Find amount after discount. a.) A dog toy is on sale for 20% off the original price. If the original price of the toy is $3.80, what is the discounted price? b.) A sweater is on sale for 35% off the original price. If the original price of the sweater is $38, what is the discounted price? Example #4: Find amount after discount and sales tax. A shirt costs $45. It is discounted for 20%, and has a sales tax of 5.5%. What is the final sale price of the shirt? Example #5: The National Football League’s (NFL) fields are 120 yards long. The Canadian Football League’s (CFL) fields are 25% longer. What is the length of a CFL field?

18

1

Date: ______________________

Notes Section 3-8: Solving Equations and Formulas

Some equations contain more than one variable. At times, you will need to solve these equations for one of the variables.

Solve For Variables:

Example #1: Solve the following equations for the variable specified. a.) 743 =− yx , for y b.) 52 +=− smtm , for m

c.) )( cbap += , for a d.) cay=

+3

, for y

Many real-world problems require the use of formulas. Sometimes solving a formula for a specific variable will help you solve the problem.

Use Formulas:

Example #2: The formula for the circumference of a circle is C = 2π r, where C represents circumference and r represents radius.

a.) Solve the formula for r.

b.) Find the radius if the circumference is 9.5 inches.

19

Algebra 1 Name ________________________

Cost, Income, and Value Notes Date ________________ Pd ______

Steps to solving Cost, Income, Value Problems:

1.

2.

3.

4.

5.

Example 1: Tickets for a concert cost $8 for adults and $4 for students. A total of 920

tickets worth $5760 were sold. How many adult tickets were sold?

Number X Price = Cost

Adult a

Student

Example 2: Tickets for the senior class play cost $6 for adults and $3 for students. A

total of 846 tickets worth $3846 were sold. How many student tickets were sold?


Adult

Student s

20

Example 3: An apple sells for 25 cents and a peach cells for 15 cents. A total of 10 pieces

of fruit were sold for a total cost of $2.10. How many apples were sold?

Number X Price per Fruit = Cost

Apple a

Peach

Example 4: Coria and Kip went to the record store during its sale. Together they spent

$38.50. If each record cost $3.50 and Kip bought one more than Coria, how many

records did each buy?


Coria r

Kip

21

Algebra 1 Name ________________________

Mixture Notes Date ________________ Pd ______

Steps to solving Mixture Problems:

1.

2.

3.

4.

5.

Example 1: A health food store sells a mixture of raisins and roasted nuts. Raisins sell for $4

/ kg and nuts sell for $6 / kg. How many kg of each should be mixed to make 40 kg of this

snack worth $4.75 / kg?

Number of kg X Price per kg = Cost

Raisins c

Nuts

Mixture

Example 2: A grocer makes a natural breakfast cereal by missing oat cereal costing $2 / kg

with dried fruits costing $9 / kg. How many kg of each are needed to make 60 kg of cereal

costing $3.75 / kg?


Cereal c

Fruit

Mixture

22

Example 3: A chemist has 60 mL of a solution that is 70% acid. How much water should be

added to make a solution that is 40% acid?

Total Amount X Percent Acid = Amount of Acid

Original Solution

Water w

New Solution

Example 4: An auto mechanic has 300 mL of battery acid solution that is 60% acid. He must

add water to this solution to dilute it so that it is only 45% acid. How much water should he

add?


Original Solution

Water w

New Solution

23

Algebra 1 Name ________________________

Rate – Time – Distance Notes Date ________________ Pd ______

Steps to solving Rate – Time – Distance Problems:

1.

2.

3.

4.

5.

Example 1: Two jets leave St. Louis at 8 am one flying east at a speed 40 km / h greater than

the other, which is traveling west. At 10 am the planes are 2480 km apart. Find their speeds.

Rate X Time = Distance

East

West r

Example 2: Bicyclists Brent and Jane started at noon from points 60 km apart and rode

toward each other, meeting at 1:30 pm. Brent’s speed was 4 km / h greater than Jane’s speed.

Find their speeds.


Brent

Jane r

24

Example 3: A helicopter leaves Central Airport and flies north at 180 mph. Twenty minutes

later a plane leaves the airport and follows the helicopter at 330 mph. How long does it take

the plane to overtake the helicopter?


Helicopter

Plane p

Example 4: A ski lift carried Marie up a slope at the rate of 6 km / h, and she skied back down

parallel to the lift at 34 km / h. The round trip took 30 minutes. How far did she ski and for

how long?


Up

Down S

25

Study Guide and Intervention

Writing Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

© Glencoe/McGraw-Hill 137 Glencoe Algebra 1

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on

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Write Equations Writing equations is one strategy for solving problems. You can use avariable to represent an unspecified number or measure referred to in a problem. Then youcan write a verbal expression as an algebraic expression.

Translate eachsentence into an equation or aformula.

a. Ten times a number x is equal to2.8 times the difference y minus z.

10 3 x 5 2.8 3 ( y 2 z)The equation is 10x 5 2.8( y 2 z).

b. A number m minus 8 is the sameas a number n divided by 2.

m 2 8 5 n 4 2The equation is m 2 8 5 .

c. The area of a rectangle equals thelength times the width. Translatethis sentence into a formula.

Let A 5 area, , 5 length, and w 5 width.Formula: Area equals length times

width.

A 5 , 3 w

The formula for the area of arectangle is A 5 ,w.

n}2

Use the Four-Step Problem-Solving Plan.The population of the United States in 2001was about 284,000,000, and the land area ofthe United States is about 3,500,000 squaremiles. Find the average number of peopleper square mile in the United States.Source: www.census.gov

Step 1 Explore You know that there are284,000,000 people. You want to knowthe number of people per square mile.

Step 2 Plan Write an equation to represent thesituation. Let p represent the number ofpeople per square mile.

3,500,000 3 p 5 284,000,000

Step 3 Solve 3,500,000 3 p 5 284,000,000.

3,500,000p 5 284,000,000 Divide each side by

p < 81.14 3,500,000.

There about 81 people per square mile.

Step 4 Examine If there are 81 people persquare mile and there are 3,500,000square miles, 81 3 3,500,000 5283,500,000, or about 284,000,000 people.The answer makes sense.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Translate each sentence into an equation or formula.

1. Three times a number t minus twelve equals forty.

2. One-half of the difference of a and b is 54.

3. Three times the sum of d and 4 is 32.

4. The area A of a circle is the product of p and the radius r squared.

WEIGHT LOSS For Exercises 5–6, use the following information.

Lou wants to lose weight to audition for a part in a play. He weighs 160 pounds now. Hewants to weigh 150 pounds.

5. If p represents the number of pounds he wants to lose, write an equation to representthis situation.

6. How many pounds does he need to lose to reach his goal?

26

Skills Practice

Writing Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


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on

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Translate each sentence into an equation.

1. Two added to three times a number m is the same as 18.

2. Twice a increased by the cube of a equals b.

3. Seven less than the sum of p and q is as much as 6.

4. The sum of x and its square is equal to y times z.

5. Four times the sum of f and g is identical to six times g.

Translate each sentence into a formula.

6. The perimeter P of a square equals four times the length of a side s.

7. The area A of a square is the length of a side s squared.

8. The perimeter P of a triangle is equal to the sum of the lengths of sides a, b, and c.

9. The area A of a circle is pi times the radius r squared.

10. The volume V of a rectangular prism equals the product of the length ,, the width w,and the height h.

Translate each equation into a verbal sentence.

11. g 1 10 5 3g 12. 2p 1 4q 5 20

13. 4(a 1 b) 5 9a 14. 8 2 6x 5 4 1 2x

15. ( f 1 y) 5 f 2 5 16. s2 2 n2 5 2b

Write a problem based on the given information.

17. c 5 cost per pound of plain coffee beans 18. p 5 cost of dinnerc 1 3 5 cost per pound of flavored coffee beans 0.15p 5 cost of a 15% tip2c 1 (c 1 3) 5 21 p 1 0.15p 5 23

1}2

27


Solve Using Subtraction If the same number is subtracted from each side of anequation, the resulting equation is equivalent to the original one. In general if the originalequation involves addition, this property will help you solve the equation.

Subtraction Property of Equality For any numbers a, b, and c, if a 5 b, then a 2 c 5 b 2 c.

Solve 22 1 p 5 212.

22 1 p 5 212 Original equation

22 1 p 2 22 5 212 2 22 Subtract 22 from each side.

p 5 234 Simplify.

The solution is 234.

Solve each equation. Then check your solution.

1. x 1 12 5 6 2. z 1 2 5 213 3. 217 5 b 1 4

4. s 1 (29) 5 7 5. 23.2 5 , 1 (20.2) 6. 2 1 x 5

7. 19 1 h 5 24 8. 212 5 k 1 24 9. j 1 1.2 5 2.8

10. b 1 80 5 280 11. m 1 (28) 5 2 12. w 1 5

Write an equation for each problem. Then solve the equation and check thesolution.

13. Twelve added to a number equals 18. Find the number.

14. What number increased by 20 equals 210?

15. The sum of a number and fifty equals eighty. Find the number.

16. What number plus one-half is equal to four?

17. The sum of a number and 3 is equal to 215. What is the number?

5}8

3}2

5}8

3}8

Study Guide and Intervention (continued)

Solving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

ExampleExample

ExercisesExercises

28

Skills Practice

Solving Equations by Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


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on

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1. y 2 7 5 8 2. w 1 14 5 28

3. p 2 4 5 6 4. 213 5 5 1 x

5. 98 5 b 1 34 6. y 2 32 5 21

7. s 1 (228) 5 0 8. y 1 (210) 5 6

9. 21 5 s 1 (219) 10. j 2 (217) 5 36

11. 14 5 d 1 (210) 12. u 1 (25) 5 215

13. 11 5 216 1 y 14. c 2 (23) 5 100

15. 47 5 w 2 (28) 16. x 2 (274) 5 222

17. 4 2 (2h) 5 68 18. 256 5 20 2 (2e)

Write an equation for each problem. Then solve the equation and check yoursolution.

19. A number decreased by 14 is 246. Find the number.

20. Thirteen subtracted from a number is 25. Find the number.

21. The sum of a number and 67 is equal to 234. Find the number.

22. What number minus 28 equals 22?

23. A number plus 273 is equal to 27. What is the number?

24. A number plus 217 equals 21. Find the number.

25. What number less 5 is equal to 239?

29


Solve Using Division To solve equations with multiplication and division, you can alsouse the Division Property of Equality. If each side of an equation is divided by the samenumber, the resulting equation is true.

Division Property of Equality For any numbers a, b, and c, with c Þ 0, if a 5 b, then 5 .b}c

a}c


Solving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Solve 8n 5 64.

8n 5 64 Original equation

5 Divide each side by 8.

n 5 8 Simplify.

The solution is 8.

64}8

8n}8

Solve 25n 5 60.

25n 5 60 Original equation


n 5 212 Simplify.

The solution is 212.

60}25

25n}25

Example 1Example 1 Example 2Example 2

ExercisesExercises


1. 3h 5 242 2. 8m 5 16 3. 23t 5 51

4. 23r 5 224 5. 8k 5 264 6. 22m 5 16

7. 12h 5 4 8. 22.4p 5 7.2 9. 0.5j 5 5

10. 225 5 5m 11. 6m 5 15 12. 21.5p 5 275

Write an equation for each problem. Then solve the equation.

13. Four times a number equals 64. Find the number.

14. What number multiplied by 24 equals 216?

15. A number times eight equals 236. Find the number.

30

Skills Practice

Solving Equations by Using Multiplication and Division

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


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on

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1. 12z 5 108 2. 27t 5 49

3. 18e 5 2216 4. 222 5 11v

5. 26d 5 242 6. 96 5 224a

7. 5 16 8. 5 9

9. 284 5 10. 2 5 213

11. 5 213 12. 31 5 2 n

13. 26 5 z 14. q 5 24

15. p 5 210 16. 5

17. 20.4b 5 5.2 18. 1.6m 5 24

Write an equation for each problem. Then solve the equation.

19. The opposite of a number is 29. What is the number?

20. Fourteen times a number is 242. Find the number.

21. Eight times a number equals 128. What is the number?

22. Negative twelve times a number equals 2132. Find the number.

23. Negative eighteen times a number is 254. What is the number?

24. One sixth of a number is 217. Find the number.

25. Negative three fifths of a number is 215. What is the number?

2}5

a}10

5}9

2}7

2}3

1}6

t}4

d}7

d}3

a}16

c}4

31


Solve Multi-Step Equations To solve equations with more than one operation, oftencalled multi-step equations, undo operations by working backward. Reverse the usualorder of operations as you work.

Solve 5x 1 3 5 23.

5x 1 3 5 23 Original equation.

5x 1 3 2 3 5 23 2 3 Subtract 3 from each side.

5x 5 20 Simplify.


x 5 4 Simplify.


1. 5x 1 2 5 27 2. 6x 1 9 5 27 3. 5x 1 16 5 51

4. 14n 2 8 5 34 5. 0.6x 2 1.5 5 1.8 6. p 2 4 5 10

7. 16 5 8. 8 1 5 13 9. 1 3 5 213

10. 5 10 11. 0.2x 2 8 5 22 12. 3.2y 2 1.8 5 3

13. 24 5 14. 8 5 212 1 15. 0 5 10y 2 40

Write an equation and solve each problem.

16. Find three consecutive integers whose sum is 96.

17. Find two consecutive odd integers whose sum is 176.


k}24

7x 2 (21)}}

28

4b 1 8}

22

g}25

3n}12

d 2 12}

14

7}8

20}5

5x}5


Solving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

ExampleExample

ExercisesExercises

32

Skills Practice

Solving Multi-Step Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4


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on

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Solve each problem by working backward.

1. A number is divided by 2, and then the quotient is added to 8. The result is 33. Find thenumber.

2. Two is subtracted from a number, and then the difference is divided by 3. The result is30. Find the number.

3. A number is multiplied by 2, and then the product is added to 9. The result is 49. Whatis the number?

4. ALLOWANCE After Ricardo received his allowance for the week, he went to the mallwith some friends. He spent half of his allowance on a new paperback book. Then hebought himself a snack for $1.25. When he arrived home, he had $5.00 left. How muchwas his allowance?


5. 5x 1 3 5 23 6. 4 5 3a 2 14 7. 2y 1 5 5 19

8. 6 1 5c 5 229 9. 8 2 5w 5 237 10. 18 2 4v 5 42

11. 2 8 5 22 12. 5 1 5 1 13. 2 2 4 5 13

14. 2 1 12 5 27 15. 2 2 5 9 16. 1 3 5 21

17. q 2 7 5 8 18. g 1 6 5 212 19. z 2 8 5 23

20. m 1 2 5 6 21. 5 3 22. 5 2

Write an equation and solve each problem.

23. Twice a number plus four equals 6. What is the number?

24. Sixteen is seven plus three times a number. Find the number.

25. Find two consecutive integers whose sum is 35.


b 1 1}

3

c 2 5}

4

4}5

5}2

2}3

3}4

w}7

a}5

d}6

h}3

x}4

n}3

33


Solve Proportions If a proportion involves a variable, you can use cross products to solve

the proportion. In the proportion 5 , x and 13 are called extremes and 5 and 10 are

called means. In a proportion, the product of the extremes is equal to the product of the means.

Means-Extremes Property of Proportions For any numbers a, b, c, and d, if 5 , then ad 5 bc.

Solve 5 .

5 Original proportion

13(x) 5 5(10) Cross products

13x 5 50 Simplify.


x 5 3 Simplify.

The solution is 3 .

Solve each proportion.

1. 5 2. 5 3. 5

4. 5 5. 5 6. 5

7. 5 8. 5 9. 5

10. 5 11. 5 12. 5

13. 5 14. 5 15. 5

Use a proportion to solve each problem.

16. MODELS To make a model of the Guadeloupe River bed, Hermie used 1 inch of clay for5 miles of the river’s actual length. His model river was 50 inches long. How long is theGuadeloupe River?

17. EDUCATION Josh finished 24 math problems in one hour. At that rate, how manyhours will it take him to complete 72 problems?

12}9

2 1 w}

6

24}k

12}k

15}3

a 2 8}

12

2y}8

3 1 y}

4

12}x

1.5}

x

4}12

4}b 2 2

p}24

5}8

18}3

3}d

18}54

9}y 1 1

3}63

x}21

8}x

4}6

3}4

x 1 1}

4

0.5}

x

0.1}2

5}3

1}t

2}8

23}

x

11}13

11}13

50}13

13x}13

10}13

x}5

10}13

x}5

c}d

a}b

10}13

x}5


Ratios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6

ExampleExample

ExercisesExercises

34

Skills Practice

Ratios and Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

3-63-6


Less

on

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Use cross products to determine whether each pair of ratios forms a proportion.Write yes or no.

1. , 2. ,

3. , 4. ,

5. , 6. ,

7. , 8. ,

Solve each proportion. If necessary, round to the nearest hundredth.

9. 5 10. 5

11. 5 12. 5

13. 5 14. 5

15. 5 16. 5

17. 5 18. 5

19. 5 20. 5

21. 5 22. 5

23. 5 24. 5

25. BOATING Hue’s boat used 5 gallons of gasoline in 4 hours. At this rate, how manygallons of gasoline will the boat use in 10 hours?

11}30

22}x

s}84

4}21

20}g

5}12

27}39

9}c

n}60

11}15

30}m

10}14

1}9

7}b

6}f

42}56

y}69

6}23

36}s

12}7

35}21

5}e

3}5

6}z

1}6

3}a

15}10

9}g

3}9

5}b

2}14

1}a

50}85

12}17

21}98

3}14

26}38

13}19

42}90

7}16

72}81

8}9

24}28

6}7

7}11

5}9

20}25

4}5

35


Solve Problems Discounted prices and prices including tax are applications of percentof change. Discount is the amount by which the regular price of an item is reduced. Thus,the discounted price is an example of percent of decrease. Sales tax is amount that is addedto the cost of an item, so the price including tax is an example of percent of increase.

A coat is on sale for 25% off the original price. If the original priceof the coat is $75, what is the discounted price?

The discount is 25% of the original price.

25% of $75 5 0.25 3 75 25% 5 0.25

5 18.75 Use a calculator.

Subtract $18.75 from the original price.

$75 2 $18.75 5 $56.25

The discounted price of the coat is $56.25.

Find the final price of each item. When a discount and a sales tax are listed,compute the discount price before computing the tax.

1. Compact disc: $16 2. Two concert tickets: $28 3. Airline ticket: $248.00Discount: 15% Student discount: 28% Superair discount: 33%

4. Shirt: $24.00 5. CD player: $142.00 6. Celebrity calendar: $10.95Sales tax: 4% Sales tax: 5.5% Sales tax: 7.5%

7. Class ring: $89.00 8. Software: $44.00 9. Video recorder: $110.95Group discount: 17% Discount: 21% Discount: 20%Sales tax: 5% Sales tax: 6% Sales tax: 5%

10. VIDEOS The original selling price of a new sports video was $65.00. Due to the demandthe price was increased to $87.75. What was the percent of increase over the originalprice?

11. SCHOOL A high school paper increased its sales by 75% when it ran an issue featuringa contest to win a class party. Before the contest issue, 10% of the school’s 800 studentsbought the paper. How many students bought the contest issue?

12. BASEBALL Baseball tickets cost $15 for general admission or $20 for box seats. Thesales tax on each ticket is 8%, and the municipal tax on each ticket is an additional 10%of the base price. What is the final cost of each type of ticket?


Percent of Change

NAME ______________________________________________ DATE ____________ PERIOD _____

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Percent of Change

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State whether each percent of change is a percent of increase or a percent ofdecrease. Then find each percent of change. Round to the nearest whole percent.

1. original: 25 2. original: 50new: 10 new: 75




Find the total price of each item.

9. dress: $69.00 10. binder: $14.50tax: 5% tax: 7%

11. hardcover book: $28.95 12. groceries: $47.52tax: 6% tax: 3%

13. filler paper: $6.00 14. shoes: $65.00tax: 6.5% tax: 4%

15. basketball: $17.00 16. concert tickets: $48.00tax: 6% tax: 7.5%

Find the discounted price of each item.

17. backpack: $56.25 18. monitor: $150.00discount: 20% discount: 50%

19. CD: $15.99 20. shirt: $25.50discount: 20% discount: 40%

21. sleeping bag: $125 22. coffee maker: $102.00discount: 25% discount: 45%

37


Use Formulas Many real-world problems require the use of formulas. Sometimes solvinga formula for a specified variable will help solve the problem.

The formula C 5 pd represents the circumference of a circle, or thedistance around the circle, where d is the diameter. If an airplane could fly aroundEarth at the equator without stopping, it would have traveled about 24,900 miles.Find the diameter of Earth.

C 5 pd Given formula

d 5 Solve for d.

d 5 Use p 5 3.14.

d < 7930 Simplify.

The diameter of Earth is about 7930 miles.

1. GEOMETRY The volume of a cylinder V is given by the formula V 5 pr2h, where r isthe radius and h is the height.

a. Solve the formula for h.

b. Find the height of a cylinder with volume 2500p feet and radius 10 feet.

2. WATER PRESSURE The water pressure on a submerged object is given by P 5 64d,where P is the pressure in pounds per square foot, and d is the depth of the object in feet.

a. Solve the formula for d.

b. Find the depth of a submerged object if the pressure is 672 pounds per square foot.

3. GRAPHS The equation of a line containing the points (a, 0) and (0, b) is given by the

formula 1 5 1.

a. Solve the equation for y.

b. Suppose the line contains the points (4, 0), and (0, 22). If x 5 3, find y.

4. GEOMETRY The surface area of a rectangular solid is given by the formula S 5 2,w 1 2,h 1 2wh, where , 5 length, w 5 width, and h 5 height.

a. Solve the formula for h.

b. The surface area of a rectangular solid with length 6 centimeters and width 3 centimeters is 72 square centimeters. Find the height.

y}b

x}a

24,900}

3.14

C}p


Solving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

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Skills Practice

Solving Equations and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

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Solve each equation or formula for the variable specified.

1. 7t 5 x, for t 2. e 5 wp, for p

3. q 2 r 5 r, for r 4. 4m 2 n 5 m, for m

5. 7a 2 b 5 15a, for a 6. 25c 1 d 5 2c, for c

7. x 2 2y 5 1, for y 8. m 1 3n 5 1, for n

9. 7f 1 g 5 5, for f 10. ax 2 c 5 b, for x

11. rt 2 2n 5 y, for t 12. bc 1 3g 5 2k, for c

13. kn 1 4f 5 9v, for n 14. 8c 1 6j 5 5p, for c

15. 5 d, for x 16. 5 d, for c

17. 5 q, for p 18. 5 a, for b

Write an equation and solve for the variable specified.

19. Five more than a number g is six less than twice a number h. Solve for g.

20. One fourth of a number q is three more than three times a number w. Solve for q.

21. Eight less than a number s is three more than four times a number t. Solve for s.

b 2 4z}

7

p 1 9}

5

x 2 c}

2

x 2 c}

2

39


Uniform Motion Problems Motion problems are another application of weightedaverages. Uniform motion problems are problems where an object moves at a certainspeed, or rate. Use the formula d 5 rt to solve these problems, where d is the distance, r isthe rate, and t is the time.

Bill Gutierrez drove at a speed of 65 miles per hour on anexpressway for 2 hours. He then drove for 1.5 hours at a speed of 45 miles perhour on a state highway. What was his average speed?

M 5 Definition of weighted average

< 56.4 Simplify.

Bill drove at an average speed of about 56.4 miles per hour.

1. TRAVEL Mr. Anders and Ms. Rich each drove home from a business meeting. Mr. Anderstraveled east at 100 kilometers per hour and Ms. Rich traveled west at 80 kilometers perhours. In how many hours were they 100 kilometers apart.

2. AIRPLANES An airplane flies 750 miles due west in 1 hours and 750 miles due south

in 2 hours. What is the average speed of the airplane?

3. TRACK Sprinter A runs 100 meters in 15 seconds, while sprinter B starts 1.5 secondslater and runs 100 meters in 14 seconds. If each of them runs at a constant rate, who isfurther in 10 seconds after the start of the race? Explain.

4. TRAINS An express train travels 90 kilometers per hour from Smallville to Megatown.A local train takes 2.5 hours longer to travel the same distance at 50 kilometers perhour. How far apart are Smallville and Megatown?

5. CYCLING Two cyclists begin traveling in the same direction on the same bike path. Onetravels at 15 miles per hour, and the other travels at 12 miles per hour. When will thecyclists be 10 miles apart?

6. TRAINS Two trains leave Chicago, one traveling east at 30 miles per hour and onetraveling west at 40 miles per hour. When will the trains be 210 miles apart?

1}2

65 ? 2 1 45 ? 1.5}}

2 1 1.5


Weighted Averages

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Weighted Averages

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SEASONING For Exercises 1–4, use the following information.

A health food store sells seasoning blends in bulk. One blend contains 20% basil. Sheilawants to add pure basil to some 20% blend to make 16 ounces of her own 30% blend. Let brepresent the amount of basil Sheila should add to the 20% blend.

1. Complete the table representing the problem.

Ounces Amount of Basil

20% Basil Blend

100% Basil

30% Basil Blend

2. Write an equation to represent the problem.

3. How many ounces of basil should Sheila use to make the 30% blend?

4. How many ounces of the 20% blend should she use?

HIKING For Exercises 5–7, use the following information.

At 7:00 A.M., two groups of hikers begin 21 miles apart and head toward each other. The firstgroup, hiking at an average rate of 1.5 miles per hour, carries tents, sleeping bags, andcooking equipment. The second group, hiking at an average rate of 2 miles per hour, carriesfood and water. Let t represent the hiking time.

5. Copy and complete the table representing the problem.

r t d 5 rt

First group of hikers

Second group of hikers

6. Write an equation using t that describes the distances traveled.

7. How long will it be until the two groups of hikers meet?

SALES For Exercises 8 and 9, use the following information.

Sergio sells a mixture of Virginia peanuts and Spanish peanuts for $3.40 per pound. Tomake the mixture, he uses Virginia peanuts that cost $3.50 per pound and Spanish peanutsthat cost $3.00 per pound. He mixes 10 pounds at a time.

8. How many pounds of Virginia peanuts does Sergio use?

9. How many pounds of Spanish peanuts does Sergio use?

41

Algebra 1 Name ________________________

COST, INCOME, AND VALUE PROBLEMS

Solve by completing the chart. Show all work. 1. Forty students bought caps at the baseball game. Plain caps cost $4 and deluxe ones cost $6 each. If the total

bill was $236, how many students bought the deluxe cap?


Deluxe d

Plain

2. Adult tickets for the game cost $6 each and student tickets cost 43 each. A total of 1040 tickets worth $5400

were sold. How many student tickets were sold?


Adult

Student s

3. A collection of 60 dimes and nickels is worth $4.80. How many dimes are there?

Number X Value of Coin = Total Value

Dimes d

Nickels

4. A collection of 54 dimes and nickels is worth $3.80. How many nickels are there?


Dimes

Nickels n

42

5. Henry paid $.80 for each bag of peanuts. He sold all but 20 of them for $1.50 and made a profit of $54. How

many bags did he buy? (Hint: Profit = selling price – buying price.)


Bought b

Sold

6. Paula paid $4 for each stadium cushion. She sold all but 12 of them for $8 each and made a profit of $400. How

many cushions did she buy? (Hint: Profit = selling price – buying price.)


Bought b

Sold

7. I have three times as many dimes as quarters. If the coins are worth $6.60, how many quarters are there?


Dimes

Quarters q

8. I have 12 more nickels than quarters. If the coins are worth $5.40, how many nickels are there?


Quarters

Nickels n

43

Algebra 1 Name ________________________

MIXTURE PROBLEMS

Solve by completing the chart. Show all work. 1. The owner of a specialty food store wants to mix cashews selling at $8.00 / kg and pecans selling at $6.00 / kg. How

many kilograms of each should be mixed to get 12 kg of nuts worth $7.50 / kg?


Cashews c

Pecans

Mixture

2. A grocer mixed 12 pounds of egg noodles costing $.80 / lb with 3 lbs of spinach noodles costing $1.20 / lb. What will

the cost of the mixture be?


Egg Noodles

Spinach Noodles

Mixture c

3. A special tea blend is made from two varieties of herbal tea, one that costs $4.00 / kg and another that costs $2.00 / kg.

How many kilograms of each type are needed to make 20 kg of blend worth $2.50 / kg?


Tea 1 t

Tea 2

Mixture

44

4. A grocer has two kinds of nuts. One costs $5 / kg and another cots $4.20 / kg. How many kilograms of each type of

nut should be mixed in order to get 60 kg of mixture worth $4.80 / kg?


Nut 1 n

Nut 2

Mixture

5. A chemist has 80 mL of a solution that is 70% salt. How much water should he add to make a solution that is 40%

salt?


Original Solution

Water w

New Solution

6. If 800 mL of a juice drink is 10% grape juice, how much grape juice should be added to make a drink that is 20% grape

juice?


Original Solution g

Water

New Solution

45

7. How many liters of water must be added to 70 L of a 40% acid solution in order to produce a 28% acid solution?


Original Solution

Water w

New Solution

8. How many mL of pure water must be added to 60 mL of a 20% salt solution to make a 12% salt solution?


Original Solution

Water w

New Solution

9. A nurse has 100 mL of a solution that is 10% salt. How much sterile water must be added to make an 8%

salt solution?


Original Solution

Water w

New Solution

46

Algebra 1 Name ________________________

RATE TIME DISTANCE PROBLEMS

Solve by completing the chart. Show all work. 1. Two jets leave Ontario at the same time, one flying east at a speed of 20km / h greater than the other, which is flying

west. After 4 hours, the planes are 6000 km apart. Find their speeds.


East Bound

West Bound r

2. Two camper vans leave Arrowhead Lake at the same time, one traveling north at a speed of 10 km / h faster than the

other, which is traveling south. After 3 hours, the camper vans are 420 km apart. Find their speeds


South Bound r

North Bound

3. Two cars traveled in opposite directions from the same starting point. The rate of one car was 10 km less than the rate

of the other. After 4 hours the cars were 600 km apart. Find the rate of each car.


South Bound r

North Bound

4. A car started out from Memphis toward Little Rock at the rate of 60 km / h. A second car left from the same point 2

hours later and drove along the same rout at 75 km / h. How long did it take the second car to overtake the first car?


Car 1

Car 2 t

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5. A tourist bus leaves Richmond at 1:00 pm fro New York City. Exactly 24 minutes later, at truck sets out in the same

direction. The tourist bus moves at a steady 60 km/ h. the truck travels at 80 km / h. How long does it take the truck to

overtake the tourist bus?


Bus

Truck t

6. Exactly 20 minutes after Alex left home, his sister, Alison set out to overtake him. Alex drove at 48 mph and Alison

drove at 54 mph. How long did it take Alison to overtake Alex?


Alex

Alison t

7. The McLeans drove from their house in Dayton at 75 km / h. When they returned, the traffic was heavier and they

drove at 50 km / h. If it took them 1 hour longer to return than to go, how long did it take them to drive home?


To Go

Come Home t

8. It takes a plane one hour less to fly from San Diego to New Orleans at 600 km / h than it does to return at 450 km / h.

How far apart are the cities?


San Diego d

New Orleans

48

Algebra 1A Unit 04 - Woodland Hills School District 04...Algebra 1A Unit 04 Chapter 3… sections...

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