Slide 3 Slide 33 Slide 48 Slide 75 Slide 90 Slide 98 Table of Contents Basic Number Problems Number...

Table of Contents

Basic Number Problems

Number and Money Problems

Age and Digit Problems

Mixture Problems

Motion (D=RT) Problems

Test Review Problems

Test Review

Systems of EquationsSystems of Equationschapter 6chapter 6

Word Problems:Word Problems:

MoneyMoney

Coins I tem Cost

Comparison

1 taco1 milk

Total $2.10

1 taco1 milk

Total $2.10

2 taco3 milk

Total $5.15

2 taco3 milk

Total $5.15

Find the cost of a tacoFind the cost of a tacoFind the cost of a tacoFind the cost of a taco

t cost of a tacom cost of a milk

Define variables:

1 taco1 milk

Total $2.10

1 taco1 milk

Total $2.10

2 taco3 milk

Total $5.15

2 taco3 milk

Total $5.15

t m 2.10

2t 3m 5.15

Write two equations:

t cost of a tacom cost of a milk

Solve the system

t m 2.10

2t 3m 5.15 3t 3m 6.30

2t 3m 5.15

t 1.15

taco $1.15

1 taco1 milk

Total $2.10

1 taco1 milk

Total $2.10

2 taco3 milk

Total $5.15

2 taco3 milk

Total $5.15

t 1.15

T cost of a tacoM cost of a milk

4r 5a4( 3.56)

Four Oranges and five apples cost $3.56. Three oranges and four apples cost $2.76. Find the cost of an orange.

Define variables:

Let r be the cost of an orange

Write two equations

Solve the system

An orange is $0.44

Let a be the cost of an Apple

4r 5a 3.56

3r 4a 2.76

3r 4a5( 2.76)

16r 20a 14.24 15r 20a 13.80

r 0.44

A jar of dimes and quarters contains $15.25. There are 103 coins in all. How many quarters are in the jar?

Ex. #3Ex. #3Ex. #3Ex. #3

number of quarq tersnumber of dd imes

Define variables:

Ex. #3Ex. #3Ex. #3Ex. #3

q d 103

Define variables:

Write two equations 0.25q 0.10d 15.25

Ex. #3Ex. #3Ex. #3Ex. #3

q d 103

0.25q 0.10d 15.25

15q 495q 33

33 Quarters

Define variables:

25q 10d 1525

10q 10d 1030

Combined, Peyton and Eli have $106.75. Peyton has $43.75 more than Eli. How much money does Peyton have?

Ex. #4Ex. #4Ex. #4Ex. #4

p e 106.75

Peyton's mp oney Eli'se money

p e 43.75

Peyton has $75.25Peyton has $75.25

2p 150.50p 75.25

Define variables:

Write two equations

Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems#1

At a football game, a popcorn and a soda purchased together costs $4.00. Three popcorns and five sodas would cost $16.50. What is the cost of a single soda?

Define variablesLet pp be the cost of a popcorn

Let s be the cost of a soda

Write two equationsE1

p s 4.00 3p 5s 16.50

At a football game, a popcorn and a soda purchased together costs $4.00. Three popcorns and five sodas would cost $16.50. What is the cost of a single soda?

3p 3s 12.00 3p 5s 16.50

2s 4.50s 2.25Soda $2.25

Let pp be the cost of a popcorn

Let s be the cost of a soda

p s 4.00 3p 5s 16.50

Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems#2#2

Four apples and five bananas cost $3.75. Six apples and two bananas cost $2.82. What is the cost of a single banana?

Define variablesLet aa be the cost of 1 apple

Let b be the cost of 1 banana

Write two equationsE1

4a 5b 3.75

6a 2b 2.82

Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems

Four apples and five bananas cost $3.75. Six apples and two bananas cost $2.82. What is the cost of a single banana?

Let aa be the cost of 1 apple

Let b be the cost of 1 banana

4a 5b 3.75

6a 2b 2.82 24a 30b 22.50

24a 8b 11.28

22b 11.22b 0.51Banana $0.51

d q 113

Define variables:Let dd be the number of dimes

Let q be the number of quarters

Write two equations

A vending machine takes only dimes and quarters. There are 113 coins in the machine totaling $17.60. How many quarters are in the machine?

0.10d 0.25q 17.60E1

d q 113

A vending machine takes only dimes and quarters. There are 113 coins in the machine totaling $17.60. How many quarters are in the machine?

0.10d 0.25q 17.60 10d 25q 1760

10d 10q 1130

15q 630q 4242 Quarters

Let dd be the number of dimes

Let q be the number of quarters

There are 40 coins in Jenny’s coin purse – all dimes and nickels. All together it adds to $2.65. How many nickels are in Jenny’s purse?

d n 40

0.10d 0.05n 2.65

Define variables:Let dd be the number of dimes

Let n be the number of nickels

Write two equations

There are 40 coins in Jenny’s coin purse – all dimes and nickels. All together it adds to $2.65. How many nickels are in Jenny’s purse?

d n 40

0.10d 0.05n 2.65 10d 5n 265

10d 10n 400

5n 135n 2727 Nickels

Let dd be the number of dimes

Let n be the number of nickels

L B 62.75

Combined, Bart and Lisa have $62.75. Lisa has $13.75 more than Bart. How much money does Bart have?

Let LL be Lisa’s money

Let B be Bart’s money

Define variables:

Write two equations

L B 13.75

2L 76.50L 38.25

Bart has $24.50

B 24.50

#6#6 Otis has three times as much money as Milo. Together they have $60.84. How much money does each one of them have?

Let tt be Otis’ money

Let m be Milo’s money

t m 60.84

t 3m 3t m

3m m 60.84

4m 60.84

m 15.21t 45.63

Otis has $45.63

Milo has $15.21

Basic Word Problems:Basic Word Problems:Basic Word Problems:Basic Word Problems:

Example 1

The sum of two numbers is 49. One number is 13 less than the other. Find the numbers.

Define variables:

Let x be the larger number

Let y be the smaller number

Write two equations

Solve the system

x y 49 x y 13

2x 62x 31

y 1831 and 18

The difference between two numbers is 16. Threetimes the larger number is seven times the smaller. What are the numbers?Define variables:

Let x be the larger numberLet y be the smaller number

Write two equations

x y 16

3 y 16 7y

3y 48 7y

x 2812 and 28

Example 2

The sum of a number and twice The sum of a number and twice anotheranother number number is 13. The first number is 4 larger than the is 13. The first number is 4 larger than the second number. What are the two numbers?second number. What are the two numbers?

Define variables:

Let xx be the first number (larger)

Let yy be the second number

Write two equations

Solve the system

Example 3

x 2y 13 x y 4

x 2y 13

3y 9y 3x 77 and 3

The sum of two numbers is 82. One number is 12 more than the other. Find the larger number.

Define variables:

Let L be the larger numberLet S be the smaller number

Write two equations

L S 82

Define variables:

Write two equations

L S 82

L S 12

L 12 S

L S L S 12

Define variables:

Write two equations

L S 82

L S 12

Solve the system

L S 12

L S 82

L 4747

The difference between two numbers is 6. Ten times the smaller number is six times the larger. Find the numbers.

Define variables:

Write two equations

10S 6L

10S 6 S 6

10S 6S 36

S 9L 15

9 and 15

Solve the system

The sum of a number and twice another number is 37. The first number is 10 larger than the second number. What are the two numbers?

Define variables:

Write two equations

L 2S 37

L S 10 L S 10

2S 0 S1 37

3S 10 37

S 9L 19

9 and 19

Solve the system

The product of 4 times the sum of a number and 3 is another number. If the sum of the numbers is 67, what is the smallest of the two numbers?

Define variables:

Let x be one numberLet y be the “other” number

Write two equations

4 x 3 y

x y 67 4x 12 y

4x 12x 67

5x 12 67

5x 55x 11

y 5611

Solve the system

More Word Problems:More Word Problems:More Word Problems:More Word Problems:

#1#1Farmer Bob had 25 animals in the barn – all of them either cows or chickens. He counted 66 legs in all. How many cows are in the barn?

w number of cowsk number of chickens

w k 25E1E2 4w 2k 66

2 w k 25 2w 2k 50

8 cows 8 cows

The price of a ticket for the AVHS basketball game is $2.75 for a student, but only $2.25 if you have a discount card. One ticket taker sold 59 tickets for $141.75. How many students didn’t use a discount card?

Let xx be the number of students w/o discount cards

Let y be the number of students with discount cards

x y 59

2.75x 2.25y 141.75 275x 225y 14,175

225x 225y 13,275

50x 900x 18

18 students w/ o discount card 18 students w/ o discount card

#3#3At Randy’s bike shop, they only work on bicycles and tricycles. When Randy disassembled all the bikes and trikes he ended up with 34 seats and 89 wheels. How many tricycles does he have in his shop?

At Randy’s bike shop, they only work on bicycles and tricycles. When Randy disassembled all the bikes and trikes he ended up with 34 seats and 89 wheels. How many tricycles does he have in his shop?

b number of bicycles

t number of tricycles

b t 34E1E2 2b 3t 89

2 b t 34 2b 2t 68

21 tricycles 21 tricycles

2b 3t 89

Sydney took a math test that had 32 questions on it and scored 111 points. Each correct answer was awarded 5 points and for each wrong answer two points were deducted. How many questions did she miss on her test?

Sydney took a math test that had 32 questions on it and scored 111 points. Each correct answer was awarded 5 points and for each wrong answer two points were deducted. How many questions did she miss on her test?c correct answers

i incorrect answers

c i 32E1

E2 5c 2i 111

7i 49i 7She missed 7 questions.She missed 7 questions.

5c 5i 160

5c 2i 111

Will set a school record by scoring 30 points in his basketball game. What was amazing is that he scored all his points without a single free-throw. Out of the 13 baskets that he made, how many were 3-point shots?

Will set a school record by scoring 30 points in his basketball game. What was amazing is that he scored all his points without a single free-throw. Out of the 13 baskets that he made, how many were 3-point shots?x 2-point shots

y 3-point shots

x y 13E1E2 2x 3y 30

y 4He made 4, 3-pointers.He made 4, 3-pointers.

2x 2y 26

2x 3y 30

100 25

Jackie’s coin purse had only dimes and quarters in it. There were 5 more dimes than quarters, and the total amount of money was $7.85. How many dimes were in the purse?

d number of dimes

q number of quarters

0.10d 0.25q 7.85 10d 25q 785

25d 25q 125

35d 910d 2626 dimes26 dimes

Let xx be the number of T/F questions

Let y be the number of “other” questions

x y 25

2x 3y 66

3x 3y 75

x 9x 99 T/ F questions. 9 T/ F questions.

A science test has 25 questions on it and is worth a total of 66 points. The true/false questions are worth 2 points each and the rest of the questions are worth 3 points each. How many true/false questions are on the test?

2x 3y 66

#8#8 At a movie theater, tickets cost $9.50 for adults and $6.50 for children. A group of 7 moviegoers pays a total of $54.50. How many adults are in the group?Let aa be the number of adults

Let c be the number of children

9.50a 6.50c 54.50 95a 65c 545

65a 65c 455

30a 90a 33 adults 3 adults

T.G.I.F.T.G.I.F.T.G.I.F.

#1#1 At the baseball game field level seats cost $9.50 each, while seats in the second deck cost $6.25. If a ticket seller sold 52 tickets and collected $425.75, how many second deck seats did she sell?

Let ff be the number of field level tickets.

Let s be the number of 2nd deck tickets.

f s 52

9.50f 6.25s 425.75 950f 625s 42,575

950f 950s 49,400

325s 6,825s 2121 tickets21 tickets

#4#4A History test has 40 questions on it and is worth a total of 174 points. The true/false questions are worth 3 points each and the rest of the questions are worth 5 points each. How many true/false questions are on the test?

t Number of T/ F questions.

r the "regular " questions.

t r 40E1E2 3t 5r 174

5 t r 40 5t 5r 200

2t 2613 T/ F questions 13 T/ F questions

3t 5r 174

#6#6 A jar contains quarters and dimes. There are 15 more quarters than dimes. The total amount of money in the jar is $23. How many quarters are in the jar?

q d 15

0.25q 0.10d 23.00

35q 2450

q 7070 Quarters

q number of quartersd number of dimes

25q 10d 2300

10q 10d 150

#7#7At the coffee shop, two bagels and three muffins cost $12.45. Three bagels and five muffins cost $20.00. What is the cost of a single bagel?

Let b be the cost of a bagel

Let m be the cost of a muffin

2b 3m 12.45

3b 5m 20.00

2b 3m 15( 2.45) 3b 5m3( 20.00)

10b 15m 62.25 9b 15m 60.00

b 2.25Bagel $2.25

#10#10 The sum of two integers is 35 and the difference between the same two integers is 81. What is the smaller integer?

Let L be the larger number

Let S be the smaller number

L S 35 L S 81

2L 116

Haley was going to be paid to unpack a box of 125 delicate crystal ornaments. She would be paid 75 cents for each ornament unpacked, but would be charged $2.50 for any that she broke. After finishing the job she was paid $74.25. How many ornaments did she break?

Let xx be the number of ornaments unpacked successfully.

Let y be the number of ornaments broken

x y 125

0.75x 2.50y 74.25 75x 250y 7425

75x 75y 9375

325y 1950y 6

She broke 6 ornaments. She broke 6 ornaments.

#9- Bonus

Basic Word Problems:Basic Word Problems:• AgeAge• Number-DigitNumber-Digit

#1 The sum of the digits of a two-digit number is 10. When the digits are reversed, the new number is 54 more than the original number. What is the original number?

Let t be the tens digit of the original number

Let u be the units (ones) digit of the original number

t u 10

Original number New number

101 t tu u0 54 9u 9t 54

9u 9t 90

18u 144u 8t 2

#2 The sum of the digits of a two-digit number is 7. When the digits are reversed, the new number is 45 less than the original number. What is the original number?

101 t tu u0 45 9u 9t 45

9u 9t 63

18u 18u 1t 6

Andy is 21 years older than Bob. In two years, Andy Andy is 21 years older than Bob. In two years, Andy will be twice as old as Bob. What is Andy’s current will be twice as old as Bob. What is Andy’s current age? age?

Let a be Andy’s current age

Let b be Bob’s current age

Age in 2 years

a 2b 2

a b 21

a 2 2b 4 a 2b 2 a 2 2 b 2

Andy is 21 years older than Bob. In two years, Andy Andy is 21 years older than Bob. In two years, Andy will be twice as old as Bob. What is Andy’s current will be twice as old as Bob. What is Andy’s current age? age?

Let a be Andy’s current age

Age in 2 years

a 2b 2

a b 21

a 2b 2 2b 2 b 21b 19

#4 Tom is 5 years older than Jerry. Last year Tom was twice as old as Jerry. How old is Tom today?

Let t be Tom’s current age

Let j be Jerry’s current age

Last year

t 1j 1

t 1 2 j 1 t 1 2j 2 t 2j 1

#4 Tom is 5 years older than Jerry. Last year Tom was twice as old as Jerry. How old is Tom today?

Let t be Tom’s current age

Let j be Jerry’s current age

Last year

t 1j 1

t 2j 1 j 5 2j 1

j 6t 11

Let xx be the cost of an adult ticket

Let y be the cost of a youth ticket

6x 9y 402

6x 4y 292 2 3 2x 3y 134

3x 2y 146

5y 110

y 22$22 f or a youth ticket$22 f or a youth ticket

100 25

#10#10

Let xx be the number of Granny Smith apples

Let y be the number of Gala apples

x y 19

0.25x 0.30y 5.10

25x 25y 475

25x 30y 510

5y 35y 77 Gala and 12 Granny Smith

Let a be the measure of angle aLet b be the measure of angle b

a b 90

b 2a 12

m a 26

#2#2#2#2 Angle a and angle b are complementary angles. The measure of angle b is 12 more than twice the measure of angle a. Find the measure of angle a.

Angle a and angle b are complementary angles. The measure of angle b is 12 more than twice the measure of angle a. Find the measure of angle a.

2a 12a 90

3a 12 90

3a 78a 26

Let xx be the cost of an adult ticket

Let y be the cost of a youth ticket

6x 9y 402

6x 4y 292 2 3 2x 3y 134

3x 2y 146

5y 110

y 22$22 f or a youth ticket$22 f or a youth ticket

100 25

Let xx be the number of Granny Smith apples

Let y be the number of Gala apples

x y 19

0.25x 0.30y 5.10

25x 25y 475

25x 30y 510

5y 35y 77 Gala and 12 Granny Smith

#1#1#1#1

#2#2#2#2

Marcus is 5 years older than Katie. Last year he was twice as old as she was. How old will Katie be next year?

#1#1#1#1 Marcus is 5 years older than Katie. Last year he was twice as old as she was. How old will Katie be next year?

Let m be Marcus’s current age

Let k be Katie’s current age

Age last year

m 1k 1

m k 5 m 1 k 12

m 2k 1

m 1 2k 2 m 2k 1

#1#1#1#1 Marcus is 5 years older than Katie. Last year he was twice as old as she was. How old will Katie be next year?

Let m be Marcus’s current age

Let k be Katie’s current age

Age last year

m 1k 1

m 2k 1

k 5 2k 1 5 k 16 kKatie will be 7 years-old next year.Katie will be 7 years-old next year.

a b 90

b 2a 12

m a 26

#2#2#2#2 Angle a and angle b are complementary angles. The measure of angle b is 12 more than twice the measure of angle a. Find the measure of angle a.

2a 12a 90

3a 12 90

3a 78a 26

#1#1A cashier is counting money at the end of the day. She has a stack that contains only $5 bills and $10 bills. There are 45 bills in the stack for a total of $290. How many $5 bills are in the stack?

F number of $5 bills

T number of $10 bills

F T 45E1E2 5F 10T 290

10F 10T 450

5T 160

32 $5-bills32 $5-bills

5F 10T 290

#2 The sum of the digits of a two-digit number is 13. When the digits are reversed the new number is 27 more than the original number. What was the original number?

t u 13

101 u ut t0 27 9t 9u 27

9t 9u 117

18t 90

u 8t 558

Let d be Danielle’s current age

Let a be Alison’s current age

Age 2 years ago

d 2a 2

d a 36 d 2 a 2

Danielle is 36 years older than her daughter Alison. Two years ago, Danielle was 5 times as old as Alison. Find Alison’s current age.

d 5a 8

d 2 5a 10

Let d be Danielle’s current age

Let a be Alison’s current age

Age in 2 years

d 2a 2

d a 36

Danielle is 36 years older than her daughter Alison. Two years ago, Danielle was 5 times as old as Alison. Find Alison’s current age.

d 5a 8

5a 8 a 36 4a 8 36

4a 44a 11

Alison is 11 years-oldAlison is 11 years-old

A jar contains 55 quarters and dimes. The total amount of money in the jar is $8.50. Find the number of dimes in the jar

q d 55

0.25q 0.10d 8.50

15d 525

d 3535 dimes

25q 10d 850

25q 25d 1375

#5#5Becky is selling tickets to a school play. Adult tickets cost $12 and student tickets cost $6. Becky sells a total of 48 tickets and collects a total of $336. How many $6 tickets did she sell?

Becky is selling tickets to a school play. Adult tickets cost $12 and student tickets cost $6. Becky sells a total of 48 tickets and collects a total of $336. How many $6 tickets did she sell?

Let a be the number of adult tickets

Let s be the number of student tickets

a s 48

12a 6s 336 12a 6s 336

12a 12s 576

6s 240

s 4040 student tickets

#6The sum of the digits of a two-digit number is 14. The first digit is 4 less than twice the second digit. What is the number?

Let t be the tens digit of the numberLet u be the ones digit of the number

t u 14

t 2u 4

2u 4 u 14

3u 4 143u 18u 6

Five years ago, Beth was three times as old as Frank. Next year she will be twice as old as Frank. How old is Beth today?

b is Beth’s current age

f is Frank’s current age

Age 5 years ago

b 5f 5

b 5 f 53

b 5 3f 15

Age next year

b 1f 1

b 3f 10

Age 5 years ago

b 5f 5

Age next year

b 1f 1

b 3f 10 b 1 f 1

b 1 2f 2

b 2f 1 b 2f 1

Age 5 years ago

b 5f 5

Age next year

b 1f 1

b 3f 10

b 2f 1

3f 10 2f 1 f 10 1

f 11Beth is 23 years-oldBeth is 23 years-old

Angle a and angle b are complementary angles. Angle b is 15 more than four times angle a. Find the measure of both angles.

a b 90

b 4a 15

m a 15

4a 15a 90

5a 15 90

5a 75a 15 m b 75

Basic Word Problems:Basic Word Problems:

• Mixture ProblemsMixture Problems

aa = amount of 12% solution

bb = amount of 20% solution

A chemist mixes a 12% alcohol solution with a 20% alcohol solution to make 300 milliliters of an 18% alcohol solution. How many milliliters of each solution does the chemist use?

12% solution

18% solution

20% solution

#1#1#1#1

a b 300

0.12a 0.20b 54

Amount of Solution

Amount of alcohol

a b 300300

0.12a 0.20b

0.18(300) = 54

12%solution

20%solution

18%solution

#1#1#1#1

Amount of Solution

Amount of alcohol

a b 300300

0.12x 0.20y

0.18(300) = 54

12%solution

20%solution

18%solution

#1#1#1#1

a b 300

0.12a 0.20b 54

Amount of Solution

Amount of alcohol

a b 300300

0.12x 0.20y

0.18(300) = 54

12%solution

20%solution

18%solution

#1#1#1#1

b 22575 milliliters of 12% solution75 milliliters of 12% solution

225 milliliters of 20% solution225 milliliters of 20% solution

A solution containing 30% insecticide is to be mixed with a solution containing 50% insecticide to make 200 Liters of a solution containing 42% insecticide. How much of each solution should be used?

0.42(200) = 84

#2#2#2#2

Amount of Solution

Pureinsecticide

a b 200200

0.3a 0.5b 84

+ =30%solution

50%solution

42%solution

a b 200 0.3a 0.5b 84

A solution containing 30% insecticide is to be mixed with a solution containing 50% insecticide to make 200 Liters of a solution containing 42% insecticide. How much of each solution should be used?

#2#2#2#2

a b 200

0.3a 0.5b 84 3a 5b 840

3a 3b 600

2b 240b 120a 80

#3#3#3#3

c c = amount of Columbian beans (lbs.).

hh = amount of Hawaiian beans (lbs.)

Amount of Beans (lbs.)

Cost ofBeans ($)

c h 100100

1.50c 3.50h 270

+ =Columbian Hawaiian ChristmasBlend

c h 100

1.50c 3.50h 270

Starbucks wants to make 100 pounds of their special Christmas blend and sell it for $2.70 per pound. They will mix Columbian beans that sell for $1.50 per pound and Hawaiian beans that cost $3.50 per pound. How many pounds of the Hawaiian beans will they need to order?

#3#3#3#3

c c = amount of Columbian beans (lbs.).

hh = amount of Hawaiian beans (lbs.)

c h 100

1.50c 3.50h 270 150c 350h 27000 150 150c 150h 15000

200h 12000h 60c 40

60 pounds of Hawaiian beans60 pounds of Hawaiian beans

Amount of Solution

Amount of Saline

a b 800800

0.3a 0.5b

0.45(800) = 360

30%solution

50%solution

45%solution

Jenny mixes a 30% saline solution with a 50% saline solution to make 800 milliliters of a 45% saline solution. How many milliliters of each solution does she use?

#1#1#1#1

aa = amount of 30% solution (ml)

bb = amount of 50% solution (ml)

a b 800

0.3a 0.5b 360

Jenny mixes a 30% saline solution with a 50% saline solution to make 800 milliliters of a 45% saline solution. How many milliliters of each solution does she use?

#1#1#1#1

aa = amount of 30% solution (ml)

bb = amount of 50% solution (ml)

a b 800

3a 5b 3600

3a 3b 2400

2b 1200b 600a 200

Amount of mix

Amount of aspirin

a b 1010

0.10a 0.25b

0.16(10) = 1.6

10% mix

25% mix

16% mix

A pharmacist wants to mix medicine that is 10% aspirin with a medicine that is 25% aspirin to make 10 grams of a medicine that is 16% aspirin. How many grams of each medicine should the pharmacist mix together?

#2#2#2#2

aa = amount of 10% mix (grams)

bb = amount of 25% mix (grams)

a b 10

0.10a 0.25b 1.6

A pharmacist wants to mix medicine that is 10% aspirin with a medicine that is 25% aspirin to make 10 grams of a medicine that is 16% aspirin. How many grams of each medicine should the pharmacist mix together?

#2#2#2#2

aa = amount of 10% mix (grams)

bb = amount of 25% mix (grams)

10a 25b 160

0.10a 0.25b 1.6

a b 10 10a 10b 100

15b 60b 4a 6

6 grams of 10% mix6 grams of 10% mix

4 grams of 25% mix4 grams of 25% mix

#3#3#3#3

p p = amount of peanuts (lbs.).

rr = amount of raisins (lbs.)

Weight of“stuff” (lbs.)

Cost of“stuff” ($)

p r 88

1.60p 2.40r 17.60

+ =Peanuts Raisins Mixture

1.60p 2.40r 17.60

Peanuts cost $1.60 per pound and raisins cost $2.40 per pound. Brad wants to make 8 pounds of a mixture that costs $2.20 per pound. How many pounds of peanuts and raisins should he use?

8(2.20) = 17.60

#3#3#3#3

p p = amount of peanuts (lbs.).

rr = amount of raisins (lbs.)

1.60p 2.40r 17.60

Peanuts cost $1.60 per pound and raisins cost $2.40 per pound. Brad wants to make 8 pounds of a mixture that costs $2.20 per pound. How many pounds of peanuts and raisins should he use?

16p 24r 176

16p 16r 128

8r 48r 6p 2

2 pounds of peanuts2 pounds of peanuts

6 pounds of raisins6 pounds of raisins

Basic Word Problems:Basic Word Problems:

• Motion ProblemsMotion Problems

Motion Problems Motion Problems Motion problems involve distance, time and rate.

The equation that links these concepts is called

The Distance Formula:

d = r td = distance

t = timer = rate

• milesmiles• kilometers kilometers • metersmeters• feetfeet• inchesinches

• hourshours• minutesminutes• secondsseconds• daysdays• yearsyears

• miles/hourmiles/hour• km/min. km/min. • m/sm/s• ft./sec.ft./sec.• inches/sec.inches/sec.

A jet airplane flies 2000 miles with the wind in four A jet airplane flies 2000 miles with the wind in four hours. The return trip against the same takes 5 hours. The return trip against the same takes 5 hours. Find the speed of the jet and the speed of hours. Find the speed of the jet and the speed of the wind. the wind.

d = 2000 miles t = 4 hours

w = speed of windw = speed of wind

j = speed of the jetj = speed of the jet

Withthe wind

Time Distance

Againstthe wind

j w 5 2000

j w 4 2000

j w 5 2000

j w 4 2000

j w 5 2000

4j 4w 2000

5j 5w 2000 20j 20w 8000

20j 20w 10,000

40j 18,000j 450w 50

Speed of the jet = 450 mphSpeed of the jet = 450 mph

Speed of the wind = 50 mphSpeed of the wind = 50 mph

Withthe current

c = speed of the current c = speed of the current (mph)(mph)

b = speed that Ben can paddle in still water b = speed that Ben can paddle in still water (mph).(mph).

Ben paddles his kayak 8 miles upstream in 4 hours. Ben paddles his kayak 8 miles upstream in 4 hours. He turns around and paddles back downstream to his He turns around and paddles back downstream to his starting point in just 2 hours. What is the speed of starting point in just 2 hours. What is the speed of the current?the current?

Time Distance

Againstthe current

b c 4 8

b c 2 8

b c 4 8

c = speed of the current (mph)c = speed of the current (mph)

b = speed that Ben can paddle in still water (mph).b = speed that Ben can paddle in still water (mph).

Ben paddles his kayak 8 miles upstream in 4 hours. Ben paddles his kayak 8 miles upstream in 4 hours. He turns around and paddles back downstream to He turns around and paddles back downstream to his starting point in just 2 hours. What is the speed his starting point in just 2 hours. What is the speed of the current?of the current?

b c 2 8

b c 4 8

2b 2c 8

4b 4c 8 4b 4c 16 4b 4c 8

8b 24b 3

Speed of current = 1 mphSpeed of current = 1 mph

Speed that ben can paddle = 3 mphSpeed that ben can paddle = 3 mph

Test ReviewTest Review

Withthe wind

c = speed of the helicopterc = speed of the helicopter

With a tailwind, a helicopter flies 300 miles in 1.5 hours. When the helicopter flies back against the same wind, the trip takes 3 hours. What is the helicopter’s speed in still air? What is the speed of the wind?

Time Distance

Againstthe wind

1.5c w

c w 3 300

c w 1.5 300

c w 3 300

1.5c 1.5w 300

3c 3w 300 3c 3w 300

3c 3w 600

6c 900c 150w 50

Speed of the heli. = 150 mphSpeed of the heli. = 150 mph

Speed of the wind = 50 mphSpeed of the wind = 50 mph

c = speed of the helicopterc = speed of the helicopter

With a tailwind, a helicopter flies 300 miles in 1.5 hours. When the helicopter flies back against the same wind, the trip takes 3 hours. What is the helicopter’s speed in still air? What is the speed of the wind?

Withthe current

c = speed of currentc = speed of current

b = speed of the bargeb = speed of the barge

A barge on the Sacramento river travels 24 miles upstream in 3 hours. The return trip take the barge only two hours. Find the speed of the barge in still water.

Time Distance

Againstthe current

b c 3 24

b c 2 24

b c 3 24

2b 2c 24

3b 3c 24 b c 8

b c 12

2b 20b 10c 2

Speed of the barge = 10 mphSpeed of the barge = 10 mph

Speed of the current = 2 mphSpeed of the current = 2 mph

c = speed of currentc = speed of current

b = speed of the bargeb = speed of the barge

2222 A barge on the Sacramento river travels 24 miles upstream in 3 hours. The return trip take the barge only two hours. Find the speed of the barge in still water.

A barge on the Sacramento river travels 24 miles upstream in 3 hours. The return trip take the barge only two hours. Find the speed of the barge in still water.

Amount of Solution

Amount of Acid

a b 500500

0.1a 0.2b

0.16(500) = 80

10%solution

20%solution

16%solution

A chemist mixes a 10% acid solution with a 20% acid solution to make 500 milliliters of a 16% acid solution. How much of the 20% solution did he use in his mixture?

#3#3#3#3

aa = amount of 10% acid solution (ml)

bb = amount of 20% acid solution (ml)

a b 500

0.1a 0.2b 80

A chemist mixes a 10% acid solution with a 20% acid solution to make 500 milliliters of a 16% acid solution. How much of the 20% solution did he use in his mixture?

#3#3#3#3

a 2b 800 −a−b −500

b =300

a 200200 ml of 10% acid solution200 ml of 10% acid solution

300 ml of 20% acid solution300 ml of 20% acid solution

aa = amount of 10% acid solution (ml)

bb = amount of 20% acid solution (ml)

a b 500

0.1a 0.2b 80

#4#4#4#4

a a = amount of apricots (lbs.).

cc = amount of cherries (lbs.)

Weight of“stuff” (lbs.)

Cost of“stuff” ($)

a c 2020

1.50a 3.50c 54

+ =apricots cherries Mixture

a c 20

1.50a 3.50c 54

At the Snack Shack, dried cherries cost $3.50 per pound. Dried apricots cost $1.50 per pound. The store’s owner wants to make 20 pounds of a mixture that costs $2.70 per pound. How many pounds of cherries will be needed to make the mixture?

20(2.70) = 54

#4#4#4#4

15a 35c 540

15a 15c 300

20c 240c 1212 pounds of cherries12 pounds of cherries

a c 20

1.50a 3.50c 54

At the Snack Shack, dried cherries cost $3.50 per pound. Dried apricots cost $1.50 per pound. The store’s owner wants to make 20 pounds of a mixture that costs $2.70 per pound. How many pounds of cherries will be needed to make the mixture?

a a = amount of apricots (lbs.).

cc = amount of cherries (lbs.)

Are You Ready For The Test?

Which system can solve the word problem given?

At the hardware store, 15 screws and 7 bolts weigh 303 grams. 12 bolts and 5 screws weigh 188 grams. What will 9 bolts weigh? b = weight of a bolt

s = weight of a screw

7b 15s 303

12b 5s 188

5b 12s 188

15b 7s 303

7b 15s 188

9b 5s 303

15b 7s 303

12b 5s 188

Andy is 21 years older than Bob. In two years, Andy will be twice as old as Bob. What is Andy’s current age? Let a be Andy’s current age

a b 21

a 2 2b 4

a b 21

a 2 2b 4

a b 21

2a 4 b 2

a b 21

a 2 2b 2

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