Ratios and Unit Rates

Ratios and Unit RatesRatios and Unit RatesPRE-ALGEBRA LESSON 6-1PRE-ALGEBRA LESSON 6-1

6-1

Write three fractions equivalent to . 412

, , 618

1 3

2 6

Sample answers


Write in simplest form.

1. 2. 3.

4. 5. 6.

3035

2440

5460

1215

1442

4024

(For help, go to Lesson 4-4.)

Check Skills You’ll Need

6-1


Solutions

1. = 2. =

= =

3. = 4. =

= =

5. = 6. =

= = = 1

3035

30 ÷ 535 ÷ 5

67

2440

24 ÷ 840 ÷ 8

35

5460

54 ÷ 660 ÷ 6

1215

12 ÷ 315 ÷ 3

910

45

1442

14 ÷ 1442 ÷ 14

4024

40 ÷ 824 ÷ 8

13

23

53

6-1

A survey asked students whether they had after-

school jobs. Write each ratio as a fraction in simplest form.


a. all students surveyed to studentswithout jobs

b. all students surveyed to studentswith jobs

all students surveyedstudents with jobs

10040

=

=52

After-School Jobs

Response Number

Have a Job 40

60

100

Don’t Have a Job

Total

all students surveyedstudents without jobs

10060

=

=53

Quick Check

6-1

The table shows prices for different packages of index

cards. Which size has the lowest unit price?


Price

$2.70

$1.30

$.75

100

50

25

Size(cards)

100 cards:price

number of cards$2.70

100 cards = $.027/card

25 cards:price


25 cards = $.03/card

The 50-card pack has the lowest unit price.

50 cards:price


50 cards = $.026/card Find the unit prices.

Quick Check

6-1

Convert 30 gal/min to cups/second.


30 gal/min = • •30 gal1 min

16 c1 gal

1 min60 s

Use conversion factors that convert gallons to cups and minutes to seconds.

1 min

1260 s

Divide the common factors and units.= • •

130 gal1 min

816 c1 gal

=8 cs

Simplify.

30 gal/min equals 8 c/s.

Quick Check

6-1


Write each ratio as a fraction in simplest form.

1. 4 out of 20 students ride the bus to school.

2. 8 out of 10 students buy lunch.

Find each unit rate.

3. You pay $4.50 for 2 gallons of orange juice.

4. You pay $21.60 for a 20-lb bag of dog food.

5. Convert 15 yd/min to inches/second.

15

45

$2.25/gal

$1.08/lb

9 in./s

6-1

ProportionsProportionsPRE-ALGEBRA LESSON 6-2PRE-ALGEBRA LESSON 6-2

6-2

Evaluate each of the following operations for n = 6.

a. Multiply n by 4 and add 6. b. Multiply n by 9 and add 12

30 66


Solve each equation.

1. 4x = 52 2. 3y = 18

3. 5b = 75 4. 7k = 21



6-2


Solutions

1. 4x = 52 2. 3y = 18

x = 13 y = 6

3. 5b = 75 4. 7k = 21

b = 15 k = 3

4x4

524=

3y3

183=

5b5

755=

7k7

217=

6-2

Solve = .


27

y14

Method 1 Multiplication Property of Equality

=27

y14

• 14 = • 1427

y14

= y287

4 = y

6-2

(continued)


Method 2 Cross products

=27

y14

2 • 14 = 7 • y

28 = 7y

4 = y

=287

7y7

Quick Check

6-2

Do the ratios and form a proportion? Explain.


105 = 105 Simplify.

Yes; the ratios do form a proportion. The cross products

are equal.

35

2135

Test by writing as a proportion.35

2135

3 • 35 5 • 21 Write cross products.

Quick Check

6-2

One hundred rods is about 275 fathoms. About how many

fathoms is 25 rods?


Let d = distance in fathoms.

length in rodslength in fathoms

100275 =

25d

distance in rodsdistance in fathoms

100d = 275(25) Write cross products.

d = 68.75 Simplify.

d = Divide each side by 100.275(25)100

25 rods is about 68.75 fathoms.

Quick Check

6-2


Solve each proportion.

1. 2.

Does each pair of ratios form a proportion? Explain.

3. 4.

5. 100 nautical miles equals about 115 statute miles. About how far in ` nautical miles is 50 statute miles? Round to the nearest whole

number.

a12

56=

6x=

38

0.41.8=

29

2150 =

1425

10 16

Yes; the cross products are equal.

No; the cross products are not equal.

about 43 nautical miles

6-2

Similar Figures and Scale DrawingsSimilar Figures and Scale DrawingsPRE-ALGEBRA LESSON 6-3PRE-ALGEBRA LESSON 6-3

6-3

Find each value of n.

a. b. 1815 =

n5

n2.7

1.54.5 =

n = 6 n = 0.9


Solve each proportion. Round to the nearest tenth where necessary.

1. 2.

3. 4.

23

f21=

38

50p=

94

15p=

163

19g=



6-3


Solutions

1.23

f21=

3 • f = 2 • 21

3f = 42

3f3

423=

f = 14

38

50p=

8 • 50 = 3 • p

400 = 3p

4003

3p3=

133.3 = p

2. 3.94

15p=

4 • 15 = 9 • p

60 = 9p

609

9p9=

6.7 = p

4.163

19g=

3 • 19 = 16 • g

57 = 16g

5716

16g16=

3.6 = g

6-3

Trapezoid ABCD ~ trapezoid EFGH. Find the

value of k.


Write a proportion for corresponding sides.6k

Side AB corresponds = Side CD corresponds to side EF. to side GH.

32

6 • 2 = k • 3 Write cross products.

= Divide each side by 3.6 • 23

3k3

4 = k Simplify. Quick Check

6-3

A flagpole casts a shadow 5 ft long. At the same time, a

yardstick casts a shadow 1.5 ft long. The triangle shown for the flagpole

and its shadow is similar to the triangle shown for the yardstick and its

shadow. How tall is the flagpole?


Corresponding sides of similar triangles are in proportion.

1.55

= 3x

1.5x = 5 • 3 Write cross products.

Divide each side by 1.5.1.5x1.5

= 5 • 31.5

x = 10 Simplify.

The flagpole is 10 ft tall.Quick Check

6-3

The scale of a map is 1 in. : 24 mi. About how far is it between

two cities that are 3 in. apart on the map?


It is about 72 mi between the two cities.

map (in.)actual (mi.)

124

3d

map (in.)actual (mi.)= Write a proportion.

1 • d = 24 • 3 Write cross products.

d = 72 Simplify.

Quick Check

6-3


Solve.

1. Parallelogram ABCD ~ parallelogram EFGH. Find the value of x.

2. A girl who is 4 feet tall casts a shadow that is 6 feet long. The tree next to her casts a shadow that is 12 feet long. How tall is the tree?

3. The scale on a map is 3 in. : 100 mi. What is the actual distance between two towns that are 9 in. apart on the map?

12

8 ft

300 mi

6-3

ProbabilityProbabilityPRE-ALGEBRA LESSON 6-4PRE-ALGEBRA LESSON 6-4

6-4

45

16

24

Forty-three boys and five girls tried out for the middle school football

team. One-fourth of the students were dropped after the first day. Of

those who were left, were dropped after the second day. Of those

who remained, made the team. How many students madethe team?


Simplify.

1. 1 – 2. 1 –

3. 1 – 4. 1 –

38

1720

611

112



6-4


Solutions

1. 1 – 2. 1 –

– = – =

= =

3. 1 – 4. 1 –

– = – =

= =

38

1720

88

38

8 – 38

2020

1720

20 – 1720

58

320

611

112

1111

611

11 – 611

1212

112

12 –112

511

1112

6-4

Find P(rolling a prime number) with one number cube.


=number of favorable outcomesnumber of possible outcomes

36

3 prime-number outcomes (2, 3, 5)6 possible outcomes

P(rolling a prime number) = , or .36

12

Quick Check

6-4

The probability that a child is an identical twin is 4 in 1,000.

Find P(not an identical twin).

P(not an identical twin) + = 1 Substitute.4

1,000

P(not an identical twin) + – = 1 – Subtract

from

each side.

41,000

41,000

41,000 4

1,000

= 249250

9961,000 P(not an identical twin) = Simplify.

The probability that a child is not an identical twin is .249250

P(not an identical twin) + P(identical twin) = 1 Write an equation.


Quick Check

6-4

You have five different coins in your pocket: a penny, a

nickel, a dime, a quarter, and a half-dollar. You pull out one coin at

random. What are the odds in favor of the coin being worth less

than ten cents?

odds in favor =23

2 are worth less than ten cents.3 are not.

The odds are , or 2 to 3, in favor of the coin being worth less than ten cents.23


Quick Check

6-4


Find each probability.

1. A letter is selected at random from the letters A, E, I, O, and U. Find the probability that the letter is an A.

2. In one class, Ms. Lang has 8 boys and 12 girls. She needs to choose one student to help pass out papers. What are the odds that a girl is chosen?

3. When rolling a number cube, what is P(6)? What is P(not 6)?

15

32

;16

56

6-4

Fractions, Decimals, and PercentsFractions, Decimals, and PercentsPRE-ALGEBRA LESSON 6-5PRE-ALGEBRA LESSON 6-5

6-5

58Samantha needs a -in. bit for her drill. Should she look in the bin

marked 62.5 in., 6.25 in., 0.625 in., or 0.063 in.?

0.625 in.


Write each fraction as a decimal.

1. 2. 3.

4. 5. 6.

58

920

56

23


34

811


6-5


1. = 5 ÷ 8 2. = 9 ÷ 20 3. = 3 ÷ 4

= 0.625 = 0.45 = 0.75

4. = 5 ÷ 6 5. = 2 ÷ 3 6. = 8 ÷ 11

= 0.83 = 0.6 = 0.72

920

811

34

Solutions

58

56

23

6-5

Write each percent as a fraction or a mixed number.


a. 30%

b. 175%

Write as a fraction with a denominator of 100.30

100

Simplify.3

10

Write as a fraction with a denominator of 100.175100

1 Write as a mixed number.34

Simplify.74

Quick Check

6-5

Express 7.3% as a decimal.


= 0.073

7.3% = Write as a fraction with a denominator of 100.7.3100

= 007.3 Divide by moving the decimal point two places to the left. You may need to write one or more zeros.

Quick Check

6-5

Express 0.412 as a percent.


Method 1 Rewrite as a fraction.

= 41.2%

Method 2 Move the decimal point.

0.412 = 41.2%

0.412 = 4121,000

= 412 ÷ 101,000 ÷ 10

= 41.2100

Quick Check

6-5

Four out of seven members of the chess club are boys.

What percent of the chess club members are boys?


0.5714 Divide the numerator by the denominator.

57.14% Write as a percent.

About 57% of the chess club members are boys.

Write a fraction.47

Quick Check

6-5


Write each percent as a fraction or mixed number, and as a decimal.

1. 325% 2. 1.1%

Write each as a percent.

3. 2.01 4.

5. Six out of fifteen students own skateboards. What percent of students own skateboards?

134

, or 3 ; 3.2514

111,000

; or 0.011

201%

35

60%

40%

6-5

Proportions and PercentsProportions and PercentsPRE-ALGEBRA LESSON 6-6PRE-ALGEBRA LESSON 6-6

6-6

A painter uses 3 cans of paint to paint four sets of shelves. How many

cans of paint does he need to buy to paint nine sets of shelves?

7 cans


Solve each proportion.

1. = 2. =

3. = 4. =

25100

98.9x


x28

43100

52100

13x

x100

27150


6-6


1. = 2. =

100 • x = 25 • 28 43 • x = 98.9 • 100

100x = 700 43x = 9890

= =

x = 7 x = 230

3. = 4. =

13 • 100 = 52 • x 100 • 27 = x • 150

1300 = 52x 2700 = 150x

= =

25 = x 18 = x

98.9x

Solutions

25100

x28

43100

100x100

700100

43x43

989043

52100

13x

x100

27150

130052

52x52

2700150

150x150

6-6

Find 23% of 158.


23(158) = 100n Write cross products.

36.34 = n Simplify.

23% of 158 is 36.34.

= Write a proportion.23

100n

158

= Divide each side by 100.23(158)

100100n100

Quick Check

6-6

What percent of 34 is 28? Round to the nearest tenth

of a percent.


34n = 100(28) Write cross products.

n = 82.35... Simplify.

= Write a proportion.n

1002834


3434n34

28 is approximately 82.4% of 34.

82.4 Round.

Quick Check

6-6

216 is 72% of what number?


72n = 100(216) Write cross products.

n = 300 Simplify.


7272n72


100216

n

216 is 72% of 300.

Quick Check

6-6

A tile floor has 90 blue tiles, which is 15% of all the tiles in

the floor. How many tiles are in the floor in all?


15x = 100(90) Write cross products.

x = 600 Simplify.

The floor has 600 tiles in all.

Check. Is the answer reasonable? The problem says the number of blue tiles is 15%. 10% of 600 is 60, so 5% of 600 is 30, and 15% is 60 + 30 = 90. The answer is reasonable.


90x


1515x15

Quick Check

6-6


Solve.

1. Find 47% of 2,400. 2. What percent of 700 is 1,498?

3. 6 is 3% of what number?

4. Water covers about 361,736,000 km2, or about 70.8% of Earth’s surface. What is the approximate surface area of Earth?

1,128

200

214%

about 511,000,000 km2

6-6

Percents and EquationsPercents and EquationsPRE-ALGEBRA LESSON 6-7PRE-ALGEBRA LESSON 6-7

6-7

Match each expression in Column 1 with an equivalent expression in Column 2.

Column 1 Column 21. 50% of 250 a. 2502. 100% of 250 b. 2.53. 1% of 250 c. 254. 10% of 250 d. 125

1. d 2. a 3. b 4. c


Write each percent as a decimal.

1. 48% 2. 5% 3. 23.8%

4. 72.25% 5. 136% 6. 178.5%



6-7


1. 48% = 2. 5% =

= 0.48 = 0.05

3. 23.8% = 4. 72.25% =

= 0.238 = 0.7225

5. 136% = 6. 178.5% =

= 1.36 = 1.785

Solutions

48100

5100

23.8100

72.25100

136100

178.5100

6-7

What is 35% of 84?


n = 0.35 • 84 Write an equation. Write the percent as a decimal.

n = 29.4 Simplify.

35% of 84 is 29.4.

Quick Check

6-7

What percent of 26 is 65?


n • 26 = 65 Write an equation.

n = 2.5 Simplify.

= 250% Change the decimal to a percent.

65 is 250% of 26.

= Divide each side by 26.26n26

6526

Quick Check

6-7

A car salesman makes a 6.5% commission on each car he

sells. How much does he make on the sale of a car for $35,000?


c = 0.065 • 35,000

= 2,275

The salesman’s commission is $2,275.

amount of commission

Words is 6.5% of $35,000

Let c = amount of commission.

Equation = 0.065 • $35,000c

Quick Check

6-7

During a telephone survey, 414 people, or 46% of those

called, said that they were watching station RFGT at the time of the

call. How many people were called?


0.46n = 414

n = 900

900 people were called.

=0.46n0.46

4140.46

Words 414 is 46% of people called

Let n = number of people called.

Equation 414 = 0.46 • n

Quick Check

6-7


Solve

1. What is 9% of 30? 2. What is 25% of 312?

3. What percent of 9 is 3? 4. What percent of 3 is 9?

5. A car dealer makes an 8% commission on each car she sells. How much does she make on a $40,000 sale?

6. During a telephone survey, 320 people, or 25% of those called, said they were listening to the same station at the time of the call. How many people were called?

2.7 78

$3,200

33 %13 300%

1,280 people

6-7

Percent of ChangePercent of ChangePRE-ALGEBRA LESSON 6-8PRE-ALGEBRA LESSON 6-8

6-8

Write these fractions to the nearest tenth of a percent.

a. b. 65255

42 52

25.5% 80.8%


Write each decimal as a percent.

1. 0.46 2. 2.47 3. 0.03 4. 5.236



6-8


1. Move the decimal 2 places to the right. 0.46 = 46%

2. 2.47 = 247%

3. 0.03 = 3%

4. 5.236 = 523.6%

Solutions

6-8

Find the percent of increase from 8 to 9.6.


amount of increase = 9.6 – 8 = 1.6

= 0.2 = 20%

The percent of increase from 8 to 9.6 is 20%.

percent of increase = amount of increase

original amount

=1.68

Quick Check

6-8

In a given year, Hillsboro had a total of 7.5 in. of rain by

March 1 and a total of 22.5 in. by July 1. Find the percent of

increase from 7.5 to 22.5.


amount of increase = 22.5 – 7.5 = 15

= 2 = 200%

The percent of increase from March 1 to July 1 was 200%.

percent of increase = amount of increase

original amount

=157.5

Quick Check

6-8

Find the percent of decrease from 1,250 to 1,120.


amount of decrease = 1,250 – 1,120 = 130

= 0.104 = 10.4%

percent of decrease = amount of decrease

original amount

= 1301,250

Quick Check

6-8


Solve.

1. Find the percent of increase from 22 to 63.

2. Find the percent of decrease from 6.3 to 2.2.

3. A surgeon performed 62 operations in 1991. He performed 342 in 2001. Find the percent of increase.

186.4%

451.6%

65.1%

6-8

Markup and DiscountMarkup and DiscountPRE-ALGEBRA LESSON 6-9PRE-ALGEBRA LESSON 6-9

6-9

Without computing, explain why 40% of 22 is less than 11.

11 is half, or 50%, of 22, so 40% of 22 is less than 11.


Write an equation and solve. Round to hundredths as needed.

1. What is 75% of $82?

2. What is 42% of $170?

3. What is 5.5% of $24?

4. What is 80% of $15.99?



6-9


1. What is 75% of $82? 2. What is 42% of $170?n = 0.75 • 82 n = 0.42 • 170n = 61.5 n = 71.475% of $82 is $61.50. 42% of $170 is $71.40.

3. What is 5.5% of $24? 4. What is 80% of $15.99?n = 0.055 • 24 n = 0.80 • 15.99n = 1.32 n = 12.795.5% of $24 is $1.32. 80% of $15.99 is $12.79.

Solutions

6-9

A grocery store has a 20% markup on a can of soup.

The can of soup costs the store $1.25. Find the markup.


markup = percent of markup • store’s cost

= 0.2 • 1.25

= 0.25 Simplify.

The markup is $.25.

Quick Check

6-9

A bookstore pays $4.50 for a novel. The percent of

markup is 45%. Find the novel’s selling price.


0.45 • 4.50 = 2.03 Multiply to find the markup.

4.50 + 2.03 = 6.53 Store’s cost + markup = selling price.

The selling price is $6.53.

Quick Check

6-9

A camera that regularly sells for $210 is on sale for 30%

off. Find the discount.


discount = percent of discount • regular price

= 0.30 • 210

= 63

The discount is $63.

Quick Check

6-9


Solve.

1. The school store has a 65% markup on each stapler. Each stapler costs the store $2.10. Find the markup.

2. A clothes store pays $40 for a skirt. The percent of markup is 25%. Find the skirt’s selling price.

3. A pair of shoes that regularly sells for $94.99 is on sale for 30% off. What is the sale price?

$1.37

$66.49

$50

6-9

Problem Solving Strategy: Make a TableProblem Solving Strategy: Make a TablePRE-ALGEBRA LESSON 6-10PRE-ALGEBRA LESSON 6-10

6-10

A theater has 25 seats in the first row. Each of the following rowshas two more seats than the row before it. How many seats are inthe 15-row theater?

585


Solve.

1. For two weeks, you double the amount of money you save each day. You save $.01 the first day. How much money will you have at the

end of the two weeks?



6-10


Solutions

After 2 weeks, $163.83 will have been saved.

Day Amount of new money saved Total amount saved

1 $.01 $.012 $.02 1 + 2 = $.033 $.04 3 + 4 = $.074 $.08 7 + 8 = $.155 $.16 15 + 16 = $.316 $.32 31 + 32 = $.637 $.64 63 + 64 = $1.278 $1.28 1.27 + 1.28 = $2.559 $2.56 2.55 + 2.56 = $5.1110 $5.12 5.11 + 5.12 = $10.2311 $10.24 10.23 + 10.24 = $20.4712 $20.48 20.47 + 20.48 = $40.9513 $40.96 40.95 + 40.96 = $81.9114 $81.92 81.91 + 81.92 = $163.83

6-10

Martin had 100 trees in his orchard the first year. Each

year after that he increased the number of trees in his orchard by

10%, rounded to the nearest whole number. How many trees did he

have in his orchard in the sixth year?


Year Tree Count at Rate of Increase in Tree Count atBeginning of Increase Tree Count End of Year

Year (10%)1 100 0.1 10 1102 110 0.1 11 1213 121 0.1 12 1334 133 0.1 13 1465 146 0.1 15 1616 161

Martin had 161 trees in the sixth year. Quick Check

6-10


Solve.

1. For an auction item, the bid starts at $100 and increases 50% every minute. In how many minutes will the bid go over $2,500?

2. Sharon uses her initials, SLF, for her password. How many different three-letter passwords can she make with these

letters?

3. In how many ways can you make 45¢ in change using only nickels, dimes, and quarters?

8 minutes

8 ways

6 passwords

6-10

Ratios and Unit Rates

Documents

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