Notesale: Turn your study into money

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Chapter 3

Decimals
3
...

There are a number of everyday occurrences in which we
use decimal numbers
...
5 MHz FM; 107
...

In a shop, a pair of trainers cost, say, £57
...
95 is
another example of a decimal number
...
95 is a decimal
fraction, where a decimal point separates the integer, i
...

57, from the fractional part, i
...
0
...
95 actually means (5 × 10) + (7 × 1)

 

1
1
+ 9×
+ 5×
10
100

3
...

Problem 1
...
375 to a proper fraction in
its simplest form
0
...
375 may be written as
i
...

1000
375
0
...
1016/B978-1-85617-697-2
...

3
Hence, the decimal fraction 0
...

Problem 2
...
4375 to a mixed number

0
...
4375 may be written as
i
...

10000
4375
0
...

7
(v) Hence, 0
...
4375 = 3
16
number
...
Express as a decimal fraction
8
To convert a proper fraction to a decimal fraction, the
numerator is divided by the denominator
...
8 7 5
8 7
...
Place the 0 above the 7
...
000
(iii) 8 into 70 goes 8, remainder 6
...

(iv) 8 into 60 goes 7, remainder 4
...

(v) 8 into 40 goes 5, remainder 0
...

7
Hence, the proper fraction = 0
...

13
Problem 4
...
Place the 5 above
the next zero
...
8125
16
13
Thus, the mixed number 5 = 5
...

Now try the following Practice Exercise
Practice Exercise 8 Converting decimals to
fractions and vice-versa (answers on
page 341)
1
...
65 to a proper fraction
...


Convert 0
...


3
...
175 to a proper fraction
...


Convert 0
...


5
...

(a) 0
...
84
(c) 0
...
282
(e) 0
...
Convert 4
...

7
...
44 to a mixed number
...


Convert 10
...


9
...
4375 to a mixed number
...
Convert the following to mixed numbers
...

 0
...
0 0 0 0

(a) 1
...
35

12
...
Place the 8 above
the first zero after the decimal point and carry the
2 remainder to the next digit on the right, making
it 20
...
Place the 1 above
the next zero and carry the 4 remainder to the next
digit on the right, making it 40
...
Place the 2 above
the next zero and carry the 8 remainder to the next
digit on the right, making it 80
...
125

5
as a decimal fraction
...

16
7
as a decimal fraction
...

16
9
Express as a decimal fraction
...
Express

(i) 16 into 13 will not go
...

(ii) Place the decimal point above the decimal point
of 13
...
275
(e) 16
...

14
...


3
...


18 Basic Engineering Mathematics
For example,
3

3
= 3
...
For
example,
5
1 = 1
...
is a non-terminating decimal
7

Note that the zeros to the right of the decimal point do
not count as significant figures
...
Express 14
...


The answer to a non-terminating decimal may be
expressed in two ways, depending on the accuracy
required:

2
...
7846 correct to 4 significant figures
...
Express 43
...


correct to a number of significant figures, or

(b) correct to a number of decimal places i
...
the
number of figures after the decimal point
...

For example,

3
...
3792 correct to 2 decimal places
...
Express 1
...

6
...
0005279 correct to 3 significant
figures
...
714285
...
714 correct to 4 significant figures
= 1
...

The last digit in the answer is increased by 1 if the next
digit on the right is in the group of numbers 5, 6, 7, 8 or
9
...
7142857
...
7143 correct to 5 significant figures
= 1
...

Problem 5
...
36815 correct to
(a) 2 decimal places, (b) 3 significant figures,
(c) 3 decimal places, (d) 6 significant figures
(a)

15
...
37 correct to 2 decimal places
...
36815 = 15
...

(c)

15
...
368 correct to 3 decimal places
...
36815 = 15
...

Problem 6
...
004369 correct to
(a) 4 decimal places, (b) 3 significant figures
(a)

0
...
0044 correct to 4 decimal places
...
004369 = 0
...


3
...
This is demonstrated in the
following worked examples
...
Evaluate 46
...
06 + 2
...
09
and give the answer correct to 3 significant figures
The decimal points are placed under each other as
shown
...

46
...
06
2
...
09
52
...
Place 5 in the hundredths column
...

(ii) 8 + 0 + 4 + 0 + 1 (carried) = 13
...
Carry the 1 into the units
column
...
Place the 2 in
the units column
...


Decimals
(iv) 4 + 1(carried) = 5
...

Hence,
46
...
06 + 2
...
09 = 52
...
4, correct to 3
significant figures

19

Now try the following Practice Exercise
Practice Exercise 10 Adding and
subtracting decimal numbers (answers on
page 341)
Determine the following without using a calculator
...
Evaluate 64
...
77 and give the
answer correct to 1 decimal place
As with addition, the decimal points are placed under
each other as shown
...
46
−28
...
69
(i) 6 − 7 is not possible; therefore ‘borrow’ 1 from
the tenths column
...
Place
the 9 in the hundredths column
...
This gives 13 − 7 = 6
...

(iii) 3 − 8 is not possible; therefore ‘borrow’ from the
hundreds column
...
Place
the 5 in the units column
...


Evaluate 37
...
6, correct to 3 significant figures
...


Evaluate 378
...
85, correct to 1 decimal
place
...


Evaluate 68
...
84 − 31
...


4
...
841 − 249
...
883, correct
to 2 decimal places
...


Evaluate 483
...
44 − 67
...


6
...
22 −349
...
336 +56
...


7
...
1
...

82
...
Place the 3 in the hundreds column
...
41

8
...
67

Hence,
64
...
77 = 35
...
7 correct to 1 decimal place
Problem 9
...
64 − 59
...
66 +
38
...
64 + 38
...
14
...
826 + 79
...
486
...
140
−139
...
654
Hence, 351
...
486 = 211
...
7, correct to 4 significant figures
...
1

3
...

This is demonstrated in the following worked examples
...


Evaluate 37
...
4

(vi) 6 × 12 = 72; place the 72 below the 82
...

(viii) Bring down the 5 to give 105
...
500
(x) 8 × 12 = 96; place the 96 below the 105
...


(i) 376 × 54 = 20304
...


(ii) As there are 1 + 1 = 2 digits to the right of
the decimal points of the two numbers being
multiplied together, 37
...
4, then

(xiii) 12 into 90 goes 7; place the 7 above the first
zero of 442
...
6 × 5
...
04

(xiv) 7 × 12 = 84; place the 84 below the 90
...

(xvi) Bring down the 0 to give 60
...
Evaluate 44
...
2, correct to
(a) 3 significant figures, (b) 2 decimal places
44
...
2 =

44
...
2

The denominator is multiplied by 10 to change it into an
integer
...
Thus,
44
...
25 × 10 442
...
2
1
...

 36
...
500
36
82
72
105
96
90
84
60
60
0
(i) 12 into 44 goes 3; place the 3 above the second
4 of 442
...
500
(iii) 44 − 36 = 8
...
500
442
...
875
(xviii) Hence, 44
...
2 =
12
So,
(a)

44
...
2 = 36
...


(b) 44
...
2 = 36
...

2
Problem 12
...
666666
...
666666
...
667 correct to 4 significant figures
...
6666

...
6 recurring and is

...
6
Now try the following Practice Exercise
Practice Exercise 11 Multiplying and
dividing decimal numbers (answers on
page 341)
In Problems 1 to 8, evaluate without using a
calculator
...


1
...
57 × 1
...
500

2
...
92 × 0
...


Evaluate 167
...
3

4
...
6 × 1
...


Evaluate 548
...
2

6
...
3 ÷ 1
...


7
...
48 ÷ 0
...


8
...
4 ÷ 1
...

4
, correct to 3 significant figures
...

10
...
1 , correct to 4 significant figures
...


5
, correct to 3 decimal places
...
13 , correct to 2 decimal places
...
8 , correct to 3 significant figures
...
53

15
...
8 ÷ 17, (a) correct to 4 significant figures and (b) correct to 3 decimal
places
...
0147
, (a) correct to 5 decimal
2
...


16
...



...
6
(b) 5
...
Evaluate (a)
1
...
A tank contains 1800 litres of oil
...
75 litres can be filled from
this tank?

21

List of formulae
Laws of indices:

Areas of plane figures:

a m × a n = a m+n
a m/n =

√
n m
a

am
an

= a m−n (a m )n = a mn

a −n =

Quadratic formula:
If ax 2 + bx + c = 0

1
an

Area = l × b

(i) Rectangle

a0 = 1
b

√
−b ± b2 − 4ac
x=
2a

then

Equation of a straight line:

l

(ii) Parallelogram Area = b × h

y = mx + c

Definition of a logarithm:
If y = a x

then

h

x = loga y

Laws of logarithms:
log(A × B) = log A + log B
 
A
= log A − log B
log
B

b

(iii) Trapezium

log An = n × log A

1
Area = (a + b)h
2
a

Exponential series:
ex = 1 + x +

x2 x3
+
+···
2! 3!

h

(valid for all values of x)
b

Theorem of Pythagoras:
b 2 = a 2 + c2

(iv) Triangle

Area =

1
×b×h
2

A

c

B

b
h

a

C
b

List of formulae
Area = πr 2 Circumference = 2πr

(v) Circle

(iii) Pyramid
If area of base = A and
perpendicular height = h then:

r

s

␪

Volume =

1
× A×h
3

r

2π radians = 360 degrees

Radian measure:

h

For a sector of circle:
θ◦
(2πr) = rθ
360

(θ in rad)

1
θ◦
(πr 2 ) = r 2 θ
360
2

(θ in rad)

s=

arc length,

shaded area =

Equation of a circle, centre at origin, radius r:
x 2 + y2 = r 2
Equation of a circle, centre at (a, b), radius r:

Total surface area = sum of areas of triangles
forming sides + area of base
(iv) Cone
1
Volume = πr2 h
3
Curved Surface area = πrl

(x − a)2 + (y − b)2 = r 2

Total Surface area = πrl + πr2

Volumes and surface areas of regular
solids:
l

(i) Rectangular prism (or cuboid)

h

Volume = l × b × h
Surface area = 2(bh + hl + lb)

r

(v) Sphere

l

h

4
Volume = πr3
3
Surface area = 4πr2

b

(ii) Cylinder
Volume = πr2 h
Total surface area = 2πrh + 2πr2
r

r

h

337

338 Basic Engineering Mathematics
Areas of irregular figures by approximate
methods:
Trapezoidal rule


 
width of 1 first + last
Area ≈
interval
2 ordinate

+ sum of remaining ordinates
Mid-ordinate rule
Area ≈ (width of interval)(sum of mid-ordinates)
Simpson’s rule



1 width of
first + last
Area ≈
ordinate
3 interval




sum of even
sum of remaining
+4
+2
ordinates
odd ordinates

For a general sinusoidal function y = A sin (ωt ± α),
then
A = amplitude
ω = angular velocity = 2π f rad/s
ω
= frequency, f hertz
2π
2π
= periodic time T seconds
ω
α = angle of lead or lag (compared with
y = A sin ωt )

Cartesian and polar co-ordinates:
If co-ordinate (x, y) = (r, θ) then

y
r = x 2 + y 2 and θ = tan−1
x
If co-ordinate (r, θ) = (x, y) then
x = r cosθ and y = r sin θ

Mean or average value of a waveform:
area under curve
length of base
sum of mid-ordinates
=
number of mid-ordinates

mean value, y =

Triangle formulae:
Sine rule:
Cosine rule:

b
c
a
=
=
sin A sin B
sin C
a 2 = b2 + c2 − 2bc cos A

B

b

a

If a = first term and d = common difference, then the
arithmetic progression is: a, a + d, a + 2d,
...

The n’th term is: arn−1
Sum of n terms, Sn =

A
c

Arithmetic progression:

a (1 − r n )
a (r n − 1)
or
(1 − r )
(r − 1)

If − 1 < r < 1, S∞ =

a
(1 − r )

C

Area of any triangle
1
= × base × perpendicular height
2
1
1
1
= ab sin C or
ac sin B or
bc sin A
2
2
2

a +b+c
= [s (s − a) (s − b) (s − c)] where s =
2

Statistics:
Discrete data:

#
mean, x¯ =

x

n
)
(
 #

(x − x¯ )2

standard deviation, σ =
n

List of formulae
Grouped data:

#
fx
mean, x¯ = #
f

)
 #$
%(

f (x − x)
¯ 2

#
standard deviation, σ =
f

Standard integrals
y
axn
cos ax

Standard derivatives
sin ax

y or f(x)

dy
= or f (x)
dx

axn

anxn−1

sin ax

a cos ax

cos ax

−a sin ax

eax

aeax

ln ax

1
x

eax
1
x

&

a

y dx
x n+1
+ c (except when n = −1)
n +1

1
sin ax + c
a
1
− cos ax + c
a
1 ax
e +c
a
ln x + c

339

Answers

Answers to practice exercises
Chapter 1

Chapter 2

Exercise 1 (page 2)
1
...

7
...

13
...

17
...
16 m
3
...
£565
6
...
−36 121
9
...
1487
12
...
−70872
15
...
25 cm
d = 64 mm, A = 136 mm, B = 10 mm

1
...

4
...

7
...


5
...

13
...


Exercise 2 (page 5)
(a) 468 (b) 868
2
...

(a) 259 (b) 56
8
...


2
...

6
...

10
...


(a) 12 (b) 360
(a) 90 (b) 2700
(a) 3 (b) 180
(a) 15 (b) 6300
(a) 14 (b) 53 900

Exercise 4 (page 8)
2
...
68

3
...
5

DOI: 10
...
00040-5

3
...

10
1
14
...
1
21
6
...


16
...


2
6
8
2
...

35
9
11
3
1
7
...

5
13
5
3
2
12
...
3
2
5
12
4
3
1
17
...
13
4
9
(a) £60, P£36, Q£16

7
...

15
...


22
9
8
25
3
16
4
27
17
60

4
...

12
...

20
...
5
9
...
−33
10
...
1
2
12
14
...


71
8
11
15
3
16
51
8
52
17
20

19
...

15
5
...
4
20
...
2880 litres

Exercise 7 (page 14)
1
18
7
6
...
2
1
...
22
11
...

(a) 4 (b) 24
(a) 10 (b) 350
(a) 2 (b) 210
(a) 5 (b) 210
(a) 14 (b) 420 420

2
...

3
...

7
...


1
7
4
11
17
30
43
77
9
1
40

1
...
4

1
9
19
7
...
2
20
2
...
1
8
...
7
3

4
...

15
5
...
0
...
14
...
1
...


13
20

7
40
21
1
141
(b)
(c)
(d)
(e)
25
80
500
11
3
7
...
10
25
200
11
1
7
(b) 4
(c) 14
(d) 15
40
8
20
2
...
4
40
41
10
...
(a)

11
...
625

9
250

3
...


6
125

9
...

4
...

10
...


2
...
11
...
185
...
8307
5
...
1581
6
...
571
5
...
1
...
0
...
068
11
...
5 ×10
12
...
5 ×103
−6
−3
4
...
202
...
18
...
6
...
0
...
11
...
0
...
14
...
1
...
2
...
65
...
0
...
329
...
18
...
43
...
72
...
12
...
−124
...
4
...
0
...
−

9
10

4
...
732
8
...
0
...
0
...
0
...
−0
...
0
...
0
...
5
...
2
...
0
...
0
...
0
...
998 2
...
544
3
...
02 4
...
42
456
...
434
...
626
...
1591
...
444 10
...
62963
11
...
563 12
...
455
13
...
8
...
(a) 24
...
812
(a) 0
...
0064 17
...
4˙ (b) 62
...
4
...
0
...
3
...
13
...
50
...
53
...
36
...
12
...
0
...
46
...
1
...
2
...
2
...
30
...
0
...
219
...
5
...
5
...
52
...
0
...
25
...
591
...
69
...
17
...

4
...

10
...
11927
6
...
0944
10
...
325

2
...
30
...
84
...
10
...
2
...


Exercise 10 (page 19)

1
...

9
...

16
...


2
...
0
...
137
...
19
...
515
...
15
...
52
...
0
...
80
...
295
...
59 cm2
159 m/s
0
...

5
...

11
...
78 mm
0
...
8 m2
281
...

6
...

12
...
5
5
...
5
2
...

4
...

10
...


£58

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