Download Simple formulae

SIMPLE FORMULA
What is a formula?
When you do a calculation, you might add numbers together, subtract numbers, multiply or
divide them.

Take addition as an example:
37 + 45 = 82
Two given numbers added together result in a third number.

This can be extended to a general case for any two numbers, when you add them
together you get a third number.
NUMBER ONE + NUMBER TWO = NUMBER THREE

Taking this a step further, rather than writing NUMBER ONE etc, which is very long
winded, you could call NUMBER ONE, a, NUMBER TWO, b and NUMBER THREE, c, giving
us:
a + b = c
So we now have a formula, which tells us that if you add any two numbers together,
you will get a third number.

This is what algebra is all about, the letters that are used, are just abbreviations, they
stand in for numbers.
A formula is an equation where the left hand side equals the right hand side.
LEFT HAND SIDE = RIGHT HAND SIDE
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Updated Feb 2013
 The area of a rectangle is the length times the breadth
You can write this as a formula:
Area = length  breadth
Rather than writing this out in full, use abbreviations
A = l b
Now, given any length and breadth you can substitute them into the formula to find
the area
Given that the length is 4 m and breadth is 2 m:
A = l  b = 4 m  2 m = 8 m2

Take the equation:
5 + 5 = 10
In this case we have added the same number together twice and got another number.
We could write the general case for this out as:
a + a = b
This formula says that when you add a number to itself you will get a second number.

Another way of writing 5 + 5 would be to write 2  5
Relating this to the general case, we have that:
a + a
2  a
is the same as
To make life easier, you omit the "" and just write 2a
i.e.

a + a = 2a
This can be extended further:
a + a + a = 3a
a + a + a + a = 4a
etc.
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Updated Feb 2013

The next step is to add expressions like 2a + 3a
Remembering that 2a = a + a and 3a = a + a + a
You have: 2a + 3a = a + a + a + a + a = 5a
So, to add 2a + 3a, you just add the 2 and the 3
 5a + 7a = 12 a
 3y + 11y = 14y
 This works the same for subtraction, you subtract the numbers
 9a - 6a = 3a
 15p - 4p = 11p
 If you were given the equation: 2a = 10, you could see that for the left hand side to
equal the right hand side, a must equal 5.
 3a = 12
is only true when a equals 4
 5b = 25
is only true when b = 5
 Given the equation, 2a + b = 7, find b when a = 2
You have been given a value for a and you need to substitute this value in the equation,
before you can find the value of b.
Remember that 2a means 2  a,
so when a = 2, 2a = 2  a = 2  2 = 4
2a + b = 7 will become 4 + b = 7
For the right hand side to equal the left hand side b must equal 3.
 3a + b = 8,
31 = 3
find b when a = 1

 2x - y = 3,
2  7 = 14
3+ b = 8

b = 5

y = 11
find y when x = 7

14 - y = 3
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 When you multiply a number by itself, 3  3, you can write it as 32
The number 2 written as a superscript indicates that you multiply the number by itself
 52 = 5  5 = 25
 72 = 7  7 = 49
A superscript of three means the product of three factors
 33 = 3  3  3 = 27
 53 = 5  5  5 = 125
The number in the superscript indicates the number of factors multiply together
 24 = 2  2  2  2 = 16
 25 = 2  2  2  2  2 = 32
 This can be extended to letters
aa
can be written as a2
a  a  a can be written as a3
a  a  a  a  a can be written as a5
 Given that a2 = 16, what does a equal?
You are being asked, what number when multiplied by itself gives 16?
4  4 = 16 so a = 4
 Given that b3 = 8 what does b equal?
2  2  2 = 8 so b = 2
 Given that s2 = 81 what does s equal?
9  9 = 81 so s = 9
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Updated Feb 2013
 Given that g4 = 81 what does g equal?
3  3  3  3 = 81 so g = 3

Sometimes you are given a formula and you need to rearrange it.
Density 
 i.e.
Mass
Volume

D 
M
V
The above formula allows you to find the density of a substance, given its mass and
volume.
Find the density of lead, given that 10 cm3 has a mass of 115 g.
D 
M
V

115 g
10 cm 3

11.5 g/cm 3
 If you were asked to find the mass of 15 cm3 of silver, given that silver has a density
of 10.5 g/cm3, you would need to rearrange the formula first.
One way of doing this is to put a triangle round the formula.
M
D
V
If you look at the triangle you can see that:
D
M
V
V
M
D
MD  V
This is a very helpful way of remembering the different formulae, all you need to do
is remember one and the others follow.
So,
M  D  V  10.5
g
 15 cm 3  157.5 g
3
cm
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Updated Feb 2013
 What volume would 120 g of cork have, given that the density of cork is 0.24 g/cm 3 ?
Looking at the triangle, you can see that the formula to find the volume is V 
M
D
so,
V 

M
D

120 g
0.24 g/cm 3
 500 cm 3
The triangle method can be used for any formulae where there are three variables
(letters) involved.
 Speed =
Distance
Time
 S 
D
T
D
D
S
T
How long would it take a car travelling at an average speed of 70 km/h to travel 350
km?
T 
D
350 km

 5 hours
S
70 km/h
 The Area of a circle =   the square of the radius
A  r 2
A
A

r2
Find the radius of a circle that has an area of 50 cm2.
r2 
i.e.
A
50

 15.9 cm2

3.142
r2 = 15.9  16 cm2
r 
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16  4 cm
 Given that the circumference of a circle equals   the diameter, what would be
the formula for the diameter, given the circumference?
C = D
D 
C
C
C


D
T
 Formulae need to be built. As an example, let's look at how you arrive at the formula to
convert oC to oF.
There are two pieces of information that you need to know.
Freezing point is 0oC and 32oF
9oF = 5oC (1oC is almost twice 1oF)
From this we have that:
5 oC
1F =
9
o
Now because they have different freezing points, you need to add on 32 oF to arrive at
the equivalent temperature.
9  oC
+ 32oF
5
o
i.e.
F =
 Convert 20oC to oF
o
F =
9  20
+ 32oF = 36 + 32 = 68oF
5
 What would be the equivalent formula for converting oF to oC ?
Using the same information we first of all need to subtract 32 from the number of oF.
o
C =
o
F - 32oF
9oC = 5(oF - 32oF)

1oC =
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Updated Feb 2013
5( o F - 32 o F)
9
So the formula is:
o
C =
5( o F - 32 o F)
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 Convert 77oF to oC
o
C =
5(77 - 32)
5(45)
=
= 25oC
9
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Formulae exercises
1. a)
4b + 7b =
b)
7x - 2x =
c)
17p - 15p =
d)
3f + 5f =
e)
8m + 2m - 7m =
f)
w + 4w + 3w =
2. a)
7x - y = 9 find y when x = 2
b)
5a - b = 11 find b when a = 3
c)
4x - y = 7 find y when x = 5
d)
11p - q = 14 find p when q = 8
e)
2g + f = 11 find g when f = 3
f)
6c - d = 4 find c when d = 2
3. a)
Given that a = 3, b = 2 and c = 4, find:
i)2a + 1 =
ii)
b2 =
iii)
a(b + c) =
iv)
12 - ab =
v)
2(a + b + c) =
b)
Given that x = 2, y = 5 and z = 3, find:
i)
x3 =
ii)
3x(y – z) =
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Updated Feb 2013
iii)
xy - xz =
iv)
xyz =
v)
3y2 =
4. Find the following:
a)
24 =
b)
72 =
c)
34 =
d)
26 =
e)
53 =
f)
43 =
g)
100 =
h)
a2 = 25
find the value of a
b)
x3 = 27
find the value of x
c)
y5 = 32
find the value of y
d)
n4 = 81
find the value of n
e)
h3 = 125 find the value of h
f)
d4 = 16
4. a)
5. a)
144 =
i)
42 =
find the value of d
If 395 g of iron has a volume of 50 cm3, find the density of iron.
b)
Copper has a density of 8.7 g/cm3, what would be the volume of a mass of 240 g?
c)
What would be the mass of 2 cm3 of gold, given that gold has a density of 19.2
g/cm3.
d)
i) What is the formula for finding the time taken to travel a certain distance at a
given speed?
ii) How long would it take to travel 70 km at an average speed of 50 km/h?
e)
How far could you travel in 5 hours at an average speed of 40 km/h?
f)
i)
The area of a triangle can be found by multiplying half the length of the base
by its perpendicular height. Write this as a formula.
ii)
What is the area of a triangle that has a base of 4 cm and a perpendicular
height of 5 cm?
iii) A triangle has an area of 20 cm2 and a base 8 cm long. What is its
perpendicular height?
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Updated Feb 2013
Formulae exercises solutions
1. a)
11b
b)
5x
c)
2p
d)
8f
e)
3m
f)
8w
y=5
b)
b=4
c)
y = 13
d)
p=2
e)
g=4
f)
c=1
2. a)
3. a) i)
7
b) i)
8
ii)
4
ii)
12
iii)
18
iii)
4
iv)
6
iv)
30
v)
18
v)
75
4. Find the following:
a)
16
b)
49
c)
81
d)
64
e)
125
f)
64
g)
10
h)
12
i)
4
4. a)
a=5
b)
x=3
c)
y=2
d)
n=3
e)
h=5
f)
d=2
5. a)
7.9 g/cm3
b) 27.6cm3
d)
i) T 
D
S
f)
i) A 
1
bh
2
c) 38.4 g
ii) 1 hour 24 minutes
ii) A = 10 cm2
e) 200 km
iii) h = 5 cm
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Updated Feb 2013