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14 Rate, Base, and Part

# 14 Rate, Base, and Part

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1.14

Example 1

Example 2

Rate, Base, and Part

81

Given: 25% of \$80 is \$20. Identify R, B, and P.

R is 25%.

25 is the number with a percent sign. Remember to change 25% to the

decimal 0.25 for use in a formula.

B is \$80.

P is \$20.

\$80 is the whole amount. It also follows the word of.

\$20 is the part. It is also the number that is not R or B.

Given: 72% of the 75 students who took this course last year are now working; find how

many are now working. Identify R, B, and P.

R is 72%.

B is 75 students.

P is the unknown.

72 is the number with a percent sign.

75 is the whole amount. It also follows the word of.

The unknown is the number that is some fractional part of the

base. It is also the number that is not R or B.

Percent Problems: Finding the Part

After you have determined which two numbers are known, you find the third or unknown

number by using one of three formulas.

Formulas for Finding Part, Base, and Rate

1. P ϭ BR

P

2. B =

R

P

3. R =

B

Use to find the part.

Use to find the base.

Use to find the rate or percent.

Note: After you have studied algebra later in the text, you will need to remember only the

first formula.

Example 3

Find 75% of 180.

R ϭ 75% ϭ 0.75

B ϭ 180

P ϭ the unknown

P ϭ BR

P ϭ (180)(0.75)

ϭ 135

Example 4

75 is the number with a percent sign.

180 is the whole amount and follows the word of.

Use Formula 1.

3

\$45 is 9 % of what amount?

4

3

R = 9 % = 9.75% = 0.0975

4

B ϭ the unknown

P ϭ \$45

P

B =

R

\$45

B =

0.0975

ϭ \$461.54

943 % is the number with a percent sign.

Use Formula 2.

\$45 is the part.

82

Chapter 1

Basic Concepts

Example 5

What percent of 20 metres is 5 metres?

R ϭ the unknown

B ϭ 20 m

Pϭ5m

P

R =

B

5m

R =

20 m

ϭ 0.25 ϭ 25%

Example 6

Aluminum is 12% of the mass of a given car. This car has 186 kg of aluminum in it. What

is the total mass of the car?

R ϭ 12% ϭ 0.12

B ϭ the unknown

P ϭ 186 kg

P

B =

R

186 kg

B =

0.12

ϭ 1550 kg

Example 7

Use Formula 3.

20 m is the whole amount and follows the word of.

5 m is the part.

12 is the number with a percent sign.

Use Formula 2.

186 kg is the part.

A fuse is a safety device with a core. When too much current flows, the core melts and

breaks the circuit. The size of a fuse is the number of amperes of current the fuse can safely

carry. A given 50-amp (50-A) fuse blows at 20% overload. What is the maximum current

the fuse will carry?

First, find the amount of current overload:

R ϭ 20% ϭ 0.20

B ϭ 50 A

P ϭ the unknown

P ϭ BR

P ϭ (50 A)(0.20)

ϭ 10 A

20 is the number with a percent sign.

50 A is the base.

Use Formula 1.

The maximum current the fuse will carry is the normal current plus the overload:

50 A ϩ 10 A ϭ 60 A

Example 8

Georgia’s salary was \$600 per week. Then she was given a raise of \$50 per week. What percent raise did she get?

R ϭ the unknown

B ϭ \$600

P ϭ \$50

P

R =

B

\$50

R =

\$600

1

1

= 0.08 = 8 %

3

3

Use Formula 3.

\$600 is the base.

\$50 is the part.

1.14

Example 9

Rate, Base, and Part

83

Castings are listed at \$9.50 each. A 12% discount is given if 50 or more are bought at one

a.

b.

c.

What is the discount on one casting?

What is the cost of one casting?

What is the total cost?

a. Discount equals 12% of \$9.50.

R ϭ 12% ϭ 0.12

B ϭ \$9.50

P ϭ the unknown (the discount)

P ϭ BR

P ϭ (\$9.50)(0.12)

ϭ \$1.14 (the discount on one casting)

b. Cost (of one casting) ϭ list Ϫ discount

ϭ \$9.50 Ϫ \$1.14

ϭ \$8.36

c. Total cost ϭ cost of one casting times the number of castings

ϭ (\$8.36)(60)

ϭ \$501.60

You may also need to find the percent increase or decrease in a given quantity.

Example 10

Mary’s hourly wages changed from \$18.40 to \$19.55. Find the percent increase in her wages.

First, let’s find the change in her wages.

\$19.55 Ϫ \$18.40 ϭ \$1.15

Then, this change is what percent of her original wage?

R =

\$1.15

P

=

= 0.0625 = 6.25%

B

\$18.40

The process of finding the percent increase or percent decrease may be summarized by

the following formula:

percent increase (or percent decrease) =

Example 11

the change

* 100%

the original value

Normal ac line voltage is 115 volts (V). Find the percent decrease if the line voltage drops

to 109 V.

the change

* 100%

the original value

115 V - 109 V

=

* 100%

115 V

ϭ 5.22%

percent decrease =

84

Chapter 1

Basic Concepts

The triangle in Figure 1.36 can be used to help you remember the three percent formulas, as follows:

P

B

1. P ϭ BR

R

P

R

P

3. R =

B

2. B =

FIGURE 1.36

To find the part, cover P; B and R are next to each other on the same line, as in

multiplication.

To find the base, cover B; P is over R, as in division.

To find the rate, cover R; P is over B, as in division.

Exercises 1.14

Identify the rate (R), the base (B), and the part (P) in each

statement 1–10 (do not solve the problem):

1. 60 is 25% of 240.

2. 3313% of \$300 is \$100.

3. 40% of 270 is 108.

4. 72 is 15% of 480.

5. At plant A, 4% of the tires made were defective. Plant

A made 28,000 tires. How many tires were defective?

6. On the last test, 25 of the 28 students earned passing

grades. What percent of students passed?

7. A girls’ volleyball team won 60% of its games. The

team won 21 games. How many games did it play?

8. A rancher usually loses 10% of his herd every winter

due to weather. He has a herd of 15,000. How many

does he expect to lose this winter?

9. An electronics firm finds that 6% of the resistors it

makes are defective. There were 2050 defective resistors. How many resistors were made?

10. The interest on a \$500 loan is \$90. What is the rate of

interest?

When finding the percent, round to the nearest tenth of a

percent when necessary:

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

What percent of \$2080 is \$208?

The number 2040 is 7.5% of what number?

What percent of 5280 ft is 880 yd?

0.35 mi is 4% of what amount?

\$72 is 4.5% of what amount?

What percent of 7.15 is 3.5?

Find 235% of 48.

What percent of 81 is 151 ?

Find 28% of 32 volts (V).

Find 110% of 50.

21. A welder needs to complete 130 welds. If 97 have been

completed so far, what is the percent completed?

22. A welder makes high-quality welds 92% of the time.

Out of 115 welds, how many are expected to be of high

quality?

23. A small airport has a Cessna 172 rental plane. In one

month, 24 h of the 65 total rental hours were for lessons.

What percent of the total rental time was the plane

rented for lessons?

24. On a cross-country trip, 1.5 h were flown under VFR

(Visual Flight Rules), and 0.4 h was flown under IFR

(Instrument Flight Rules). What percent of the trip was

flown under IFR?

25. A car oil filter holds 0.3 qt of oil. The car holds 4.5 qt

of oil including the filter. What percent of the oil is in

the filter?

26. Air enters an air conditioner at the rate of 75 lb/h, and

the unit can remove 1.5 lb/h of moisture. If the air entering contains 2 lb/h moisture, what percent of the

moisture is removed?

27. Air flows through a duct at 2400 cubic feet per minute

(CFM). After several feet and a few vents, the airflow

decreases to 1920 CFM. What is the percent drop that

has occurred?

28. A building being designed will have fixed windows. Including the frame, the windows are 2 ft wide and 6 ft

high. The south wall is 78 ft 6 in. wide by 12 ft 2 in. high.

Local codes allow only 20% window area on south

walls. How many windows can you draw on this wall?

29. The embankment leading to a bridge must have a

maximum 3% slope. The change in elevation shown in

Illustration 1 must be dimensioned to meet these criteria. Find dimension A to complete the drawing.

1.14

Embankment

3%

A

Concrete

abutment

100 ft

ILLUSTRATION 1

30. A rectangular tank is being designed with an internal

catwalk around the inside as shown in Illustration 2. For

functional reasons, the walkway cannot be more than

3 in. above the liquid. The liquid level in the tank will

be maintained 43 full at all times. What height dimension

for the walkway should be put on the drawing?

86 ft

91 ft

Catwalk

66 ft

? ft

ILLUSTRATION 2

31. The application rate of a chemical is 243 lb/acre. How

many pounds are needed for 160 acres of corn? If the

chemical contains 80% active ingredients by weight,

how many pounds of active ingredients will be applied?

How many pounds of inert ingredients will be applied?

32. U.S. soybeans average 39% protein. A bushel of soybeans weighs 60 lb. How many pounds of protein are in

a bushel? A 120-acre field yields 45 bu/acre. How many

pounds of protein does that field yield?

33. A dairy cow produced 7310 lb of milk in a year. A gallon of milk weighs 8.6 lb. How many gallons of milk

did the cow produce? The milk tested at 4.2% butterfat.

How many gallons of butterfat did the cow produce?

34. You need 15% of a 60-mg tablet. How many mg would

you take?

35. Mary needs to give 40% of a 0.75-grain tablet. How

many grains does she give?

36. You need 0.15% of 2000 mL. How many millilitres do

you need?

37. What percent of 0.600 grain is 0.150 grain?

38. During a line voltage surge, the normal ac voltage increased from 115 V to 128 V. Find the percent increase.

Rate, Base, and Part

85

39. During manufacturing, the pressure in a hydraulic line

increases from 75 lb/in2 to 115 lb/in2. What is the percent increase in pressure?

40. The value of Caroline’s house decreased from \$93,500

to \$75,400 when the area’s major employer closed the

local plant and moved to another state. Find the percent

decrease in the value of her house.

41. Due to wage concessions, Bill’s hourly wages dropped

from \$25.50 to \$21.88. Find the percent decrease in his

wages.

42. A building has 28,000 ft2 of floor space. When an addition of 6500 ft2 is built, what is the percent increase in

floor space?

43. Two different items both originally selling for \$100.00

are on sale. One item is marked down 55%. The second

item is first marked down 40%, then an additional 15%.

Find the final sale price for each item.

44. A machinist is hired at \$22.15 per hour. After a 6-month

probationary period, the wage will increase by 32%. If

the machinist successfully completes the apprenticeship, what will the pay be per hour?

45. A homeowner harvests a tree to use for firewood. The

entire tree weighs 1640 lb. Of that, 95% is cut and split

into sticks of firewood. The rest is leaves and branches

too small to use for firewood. How much did the firewood weigh?

46. A fisherman catches a total of 125 lb of fish. When the

fish are cleaned, 59 lb of fillet remain and the rest is discarded. What percent of the fish was usable as fillets?

47. A flock of mallards (ducks) is called a sord. One particular mallard sord has 250 live mallards when the last

hatchling emerges in the spring. At the end of the following winter, the sord has 187 birds remaining before

the first egg hatches. What was the survival rate for the

sord?

48. In a local community, wildlife biologists estimate a

deer population of 135 on January 1. Over the following 12 months, there are 42 live births of deer fawn, 7

are killed by vehicles on the highway, 3 fawns are killed

by dogs, 5 are killed by hunters, and 10 die of disease

or other injury. As of December 31, what was the deer

population and what was the percentage change?

49. Populations of any organism increase when births exceed deaths. In a suburban area in the upper Midwest, the

large number of deer was becoming a problem. A Deer

Task Force survey suggested in 2006 that 20 deer per

square mile might be an acceptable population level for

the citizens in that area. Assume a current population

density of 25 deer per square mile and a population

86

Chapter 1

Basic Concepts

growth rate (births minus deaths) of 40% per year. a. If

there are no significant predators and hunting is not allowed, how many deer can the town planners expect per

square mile in the following year? b. How many deer can

the town planners expect per square mile in the following second year with the same population growth rate?

50. A community has a goal to decrease its municipal solid

waste (MSW) by 25% over a 5-year period. Assuming

the community has 75,000 residents who each average

4.6 lb of MSW each day, a. how much MSW would

each resident average each day if the goal were met?

b. How many tons of MSW would the community generate annually if the goal were met? c. Another nearby

larger community had decreased its annual MSW by

30% to 73,500 tons. How much was its previous annual

amount of MSW?

51. An invoice is an itemized list of goods and services

specifying the price and terms of sale. Illustration 3

shows an invoice for parts and labor for an addition to

a home for the week indicated. Complete the invoice.

Jose’s Plumbing Supply

120 East Main Street

Poughkeepsie, NY 12600

Satisfaction

Satisfaction Guaranteed

Guaranteed

Date:

6/25 — 6/29

Name:

Gary Jones

2630 E. Elm St.

City:

Poughkeepsie, NY 12600

Quantity

Quality

Quality Since

Since 1974

1974

Item

Cost/Unit

22 ea

3/4" fittings

\$1.33

14 ea

3/4" nozzles

\$3.89

12 ea

3/4" 90Њ ells

\$6.49

6 ea

3/4" faucets

\$7.43

6 ea

3/4" valves

\$8.76

6 ea

3/4" unions

\$5.54

5 ea

3/4" T-joints

\$6.45

4 ea

3/4" 45Њ ells

\$2.09

120 ft

3/4" type K copper pipe

\$1.69/ft

32 h

Labor

48.00/h

Total Cost

Total

Less 5%

Cash Discount

Net 30 Days

Net Total

ILLUSTRATION 3

1.14

Rate, Base, and Part

52. Illustration 4 shows an invoice for grain sold at a local

elevator. Complete the invoice.

Beardstown, Illinois 62618

Customer name:

Shaw Farms, Inc.

Since 1893

Account No.

Date

Gross wt–

pounds

Weight of

empty truck

Net wt–

pounds

7/2

21560

9160

12400

7/3

26720

9240

7/5

20240

7/6

7/8

Type of

grain

No. of

bushels*

3786

Price/bu

Amount

Wheat

\$ 5.67

5.71

\$1173.69

7480

Wheat

5.74

28340

9200

Wheat

5.81

26760

9160

Wheat

5.76

7/8

17880

7485

Wheat

5.76

10/1

25620

9080

Soybeans

11.72

10/1

21560

7640

Soybeans

11.69

10/2

26510

9060

Soybeans

11.68

10/2

22630

7635

Soybeans

11.65

10/4

22920

9220

Soybeans

11.72

10/5

20200

7660

Soybeans

11.81

10/6

25880

9160

Soybeans

11.90

10/7

21300

7675

Soybeans

11.84

10/8

18200

7665

Soybeans

11.79

10/12

26200

9150

Corn

4.68

10/12

22600

7650

Corn

4.65

10/13

27100

9080

Corn

4.66

10/15

22550

7635

Corn

4.61

10/15

23600

7680

Corn

4.59

10/17

26780

9160

Corn

4.63

10/18

28310

9200

Corn

4.69

10/21

21560

7665

Corn

4.67

10/22

25750

9160

Corn

4.65

Wheat

207

TOTAL

\$ 46,363.83

*Round to the nearest bushel. Note: Corn weighs 56 lb/bu; soybeans weigh 60 lb/bu; wheat weighs 60 lb/bu.

ILLUSTRATION 4

87

88

Chapter 1

Basic Concepts

53. Many lumberyards write invoices for their lumber by

the piece. (See Illustration 5.) Complete the invoice,

which is for the rough framing of the shell of a home.

KURT’S LUMBER

SOLD TO Robert Bennett

400 WEST OAK

32 Park Pl

QUANTITY

AKRON, OHIO 44300

E. Akron 44305

DATE 5/16

DESCRIPTION

UNIT PRICE

66

2ỵ x 4 ” x 16’, fir, plate material

30

2” x 4” x 10’, fir, plate studs

3.95

14

2” x 4” x 8’, fir, knee wall studs

3.39

17

2” x 6” x 12’, fir, kit. ceiling joists

6.59

4

2” x 12” x 12’, fir, kitchen girders

12.10

9

2” x 6” x 10’, fir, kitchen rafters

\$ 7.97

5.39

7

2” x 4” x 12’, fir, collar beams

10

2” x 8” x 12’, fir, 2nd floor joists

11.97

6

2” x 8” x 16’, fir, 2nd floor joists

16.89

11

2” x 8” x 20’, fir, 2nd floor joists

18.55

15

4’ x 8’ x 3/4”, T & G plywood

24.25

27

2” x 8” x 18’, fir, kitchen and living room rafters

16.95

7

2” x 8” x 16’, fir, kitchen and living room rafters

14.39

1

2” x 10” x 22’, fir, kitchen and living room ridge

24.96

10

1” x 8” x 14’, #2 white pine, sub facia

10.37

27

2” x 8” x 22’, fir, bedroom rafters

19.85

7

2” x 8” x 16’, fir, dormer rafters

12.25

1

2” x 10” x 20’, fir, bedroom ridge

17.85

7

2” x 6” x 20’, fir, bedroom ceiling joists

12.19

8

2” x 6” x 8’, fir, bedroom ceiling joists

3

2” x 12” x 14’, fir, stair stringers

17.65

80

4’ x 8’ x 1/2”, roof decking

17.29

7

rolls #15 felt building paper

20.65

1

50 lb #16 cement nails

33.59

3

30 lb galvanized roofing nails

34.97

250

TOTAL

4.97

3.49

precut fir studs

2.18

Subtotal

Less 2% cash discount

Subtotal

KURT’S LUMBER

5 3/4% sales tax

NET TOTAL

ILLUSTRATION 5

1.15

Powers and Roots

89

54. Complete the electronics parts invoice shown in Illustration 6.

APPLIANCE DISTRIBUTORS INCORPORATED

1400 West Elm Street

Sold to:

St. Louis, Missouri 63100

Maria’s Appliance Repair

9/26

Date:

1915 W. Main, Florissant, MO 63031

Quantity

Description

Unit price

Discount

3

67A76-1

\$ 18.58

40%

5

A8934-1

65.10

25%

5

A8935-1

73.95

25%

8

A8922-2

43.90

25%

2

A8919-2X

124.60

20%

5

700A256

18.80

15%

Net amount

SUBTOTAL

Appliance

Less 5% if paid

in 30 days

Distributors

Incorporated

TOTAL

ILLUSTRATION 6

1.15

Powers and Roots

The square of a number is the product of that number times itself. The square of 3 is 3 # 3

or 32 or 9. The square of a number may be found with a calculator as follows.

Example 1

Find 73.62 rounded to three significant digits.

73.6

x2

ϭ

5416.96

Thus, 73.62 ϭ 5420 rounded to three significant digits.

90

Chapter 1

Basic Concepts

Example 2

Find 0.1352 rounded to three significant digits.

x2

.135

ϭ

0.018225

Thus, 0.1352 ϭ 0.0182 rounded to three significant digits.

The square root of a number is that positive number which, when multiplied by itself, gives the original number. The square root of 25 is 5 and is written as 125. The symbol 2 is called a radical.

Example 3

Find the square roots of a. 16, b. 64, c. 100, and d. 144.

a.

b.

c.

d.

216 ϭ 4 because 4 # 4 ϭ 16

264 ϭ 8 because 8 # 8 ϭ 64

1100 ϭ 10 because 10 # 10 ϭ 100

1144 ϭ 12 because 12 # 12 ϭ 144

Numbers whose square roots are whole numbers are called perfect squares. For example, 1, 4, 9, 16, 25, 36, 49, and 64 are perfect squares.

The square root of a number may be found with a calculator as follows.

Example 4

Find 121.4 rounded to three significant digits.

͙ළ

21.4

ϭ

4.626013402

Thus, 121.4 ϭ 4.63 rounded to three significant digits.

Example 5

Find 10.000594 rounded to three significant digits.

͙ළ

.000594

ϭ

0.024372115

Thus, 10.000594 = 0.0244 rounded to three significant digits.

The cube of a number is the product of that number times itself three times. The cube

of 5 is 5 # 5 # 5 or 53 or 125.

1.15

Example 6

Powers and Roots

91

Find the cubes of a. 2, b. 3, c. 4, and d. 10.

a.

b.

c.

d.

23 ϭ 2 # 2 # 2 ϭ 8

33 ϭ 3 # 3 # 3 ϭ 27

43 ϭ 4 # 4 # 4 ϭ 64

103 ϭ 10 # 10 # 10 ϭ 1000

The cube of a number may be found with a calculator as follows:

Example 7

Find 123.

ϭ

*3

12

1728

Thus, 123 ϭ 1728.

yx

*Some calculators use the

Example 8

button to find a power.

Find 4.253 rounded to three significant digits.

4.25

ϭ

3

76.765625

Thus, 4.253 ϭ 76.8 rounded to three significant digits.

The cube root of a number is that number which, when multiplied by itself three

3

times, gives the original number. The cube root of 8 is 2 and is written as 18. (Note:

3

2 # 2 # 2 ϭ 8. The small in the radical is called the index.)

Example 9

Find the cube roots of a. 8, b. 27, and c. 125.

3

a. 18 ϭ 2 because 2 # 2 # 2 ϭ 8

3

b. 127 ϭ 3 because 3 # 3 # 3 ϭ 27

3

c. 1125 ϭ 5 because 5 # 5 # 5 ϭ 125

Numbers whose cube roots are whole numbers are called perfect cubes. For example,

1, 8, 27, 64, 125, and 216 are perfect cubes.

The cube root of a number may be found with a calculator as follows.

Example 10

3

Find 1512.

3

a. If your calculator has a 1

3

͙ළ

512

button,

ϭ

8

3

Thus, 1512 ϭ 8.