Notesale: Turn your study into money

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---

Chapter 5

Percentages
5
...
The
use of percentages is very common in many aspects
of commercial life, as well as in engineering
...

For this chapter you will need to know about decimals
and fractions and be able to use a calculator
...
e
...

• Interest rates indicate the cost at which we can borrow money
...
5% interest
rate for a year, it will cost you 6
...
If you repay
the loan in 1 year, how much interest will you have
paid?
• A pair of trainers in a shop cost £60
...
How much will you
pay?
• If you earn £20 000 p
...
and you receive a 2
...
What will be its cost?
When we have completed his chapter on percentages
you will be able to understand how to perform the above
calculations
...
For example, the fraction
100
and is read as ‘forty per cent’
...


DOI: 10
...
00005-3

5
...
2
...

Problem 1
...
015 as a percentage
To express a decimal number as a percentage, merely
multiply by 100, i
...

0
...
015 × 100%
= 1
...

Problem 2
...
275 as a percentage
0
...
275 × 100%
= 27
...
2
...

Problem 3
...
5% as a decimal number
6
...
5
= 0
...


34 Basic Engineering Mathematics
Problem 4
...
5% as a decimal number
17
...
5% =
100
= 0
...
2
...

Problem 5
...
5%
5
Problem 6
...
3157889
...
32% correct to 2 decimal places
Problem 7
...
Is the second
mark better or worse than the first?

57/79 =

57 57
5700
=
× 100% =
%
79 79
79

= 72
...
13% correct to 2 decimal places
Hence, the second test is marginally better than the
first test
...


5
...
4 To convert a percentage to a fraction
A percentage is converted to a fraction by dividing by
100 and then, by cancelling, reducing it to its simplest
form
...


Express 75% as a fraction

75
100
3
=
4
75
The fraction
is reduced to its simplest form by can100
celling, i
...
dividing both numerator and denominator
by 25
...

37
...
5% as a fraction

37
...

1
...
0032

2
...
734

3
...
057

4
...
374

5
...
285
6
...

7
...
25% as a decimal number
...

16
5
9
...

8
...
Express as percentages, correct to 3 significant
figures,
19
11
7
(b)
(c) 1
(a)
33
24
16
11
...

12
9
5
6
(a)
(b)
(c)
(d)
21
17
9
11
12
...

13
...
25% as a fraction in its simplest
form
...
Express 56
...

15
...

Decimal number Fraction Percentage
0
...
5
× 32
100
= 4 minutes

12
...

Alternatively, if the time is reduced by 12
...
5% = 87
...
e
...
5
× 32
100
= 28 minutes

87
...
A 160 GB iPod is advertised as
costing £190 excluding VAT
...
5%, what will be the total cost of the iPod?
17
...
25
100
Total cost of iPod = £190 + £33
...
25
VAT = 17
...
175 ×
£190 = £223
...
3
...

Problem 13
...
3

Further percentage calculations

5
...
1 Finding a percentage of a quantity
To find a percentage of a quantity, convert the percentage
to a fraction (by dividing by 100) and remember that ‘of’
means multiply
...
Find 27% of £65
27
× 65
27% of £65 =
100
= £17
...
In a machine shop, it takes 32
minutes to machine a certain part
...
5%
...
94444
...
Express 47 minutes as a percentage
of 2 hours, correct to 1 decimal place
Note that it is essential that the two quantities are in the
same units
...
2% correct to
1 decimal place

36 Basic Engineering Mathematics
5
...
3 Percentage change
Percentage change is given by
new value − original value
× 100%
...
A box of resistors increases in price
from £45 to £52
...
6% = percentage change in cost

% change =

Problem 16
...
The nearest speed available on the
machine is 412 rev/min
...


Calculate 43
...


2
...


3
...


A block of Monel alloy consists of 70%
nickel and 30% copper
...
2 g
of nickel, determine the mass of copper in the
block
...


An athlete runs 5000 m in 15 minutes 20
seconds
...
5%
...


8
...
5%
iron and the remainder aluminium
...
8 kg
mass of the alloy
...


A computer is advertised on the internet at
£520, exclusive of VAT
...
5%, what is the total cost of the computer?

10
...

11
...
Express this as a percentage of the
whole day, correct to 1 decimal place
...
Express 408 g as a percentage of 2
...

13
...
Calculate the percentage
pay increase, correct to 3 significant figures
...
A metal rod 1
...
6 mm
...

15
...
5% of a length of wood is 70 cm
...
A metal rod, 1
...
Calculate the
percentage increase in length
...
42 grams
(c)

147% of 14
...


4
...
Determine the percentage that
is unsatisfactory
...


Express
(a) 140 kg as a percentage of 1 t
...
4

More percentage calculations

5
...
1 Percentage error
Percentage error =

error
× 100%
correct value

(b) 47 s as a percentage of 5 min
...
4 cm as a percentage of 2
...


Problem 17
...
5 mm
...
What is the percentage error in the
measurement?
error
× 100%
correct value
64
...
5
150
=
× 100% =
%
63
63
= 2
...
A couple buys a flat and make an
18% profit by selling it 3 years later for £153 400
...
e
...

new value
× 100
Original cost =
100 + % change

The percentage measurement error is 2
...
38% error
...
The voltage across a component in
an electrical circuit is calculated as 50 V using
Ohm’s law
...
4 V
...
4 − 50
× 100%
=
50
...
4
40
=
× 100% =
%
50
...
4
= 0
...
79% too
low, which is sometimes written as −0
...


In this case, it is a 35% reduction in price, so we
new value
use
× 100, i
...
a minus sign in the
100 − % change
denominator
...
5
× 100
=
100 − 35
149
...
An electrical store makes 40% profit
on each widescreen television it sells
...
e
...

new value
Dealer cost =
× 100
100 + % change

new value
× 100%
100 ± % change

Problem 19
...
50 in a sale for a
DVD player which is labelled ‘35% off’
...
4
...


5
...
3 Percentage increase/decrease and
interest
New value =

100 + % increase
× original value
100

Problem 22
...
25% interest per annum
...
25
× £3600
100
106
...
0625 × £3600

Value after 1 year =

annum
...

8
...
5%
...
Calculate the increase he will actually receive per
month
...


Five mates enjoy a meal out
...
They add a 12
...
How much does each pay?

= £3825
Problem 23
...
5%
...
What is the new price?
100 + 6
...
5
=
× £2, 400 = 1
...
In December a shop raises the cost of a 40
inch LCD TV costing £920 by 5%
...
What is the sale price of
the TV?
11
...
What did he pay originally for the
business?

1
...
The
length is incorrectly measured as 36
...

Determine the percentage error in the measurement
...
A drilling machine should be set to
250 rev/min
...
Calculate the
percentage overspeed
...


When a resistor is removed from an electrical circuit the current flowing increases from
450 μA to 531 μA
...


13
...
Determine the masses of the three
elements present
...


In a shoe shop sale, everything is advertised
as ‘40% off’
...


Over a four year period a family home
increases in value by 22
...

What was the value of the house 4 years ago?

14
...

Determine the percentage of each of these
three constituents correct to the nearest 1%
and the mass of cement in a two tonne dry
mix, correct to 1 significant figure
...


An electrical retailer makes a 35% profit on
all its products
...


The cost of a sports car is £23 500 inclusive
of VAT at 17
...
What is the cost of the car
without the VAT added?

7
...
75% per

15
...
How
much ore is needed to produce 3600 kg of
iron?
16
...
5 ± 8% mm
...

17
...
If
the efficiency of the engine is 75%, determine
the power input
...
The marks available are shown in brackets at the
end of each question
...


Convert 0
...


(2)

2
...
4375 to a mixed number
...


Express

4
...

32
Express 0
...


(2)
(2)

5
...
0572953 correct to 4 significant
figures
...


Evaluate
(a) 46
...
085 + 6
...
07
(b) 68
...
34

(4)

7
...
37 × 1
...


Evaluate 250
...
1 correct to 1 decimal
place
...
2 × 12
(2)

9
...
Evaluate the following, correct to 4 significant
(3)
figures: 3
...
73 + 1
...
Evaluate 6
...
54 correct to 3 decimal
places
...
Evaluate
−
correct to 4 significant
0
...
065
figures
...
Evaluate 6 − 4 as a mixed number and as a
7
9
decimal, correct to 3 decimal places
...
Evaluate,
correct


1
...
673
√
4
...
If a = 0
...
85, c = 0
...
7 and
e = 0
...
Evaluate the following, each correct to 2 decimal
places
...
22 × 0
...
8 × 12
...
692
(4)
(b)
√
17
...
98
21
...
6 km = 1 mile, determine the speed of
45 miles/hour in kilometres per hour
...
The area A of a circle is given by A = πr 2
...
73 cm, correct
to 2 decimal places
...
The potential difference, V volts, available at battery terminals is given by V = E − I r
...
23, I = 1
...
60
(3)

23
...
20, v = 10
...
81, given that
W v2
B=
(3)
2g

4 1 3
14
...

(3)

24
...
25% as a fraction in its simplest
form
...
Evaluate
form
...
Evaluate resistance, R, given
=
+
+
R
R1 R2
1
when R1 = 3
...
2 k and
R3
R3 = 13
...

(3)

25
...
5% of a length of wood is 70 cm
...
A metal rod, 1
...
Calculate the percentage increase in length
...
A man buys a house and makes a 20% profit when
he sells it three years later for £312 000
...

The n’th term is: a + (n − 1)d
n
Sum of n terms, Sn = [2a + (n − 1)d]
2

Geometric progression:
If a = first term and r = common ratio, then the geometric progression is: a, ar, ar2 ,
...

4
...

10
...

16
...


Exercise 5 (page 11)

19 kg
2
...
479 mm
−66
5
...
−225
−2136
8
...
£10 7701
−4
11
...
5914
189 g
14
...
$15 333
89
...

3
...

5
...

9
...

9
...

17
...

(a) 8613 kg (b) 584 kg
(a) 351 mm (b) 924 mm
(a) 10 304 (b) −4433
6
...

(a) 8067 (b) 3347
10
...

4
...

8
...


(a) 48 m (b) 89 m
(a) 1648 (b) 1060
18 kg

1
...
14
7
...
88
8
...
1016/B978-1-85617-697-2
...


3 4 1 3 5
, , , ,
7 9 2 5 8
9
10
...
1
15
16
18
...


6
...

21
...
2
3
...
11
8
...

13
...

18
...

11
...

19
...

8
...

16
...


4
...
2

5
...
5

5
12
1
9
...

23
4
...
15

3
28
8
10
...


15
...
400 litres
22
...

15

1
...
59
6
...
−1

2
5

Exercise 6 (page 13)

11
...
7

(a) £1827 (b) £4158

Exercise 3 (page 6)
1
...

5
...

9
...
2

11
...
4
20
17
12
...
−

3
...
2

1
6

3
4
1
9
...
4

13
20
1
10
...


Answers to practice exercises
Exercise 14 (page 25)

Chapter 3

1
...
571
5
...
96
8
...
0871

Exercise 8 (page 17)
1
...
23
8
...


13
20
21
6
...
(a) 1
50
5
...
0
...


4
...
6

7
16

(e) 16

17
80

1
...

7
...

13
...
182
2
...
122
3
...
82
0
...
0
...
2
...
273
8
...
256
9
...
30366
6
1
...
3
...
37
...
2 × 10
14
...
767 ×10
15
...
32 ×106

12
...
6875 13
...
21875 14
...
1875

Exercise 16 (page 27)
1
...
4667

Exercise 9 (page 18)
1
...
18
5
...
297

2
...
785
3
...
38
6
...
000528

2
...
3
6
...
3

4
...
27

3
...
54
7
...
52 mm

4
...
83

13
14

3
...
458

6
...
7083

7
...
2
...
3

1
3

10
...
0776

1
...
9205
5
...
4424
9
...
6992

2
...
7314
6
...
0321
10
...
8452

3
...
9042
7
...
4232

4
...
2719
8
...
1502

Exercise 18 (page 28)

4
...
47
...
385
...
582
...
9 6
...
82
7
...
1
8
...
6
0
...
0
...
1
...
53
...
84 14
...
69
15
...
81 (b) 24
...
00639 (b) 0
...
(a) 8
...
6˙
2400

1
...
995
5
...
6977
9
...
520

Exercise 12 (page 23)
3
...
62
7
...
330

4
...
832
8
...
45

Exercise 13 (page 24)
1
...
25
2
...
0361 3
...
923 4
...
296 × 10−3
5
...
4430 6
...
197 7
...
96 8
...
0549
9
...
26 10
...
832 × 10−6

2
...
782
6
...
92
10
...
3770

3
...
72
7
...
0

4
...
42
8
...
90

Exercise 19 (page 29)
1
...

7
...


Chapter 4

2
...
1
...
12
...


Exercise 17 (page 27)

Exercise 11 (page 20)

1
...
797
5
...
42
9
...
59

1
21
9
...
567
5
...

5
...

13
...

18
...
40
3
...
13459
4
...
9
6
...
4481 7
...
36 × 10−6
9
...
625 × 10−9
10
...
70

Exercise 15 (page 25)

3
125

15
...
28125

1
...
3
5
...
3

341

A = 66
...
144 J
14 230 kg/ m3

2
...

8
...


C = 52
...
407 A
628
...
1 m/s

3
...

9
...


R = 37
...
02 mm
224
...
526 

Exercise 20 (page 30)
1
...

7
...

12
...
27
2
...
1 W
3
...
61 V
F = 854
...
I = 3
...
T = 14
...
96 J
8
...
77 A 9
...
25 m
A = 7
...
V = 7
...
53 h (b) 1 h 40 min, 33 m
...
h
...
02 h (d) 13
...
£556 2
...
264 kg 4
...
14
...
(a) 0
...
(a) 440 K (b) 5
...
0
...
173
...
5
...
37
...
128
...
0
...
0
...
0
...
38
...
(a) 21
...
2% (c) 169%
13
5
9
11
...

13
...

20
16
16
1
15
...
25, D = 25%, E = 0
...
60, H = 60%, I = 0
...
(a) 2 mA (b) 25 V 2
...
685
...
83 lb10 oz
5
...
1 litres (b) 16
...
29
...
584
...
$1012

Exercise 28 (page 46)
Exercise 22 (page 36)
1
...

5
...

10
...


21
...
9
...
4 t (b) 8
...
67%
14 minutes 57 seconds
37
...
39
...
7%
15
...
60 m

(c) 20
...

(c) 5
...

8
...

12
...

16
...
5%

2
...
8 g
£611
38
...
3
...
20 days
3
...
18 (b) 6
...
3375 4
...
(a) 300 × 103 (b) 0
...

5
...

13
...

15
...


2
...
18%
3
...
£175 000
£260
6
...
£9116
...
£50
...
60 10
...
70 11
...
7
...
6 kg, B 0
...
5 kg
54%, 31%, 15%, 0
...
5 mm, 11
...
600 kW

Chapter 6

1
...
±5

2
...
±8

3
...
100

5
...
64

4
...
1

Exercise 30 (page 50)
1
5
...
8
3
...

9
1
or 0
...
5 11
...
100 8
...

100
12
...
36
14
...
1 16
...
5 or
18
...

or 0
...
1
3
243
2
1
...
39

Exercise 24 (page 41)
1
...
3
...
47 : 3
1
4
...
5 hours or 5 hours 15 minutes
4
6
...
12 cm
8
...


1
3 × 52

5
...
1 : 15 2
...
25% 4
...
6 kg
5
...
3 kg 6
...
−
18
9
...


1
3
7 × 37

6
...
−1
14
...
±
2

3
...
−3
15
...


1
210 × 52

8
...

45
9

2
3

344 Basic Engineering Mathematics
Chapter 10

Chapter 11

Exercise 39 (page 69)
1
...

5
...

9
...

13
...

17
...

21
...

25
...


x 2 + 5x

+6
+9
4x 2 + 22x + 30
a 2 + 2ab + b2
a 2 − 2ac + c2
4x 2 − 24x + 36
64x 2 + 64x + 16
3ab − 6a 2
2a 2 − 3ab − 5b2
7x − y − 4z
x 2 − 4x y + 4y 2
0
4ab − 8a 2
2 + 5b2
4x 2 + 12x

Exercise 42 (page 75)

2
...

6
...

10
...

14
...

18
...

22
...

26
...


2x 2 + 9x

+4
− 12
2 pqr + p2 q 2 + r 2
x 2 + 12x + 36
25x 2 + 30x + 9
4x 2 − 9
r 2 s 2 + 2rst + t 2
2x 2 − 2x y
13 p − 7q
4a 2 − 25b2
9a 2 − 6ab + b2
4−a
3x y + 9x 2 y − 15x 2
11q − 2 p
2 j2 +2 j

Exercise 40 (page 71)
2(x + 2)
p(b + 2c)
4d(d − 3 f 5)
2q(q + 4n)
bc(a + b2 )
3x y(x y 3 − 5y + 6)
7ab(3ab − 4)


2x y x − 2y 2 + 4x 2 y 3
3x
17
...
(a + b)(y + 1)
22
...

3
...

7
...

11
...

15
...
0 19
...
( p + q)(x + y)
23
...

4
...

8
...

12
...

16
...
1

2
...
1

7
...
2
16
...
−4

3
...

2
13
...
−3

12
...
6

9
...
−2

2
...
4 + 3a
6
...
10y 2 − 3y +
9
...


1
− x − x2
5

1
7

1
4

8
...
5

2
...
−4

6
...
−4

8
...
−10

12
...
9

17
...
±12

22
...
−15t
12
...
2
...
2

5
...
2

10
...
3

14
...
−6

18
...
4

20
...
±3

24
...

4
...

6
...
8 m/s2
3
...
472
(a) 1
...
30 m/s2

Exercise 45 (page 80)
1
...
45◦ C
7
...
0
...
50
8
...
30
6
...
3
...
d = c − e − a − b

1
− 4x
3

10
...
2x + 8x 2
3
...

− 4x
2

5
...
R =
I
c
7
...
v =

y
7
v −u
4
...
y = (t − x)
3
y−c
8
...
x =

Answers to practice exercises
I
PR
E
11
...
C = (F − 32)
9
9
...
L =

XL
2π f

12
...
x = a(y − 3)
14
...
64 mm

1
2π CX C

*

1
√

Z2 −

14
...
1 × 10−6

aμ
ρCZ 4 n

2

Chapter 13
Exercise 47 (page 87)

Exercise 49 (page 92)

S −a
a
1
...
x =

yd
d
(y + λ) or d +
λ
λ

3
...
D =

AB 2
5E y

5
...
R2 =

R R1
R1 − R

E −e
E − e − Ir
or R =
−r
I
I

y

ay
8
...
x = 
2
4ac
(y 2 − b2 )

7
...
R =
πθ
√
Z 2 − R2
, 0
...
L =
2π f
10
...
u =




xy
1
...
r =
(1 − x − y)
c
5
...
b =
2( p2 + q 2 )
9
...
L =

8S 2
3d

Q
, 55
mc
+ d, 2
...
L =
μ−m


x−y
6
...
R = 4

uf
, 30
u− f


2dgh
, 0
...
v =
0
...
v =

x = 4, y = 2
x = 2, y = 1
...
5, n = 0
...

4
...

8
...

12
...

16
...
a = N 2 y − x

Exercise 48 (page 89)

1
...

5
...

9
...

13
...


1
...

5
...


p = −1, q = −2
a = 2, b = 3
x = 3, y = 4
x = 10, y = 15

2
...

6
...


x = 4, y = 6
s = 4, t = −1
u = 12, v = 2
a = 0
...
40

Exercise 51 (page 96)
1
1
1
...
p = , q =
4
5
5
...
x = 5, y = 1
4

1
1
2
...
x = 10, y = 5
1
6
...
1

Exercise 52 (page 99)
1
...

5
...

8
...
2, b = 4
u = 12, a = 4, v = 26
m = −0
...
00426, R0 = 22
...
I1 = 6
...
62
4
...
a = 12, b = 0
...
F1 = 1
...
5

Exercise 53 (page 100)
1
...
x = 5, y = −1, z = −2

2
...
x = 4, y = 0, z = 3

346 Basic Engineering Mathematics
5
...

9
...

11
...
x = 1, y = 6, z = 7
x = 5, y = 4, z = 2 8
...
5, y = 2
...
5
i1 = −5, i2 = −4, i3 = 2
F1 = 2, F2 = −3 F3 = 4

Exercise 57 (page 109)
1
...

6
...

10
...
191 s 2
...
345 A or 0
...
619 m or 19
...
066 m
1
...
165 m
12 ohms, 28 ohms

3
...

7
...


7
...
0133
86
...
4 or −4
4
...
5 or 1
...

10
...

16
...
−2 or −

2
3

2
...
0 or −
3
8
...
−3 or −7
14
...
−3
20
...
5

1
...
2 or −2

2
1
2
1
2
...

12
...

18
...
−1 or 1
...

or −
2
5
2
or −3
28
...
1 or −
3
7
4
1
29
...

2
3
27
...

4
21
...
4 or −7

31
...
x 2 + 5x + 4 = 0
35
...
2 or −6

or −
or

or −

Chapter 15
Exercise 59 (page 112)

1
3

1
3
1
1
6
...
2
8
...
1 10
...
2
12
...
100 000 14
...

32
1
16
...
01 17
...
e3
16
1
...
x 2 + 3x − 10 = 0
34
...
x 2 − 1
...
68 = 0

2
...
3

4
...


Exercise 60 (page 115)

Exercise 55 (page 106)
1
...
732 or −0
...
1
...
135
5
...
443 or 0
...
x = 0, y = 4 and x = 3, y = 1

2
...
137 or 0
...
1
...
310
6
...
851 or 0
...
log 6
5
...
log 15
6
...
log 2
7
...
log 3
8
...
log 10 10
...
log 2
12
...
log 16 or log24 or 4 log2
14
...

3
...

7
...

11
...


0
...
137
2
...
719
3
...
108
0
...
351
1
...
081
4 or 2
...
562 or 0
...

4
...

8
...

12
...
296 or −0
...
443 or −1
...
434 or 0
...
086 or −0
...
176 or −1
...
141 or −3
...
0
...
1

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