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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.
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82
Chapter 1
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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.
■
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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
time. We buy 60 castings.
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 =
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■
84
Chapter 1
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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.
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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
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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
Address:
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
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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
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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
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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.
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■
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.
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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.
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