Powerful Mathematics
The great power of mathematics is in the use of symbols and formulae. We
can take a general form and apply it to any specific case we wish.
A simple example:
To change a temperature from Celsius into Fahrenheit we first divide by five,
multiply the answer by nine, and then add thirty two. And this works for any
value.
But we can make this even more simple by the use of symbols to represent
each part of our formula. For example:
Let F represent the temperature in Fahrenheit
Let C represent the temperature in Celsius
Dividing by five and multiplying by nine is the same as multiplying by the
9
fraction .
5
And ‘add thirty two’ is represented by +32.
So our new way of looking at things is now:
F=
9
C + 32
5
To go back the other way requires us to ‘undo’ what we have done, in the
reverse order:
1
2
3
subtract 32
divide by 9
multiply by 5
Which, as a formula, we write as:
C=
5
F  32
9
By putting the F – 32 in brackets we are in effect saying ‘do this first’.
The three important aspects of formulae are:
I. Constructing a formula
II. Substituting values into the formula
III. Rearranging the formula
The second one is the emphasis of this booklet, being able to construct your
own formula and being able to rearrange formula are future skills to develop.
Page - 1
1.
Lets start with some of the most simple, and common, formulas in use
– the area and perimeter of a rectangle:
b
w
Area = b x w
Perimeter = b + b + w + w
or P = 2b + 2w or P = 2(b + w)
(a)
Find the area and perimeter of a rectangle measuring 3cm by 5 cm.
Try each of the three versions of perimeter to make sure you can get the
same answer.
Area = ……………………
Perimeter = ……………………
(b)
Find the length of a rectangle which has a width of 6cm and an area of
42 cm2.
Length = ……………………
(c)
Find the width of a rectangle which has a perimeter of 50 m and a
length of 15 m.
Width = ……………………
2.
(a)
Find the area of a triangle with base 3m and height 4m.
Area of a triangle =
1
(b  h)
2
h
b
Area = ……………………
(b)
Find the height of a triangle which has an area of 40 cm and a
base of 20 cm.
Height = ……………………
Page - 2
3.
(a)
What is the length of a square which has an area of 64 square
inches?
Length = ……………………
(b)
What is the length of a square which has an area of 60 square
inches?
Length = ……………………
4.
(a)
Find the area and circumference of a circle of radius 12 mm.
Circumference of a circle = d (where diameter is twice the radius)
Area of a circle = r 2
r
Area = ……………………
Circumference = ……………………
(b)
What is the diameter of a circle which has circumference of
40cm? Give your answer to two decimal places.
Diameter = ……………………
(c)
What is the radius of a circle which has an area of 50 cm 2?
Give your answer to two decimal places.
Radius = ……………………
Page - 3
5.
The area of a trapezium is found using the formula
Area = ½ (a + b) h
a
where a and b are the parallel sides and h is
the distance between them.
h
b
(a)
If a = 5 mm, b = 9 mm and h = 6 mm, find the area of the trapezium.
Area = ……………………
(b)
If a = 16 m, b = 24 m, and the total area is 80 m 2, find the height of the
trapezium.
Height = ……………………
6.
7.
(a)
Write down a formula for the volume of a cuboid:
You could use lengths a, b and c, or three of your own.
(b)
Try to give a formula for the surface area of a cuboid.
(a)
What is the length of a cube which has a volume of 64 cm 2?
Length = ……………………
(a)
What is the length of a cube which has a volume of 50 cm 2?
Give your answer to two decimal places.
Length = ……………………
Page - 4
8.
Find the surface area and volume of a sphere which has radius 4 mm.
Give your answer to two decimal places.
Surface area of a sphere = 4r 2
4
Volume of a sphere = r 3
3
r
Surface area = ……………………
Volume = ……………………
9.
Find the radius of a sphere which has volume 100 m 3.
Give your answer to two decimal places.
Radius = ……………………
10.
The formula for simple interest (I) is given by:
I
PRT
100
Where P = principal (amount invested), R = rate, and T = time.
(a)
Find the interest gained on £400 invested for 3 years at 5%.
Interest = ……………………
(b)
For how long must £500 be invested at 4% to gain an interest of
£60?
Time = ……………………
Page - 5
11.
The formula for compound interest, where A is the amount of money
accruing after n years, is given by:
R 

A  P 1 

 100 
Where
n
P = principal (amount invested)
R = rate per cent per annum
n = the number of years for which the money is invested.
Find the amount accruing after £400 invested for 3 years at 5%.
Amount = ……………………
-----------------------------------------------------------Speed, distance, time and acceleration of objects are connected by a series
of formulas:
v 2  u 2  2as
v  u  at
s  ut 
1 2
at
2
s
1
u  v t
2
where u = initial speed
v = final speed
s = distance travelled
t = time taken
a = acceleration
12.
Use v  u  at to find v if
(a)
u = 20 ms-1, a = 6 ms-2, and t = 3 seconds.
Final velocity = ……………………
(b)
u = 0 ms-1, a = -9.8 ms-2, and t = 2.5 seconds.
Final velocity = ……………………
Page - 6
13.
Use v 2  u 2  2as to find v if
(a)
u = 12 ms-1, a = 5 ms-2, and s = 20 metres.
Final velocity = ……………………
(b)
u = -10 ms-1, a = 9.8 ms-2, and s = 300 metres.
Final velocity = ……………………
14.
Use s  ut 
(a)
1 2
at to find s if
2
u = 20 ms-1, a = 6 ms-2, and t = 3 seconds.
Distance = ……………………
(b)
u = 0 ms-1, a = -9.8 ms-2, and t = 2.5 seconds.
Distance = ……………………
15.
Use s 
(a)
1
u  v t to find s if
2
u = 0 kmh-1, v = 80 kmh-1, and t = 5 hours.
Distance = ……………………
(b)
u = 100 kmh-1, v = 20 kmh-1, and t = 2 hours.
Distance = ……………………
Page - 7
Answers
1.
(a)
(b)
(c)
Area = 15 cm2
Length = 7 cm
Width = 10 cm
Perimeter = 16 cm
2.
(a)
Area = 6 m2
(b)
Height = 4 cm
3.
(a)
Length = 8 inches
(b)
Length = 7.75 inches
4.
(a)
(b)
(c)
Area = 452.39 cm2
Diameter = 12.73 cm
Radius = 3.99 cm
Circumference = 75.40 mm
5.
(a)
Area = 42 mm2
Height = 4 m
6.
Volume = a x b x c or
(b)
V = abc
Surface area = 2ab + 2ac + 2bc
or
7.
(a)
Length = 3.68 cm
8.
Surface area = 201.06 mm2
9.
Radius = 2.88 m
10.
(a)
11.
Amount = £463.05
12.
(a)
13.
Length = 8 cm
Interest = £60
(b)
SA = 2(ab+ac+bc) etc.
Volume = 268.08 mm3
(b)
Time = 3 years
38 ms-1
(b)
-24.5 ms-1
(a)
10.55 ms-1
(b)
77.33 ms-1
14.
(a)
9.33 m
(b)
-30.625 m
15.
(a)
200 km
(b)
120 km
Page - 8