TRADE MATHEMATICS
TUTORIAL BOOKLET
Chapter 1 - TRANSPOSITION
Transposition is the rearrangement of a formula so that the “subject” or “quantity” to be found is placed on one side by itself.
The easiest way to deal with transposition is a simple rule of -Whatever process is applied to one side of the formula, the same process must be applied to the other side of the formula.
This holds true for addition, subtraction, multiplication, division, square rootand powers.
Example 1:
Transpose = to find
Multiply both sides by ‘I’ =
Cancel ‘I’ on right =
Or… =
......
Example 2:
The area of a triangle is: A = ½bh where b is breadth and h is height.
Transpose = ½to find
Multiply both sides by ‘2’ = ½
Cancel ‘2’ on right =
Divide both sides by ‘b’ =
Cancel ‘b’ on right =
Therefore, if A is 6m2 and b = 3;
Then = = = 4m.
Example 3:
Transpose = to find
Add ‘5’ to both sides =
Cancel the ‘5’ on the right =
Divide both sides by 6 =
Cancel the 6 on right =
......
Example 4:
Transpose = to find
Divide both sides by r2 =
Cancel r2 on right
......
Example 5:
Transpose = to find
Divide both sides by h =
Cancel h on right =
Square root both sides =
Cancel the square =
and square root on right
Exercises:
Transpose the following:
1. to find2. =to find
3. to find 4. to find
5. to find6. to find
7. to find 8. to find
9. to find 10. to find
11. to find 12. to find
13. to find 14. to find
15. to find 16. to find
17. to find 18. to find
19. to find 20. to find
Chapter 2 - TRIGONOMETRY
Pythagoras’ theorem:
a2 = b2 +c2
i.e. the square of the hypotenuse is equal to the sum of the squares on the other two sides.
This formula can be used to calculate the sides of a right-angled triangle.
Example 1: Calculate aa2 = b2 + c2
a2 = 422 + 312
aa2 = 1764 + 961
a2 = 2725
a =2725 = 52.2
Example 2: Calculate ba2 = b2 + c2
b2 = a2 – c2 (by transposition)
b2 = 1102 - 402
b2 = 12100 - 1600
b = 10500, 10500 = 102.47
Example 3: Calculate ca2 = b2 + c2
c2 = a2 – b2 (by transposition)
c2 = 332 - 282
c2 = 1089 – 784
c = 305, 305 = 17.46
H is the Hypotenuse
O is the Opposite to the angle
A is the Adjacent to the angle
Example 4:
Use the calculator to find the values of:
a) cos 250d) sin 650
b) cos 35.770e) tan 700
c) sin 480f) tan 350
Example 5:
Use the calculator to find the angle of each of the following:
a) sin-1 0.819d) cos-1 0.766
b) sin-1 0.94e) tan-1 1.732
c) cos-1 0.5f) tan-1 0.7
Example 6:
Find the unknown to the following:
Example 7:
Find the unknown to the following:
Example 8:
Find the unknown to the following:
Example 9:
`
Find the unknown to the following:
40
tan-11.33
Example 10:
Find the unknown to the following:
Example 11:
Find the unknown to the following:
cos-10.833
…………………………………………………………………..
Exercise 1:
Find the unknown lengths to the following:
a)c)
b)d)
Exercise2:
Find the lengths or angles marked ‘x’ and in the following right-angled triangles:
a)b)
c)d)
e)f)
Chapter 3 – SI UNITS
Chapter 4 – POWERS and SUBMULTIPLES
TABLE OF POWERS OF 10Number / Power of 10 / Number / Power of 10
1000 000 / 106 / 1 / 100
100 000 / 105 / 0.1 / 10-1
10 000 / 104 / 0.01 / 10-2
1 000 / 103 / 0.001 / 10-3
100 / 102 / 0.000 1 / 10-4
10 / 101 / 0.000 01 / 10-5
1 / 100 / 0.000 001 / 10-6
PREFERRED MULTIPLES AND SUBMULTIPLES
Prefix / Symbol / Factor / Magnitude
tera / T / 1012 / 1000 000 000 000
giga / G / 109 / 1000 000 000
mega / M / 106 / 1000 000
kilo / k / 103 / 1000
milli / m / 10-3 / 0.001
micro / / 10-6 / 0.000 001
nano / n / 10-9 / 0.000 000 001
femto / f / 10-12 / 0.000 000 000 001
atto / a / 10-15 / 0.000 000 000 000 0001
According to the laws of mathematics and physics, the derived units are obtained by combining and interrelating base symbols.
You can indicate multiplication by joining the symbols and omitting spaces:
Newtons meters = N m = Nm
Division is shown by any of the following methods:
If division is to be carried out more than once, only one stroke is used. Therefore, meters per second per second is shown as:
MULTIPLE AND SUBMULTIPLE UNITS
We can use and index or power as a method of writing the multiples and submultiples of ten. This index indicates the number of times ten is multiplied or divided by itself to obtain the desired number.
10 = 1 x 10 = 101
100 = 10 x 10 = 1 x 102 = 102
1000 = 10 x 10 x 10 = 1 x 103 = 103
0.1 = 1 x 10-1 = 10-1
0.01 = 1 x 10-2 = 10-2
0.001 = 1 x 10-3 = 10-3
MULTIPLICATION OF POWERS OF 10
102 x 103
i.e. 10 x 10 multiplied by 10 x 10 x 10
which is 10 x 10 x 10 x 10 x10
which equals 105
103 x 10-2
i.e. 10 x 10 x 10 x 0.01
which equals 10
DIVISION OF POWERS OF 10
105/102
i.e.
10 x 10 x 10 x 10 x 10
10 x 10
by cancellation the solution is:
10 x 10 x 10= 103
Other numbers may also have powers of 10 in their expressions.
eg: 4 758 897 = 4.758 897 x 106
eg: 0.00032 = 0.32 x 10-3 = 3.2 x 10-4 = 32 x 10-5
TRADE APPLICATION OF UNITS
Let’s choose a unit. eg: Length.
The base unit for length is the meter (m).
Multiples and submultiples of the meter are used where appropriate. This allows the dimensions or distances are neither too large or too small.
1 meter = 1000 millimeters
or 1m = 1000mm
1 kilometer = 1000 meters
or 1km = 1000m
EXERCISES:
1. Express the following as powers of 10:
a) 1000 000
b) 10 000
c) 1000
d) 10
e) 0.01
f) 0.000 01
g) 0.000 001
2.What prefix and symbol is given to the following multiples or submultiples?
a) 1000 000
b) 1000
c) 0.001
d) 0.000 001
3.Express the following as millimeters:
a) 6 436m
b) 245m
c) 56.44m
d) 0.000 18km
4.Express the following pressures as pascals:
a) 5.78MPa
b) 1 255.75kPa
c) 0.0166Mpa
5.Calculate:
a) 103 x 105d) 106/102
b) 104 x 10-2e) 10-2/102
c) 103 x 10-3f) 103/10-2
6.Express the following quantities using the appropriate power of 10 multiple or submultiple:
a) 367 876mm
b) 1 789mm
c) 300 780g
d) 0.0067 litres
7.Multiply the following and express with a suitable power of 10:
a) 8.98 x 106 x 10-5
b) 2 678 x 10-3 x 106
c) 0.0098 x 102 x 106
d)4.5 x 106 x 10-5
e)3.05 x 10-4 x 10-1
8.Divide the following and express with a suitable power of 10:
a) 4.8 x 106 by 10-5
b) 2 498 x 10-4 by 10-2
c) 0.378 x 103 by 10-2
d) 423.834 x 106 by 10-3
9.What derived unit is obtained when the following are multiplied?
a) meters by meters
b) newtons by meters
c) kilograms by meters
10. What derived unit is obtained when the following are divided?
a) newtons by square meters
b) joules by seconds
11.Express the following in meters:
a) 2.908km
b) 0.066km
c) 2 500 000mm
d) 7 776mm
e) 45 900mm
f) 0.8 x 10-3km
12. Express the following in grams:
a) 0.348kg
b) 0.008kg
c) 7.888kg
d) 5 900mg
e) 10 700mg
13.Express the following to the units shown:
a) 0.562km to meters
b) 0.095 m to millimeters
c) 4 879g to kilograms
d) 2 760s to minutes
14.Express the following to the units shown:
a) 5 812N to kilonewtons
b) 5.18 x 106N to meganewtons
c) 1.45 x 109N to meganewtons
d) 0.088kN to newtons
e) 2.05 x 10-6MN to newtons
15.Express the following to the units shown:
a) 12250 x 10-3 J to kilojoules
b) 152350 x 106 J to kilojoules
c) 56.56 x 109 J to megajoules
d) 1.66 x 103 kJ to joules
e) 0.88 x 10-6 kJ to joules
f) 6 784 W to kilowatts
g) 77 895W to kilowatts
h) 79 x 10-3 MW to watts
i) 6.1 x 10-3MW to kilowatts
Answers to student exercises
Transposition
1.C = Q/V11.R = L/
2.n = (60Po)/2T12.C = /R
3.a = (v – u)/t13.l = RA/
4.Pi = Po100)/14.Q3 = QT – Q1 – Q2
5.c =15. = cos-1(R/Z)
6.R = V2/P16.f = V/4.44Npmax
7.f = XL/2L17.Ns = NpVs/Vp
8.C = 1/XC2f18.Vp = Vrms/0.707
9.X = Z2 – R219.Vp = Vrms/1.414
= sin-1vVmax20.Rt = 1/(1/R1 + 1/R2)
Trigonometry
1a.42.431c.185.47
1b.41.531d.114.86
2a.58.29mm2d.60.70
2b.2.97m2e.69.330
2c.5.29m2f.53.330
Multiples and submultiples
1a.1061e.10-2
1b.1041f.10-5
1c.1031g.10-6
1d.101
2a.Mega (M)2c.milli (m)
2b.kilo (k)2d.micro ()
3a.6 436 000mm3c.56 440mm
3b.245 000mm3d.180mm
4a.5 780 000Pa4c.16 600Pa
4b.1 255 750Pa
5a.1085d.104
5b.1025e.10-4
5c.100 or 15f.105
6a.367.816km6c.300.78kg
6b.1.789km6d.6.7ml
7a.89 800 x 1037d.0.045 x 103
7b.2.678 x 1067e.30.5 x 10-6
7c.80 x 103
8a.0.48 x 10128c.37.8 x 103
8b.0.02498 x 1038d.0.423 x 1012
9a.m29c.kgm
9b.Nm
10a.N/m210b.J/s
11a.2 908m11d.7.776m
11b.66m11e.45.9m
11c.2 500m11f.0.8m
12a.348g12d.5.9g
12b.8g12e.10.7g
12c.7 888g
13a.562m13c.4.879kg
13b.95mm13d.46min
14a.5.812kN14d.88N
14b.5.18MN14e.0.000002N
14c.1 450MN
15a.0.13kJ15f.6.784kW
15b.1.61kJ15g.77.895kW
15c.0.0062MJ15h.7 900W
15d.170 980J15i.6.1kW
15e.88 000J
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