This document is intended as a study help guide for students of general mathematics principles and processes. It is also a set of useful reminder sheets. Started as tutoring notes for a Reconstructing Maths course, they are often added to.
The value of each number changes with its position in relation to the dot marking the difference between whole
units and part units, because the face value of the number is multiplied by the value of its position. The value
of the Units position is always its face value, and the other positions are calculated as a multiple of the base value.
In the decimal system (base 10), each number position is worth 10 times that of the one on its right. The dot between the whole units and fractions is known as the decimal dot. The most often used values are:
million.......ten thousand.......hundred..........unit..........tenth.....thousandth....hundred thousandth
___ ___ ___ ___ ___ ___ ___ . ___ ___ ___ ___ ___ ___
hundred thousand...thousand...........ten.....decimal dot...hundredth....ten thousandth....millionth
To make large whole numbers easier to read, they are shown in divisions of 3 positions that are divided by a comma (,), thus a million is shown as 1,000,000 - not as 1000000.
|Numbers||The only numbers or numerals used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.|
|Base||Unless specifically informed otherwise (eg 5 8) you will use base 10.|
|Extra Zeroes||Zeroes before the decimal dot, or first number, are assumed.|
|Zeroes after the decimal dot, or last decimal place, are assumed.|
In all mathematical equations the basic principle is that the final result of all the actions and values
on one side of the equals symbol, =, is exactly the same as on the other side. Thus what is on one
side will always exactly balance what is on the other.
In most mathematical situations we are seeking to calculate the final result, but sometimes we know some values on both sides and this principle can be used to help find the missing values.
The main function processes used are:
|Addition||Increasing the value of one number by one or more numbers.
.....Example:....1 + 2 = 3....one plus two equals three.
|Subtraction||Decreasing the value of one number by one or more numbers.
.....Example:....3 – 2 = 1....three minus 2 equals one.
|Multiplication||Increasing the value of one number by another as if you had added
the first number to itself as often as the second number is. The term times is
often used to express the multiplication function.
.....Example:.....2 x 3 = 6.....two times three equals six.
|Division||Finding out how often one number will go into another number.
.....Example:....6 ÷ 3 = 2.....six divided by three equals two.
|Square||The square of a number is the value found when you multiply that
number by itself. This is expressed by a little 2 after and above it.
.....Example:.....32 = 9.....three squared equals nine, or 3 x 3 = 9.
|Square Root||The square root of a number is the value that, when multiplied by
itself will give you the number you started with. This is expressed by the root
symbol √, this is usually shown with a value as 2√
(square root) or 3√ (cube root) etc; if no value it is assumed
.....Example:.....2√ 9 = 3.....square root of nine equals three.
|Cube||The cube of a number is the value found when you multiply that number
by itself and then multiply the answer by the original number again. This is express as
a little 3 after and above it.
.....Example:.....23 = 8.....two cubed equals eight, or 2 x 2 x 2 = 8.
|Cube Root||The cube root of a number is the value that, when multiplied by itself
and then that answer is again multiplied by the value, will give you the number you
started with. This is expressed by the root symbol with the value of three, 3√.
.....Example:.....3√ 27 =3.....cube root of twenty-seven equals three
...........................................27 = 3 x 3 x 3, 27 = 9 x 3, or 3 x 3 x 3.
|Powers||The process used in squares and cubes can be extended infinitely and the small raised number is known as a power. Thus 22 is also known as two to the power of two. The number 26 is expressed as two to the power six, or 2 x 2 x 2 x 2 x 2 x 2 = 64. Often very large large numbers are written as 1.325 x 1013, this is called scientific notation as it is shorter than 13,250,000,000,000.000|
|NB:||This same power numbering process can also be applied to roots and thus you could have a number 9√ 512. Luckily all you are likely to come across in daily usage are square roots and cube roots|
Although 2 x 6 is easy to do in the mind, 453 x 278 is not so easy, thus breaking it down into smaller
steps makes it faster and easier. Long Multiplication breaks it into a series of single number times single
number calculations that you can then add up, remember to carry forward higher position values in the
single digit multiplications. This process also makes the workings easier to see and check. Two main methods:
|3,624||multiply each number in the top line by 8|
|31,710||write a zero then multiply each number in the top line by 7|
|90,600||write 2 zeroes then multiply each number in the top line by 2|
|125,934||add up the multiplication lines|
|24||multiply 3 x 8|
|400||write a zero then multiply 5 x 8 (equal to 50 x 8)|
|3,200||write 2 zeroes and multiply 4 x 8 (equal to 400 x 8)|
|210||write a zero and multiply 3 x 7 (equal to 3 x 70)|
|3,500||write 2 zeroes and multiply 5 x 7 (equal to 50 x 70)|
|28,000||write 3 zeroes and multiply 4 x 7 (equal to 400 x 70)|
|600||write 2 zeroes and multiply 3 x 2 (equal to 3 x 200)|
|10,000||write 3 zeroes and multiply 5 x 2 (equal to 50 x 200)|
|80,000||write 4 zeroes and multiply 4 x 2 (equal to 400 x 200)|
|125,934||add up the multiplication lines|
Although 6 ÷ 3 is easy to do in the mind but 46,480 ÷ 35 is not so easy thus
a process to handle smaller steps is easier and faster. Long Division breaks it into a
series of smaller calculations. Remember to keep position values aligned properly. It
also makes the workings easier to see and check. If the answer goes into decimal places
this is easy to calculate as you can just keep adding zeroes onto the end.
The number being divided into the other number is called the divisor. (A) Write the equation down as 35)46,480 and draw a line above the number. (B) Then see if the divisor will go into the first digit of the number, if not try the first two digits of the number; this will work as 46 is larger than 35. (C.) Work out how many times it will go and write that number above the line. (D.) Then multiply that number by the divisor and write it below the number, (E.) draw a line and subtract it from the digits of the number being used, writing the left over under that. (F.) Then bring the next digit of the number down and write it beside the remainder you have. Repeat steps B to F until you have a remainder of zero or you have enough decimals to suit you. If needed when getting into decimals you can just add a zero onto the remainder and continue going through the steps. Make sure you have your decimal dot right.
|35||)46480||First pass 46 divided by 35|
|114....||subtract the 35 from 46 and bring down the next digit|
|35||)46480||First pass 46 divided by 35|
|35......||subtract the 35 from 46|
|114....||bring down the next digit and second pass 114 divided by 35|
|105....||subtract the 105 from 114|
|98...||bring down the next digit and third pass 98 divided by 35|
|70...||subtract the 70 from 114|
|280.||bring down the next digit and fourth pass 280 divided by 35|
|280.||subtract the 280 from 280|
|nil||finished no more to do.|
Sometimes it is also possible to make the calculation easier by simplifying it by dividing both the number and the divisor by a number which goes evenly into both. In this example both are multiples of 5 and can be divided by 5. This would then make the equation 9,296 divided by 7, which will give the same result as both sides of the calculations have been altered to the same extent.
A prime number is any number that will give a whole number answer when
divided by itself or one (1); that is, it is only divisible by itself or one. The
only prime numbers under 10 are 1, 2, 3, 5, and 7. The number 2 is the only
even prime number as all other even numbers can be divided by two; in a
similar fashion any number ending with the digit 0 or 5 (except 5) is not a
prime number as it can be divided by 5.
Most numbers can be broken down into their primes and expressed as a function of their primes, the number 30 can be expressed by its prime factors as 2 x 3 x 5. The convention in listing primes is to list the lowest number first and the rest in increasing value. Many numbers will be expressed as prime factors by the use of powers (see Functions), thus the prime factors expression of 60 is 22 x 3 x 5 which is the same as 2 x 2 x 3 x 5, and 120 is 23 x 3 x 5 and 180 is 22 x 32 x 5.
When calculating the prime factors of a number you should try dividing the number by the lowest number possible that gives an answer in whole numbers, then repeat the process until you cannot divide it by anything else. The easiest way to work out the prime factors is to start at the bottom left of a piece of paper and then do a series of long division calculations working your way up the page, the prime factors are then shown as the division numbers on the left and the final number on top. If a number is not suitable at any step it is not suitable for any later step as it would have worked in the earlier step if it could. Start with the lowest number possible and work upwards, this simplifies the task as you go, use 2 if the number is even, 3 if odd. For example establishing the primes in the number 63,063. Go to the bottom left corner, step 1:
|...143||uneven try 7 again, no; try 11|
|1001||uneven number try 7 again|
|Step 4||7)7007||(NB: 3 was no good last step so don’t try it)|
|..7007||uneven number try 3 again, no; try 7|
|Step 3||3)21021||(NB Skipped 5 as it did not end in 5 or 0)|
|21021||uneven number try 3 again|
|Step 1||63063||uneven number try 3 first.|
|In real life it will show like this, primes are in bold numbers.
The answer is 3 x 3 x 7 x 7 x 11 x13 or 32 x 7 2 x 11 x 13.
The number below the line is known as the denominator and in many cases it is necessary to bring the fractions
into a common denominator in order to complete the calculation. This makes the calculation easier as you can
then handle the above the line numbers as whole numbers. Adding 1/2
and 1/4 is not readily possible, but changing the
1/2 to 2/4 makes
it easier to add as they are both now / 4 and it becomes a case of
(2 + 1)/4 = 3/4. The same
applies to any calculation with fractions.
In some cases the only way to bring fractions into a common denominator is to multiply both halves of each side of the calculation by the denominator of the other side. Thus the common denominator for 4/9 + 2/5 would be converted by multiplying the 4 x 5 and the 9 x 5 to become 20/45 then multiplying the 2 x 9 and the 5 x 9 to be 18/45 making the calculation into 20/45 + 18/45 or 38/45.
It is standard practice to convert fraction answers into whole numbers and fractions where relevant and to simplify the fraction to its lowest possible number. Thus an answer of 48/45 changes to 1 3/45 and would end up as 1 1/15.
When the calculation includes a number that is a whole and a fraction like 1 2/5 then it is best to convert the whole number into fraction and this would become 6/ 5 prior to doing the calculation.
Fraction divisions can be simplified by turning the divisor upside down and changing the division to multiplication. Thus 4/5 ÷ 2/10 becomes 4/5 x 10/2 or (4 x 10)/(5 x 2) which equals 40/ 10 or 4.
Fraction multiplications can be simplified by drawing a long line between the top and bottom, you can also make the same changes to both halves. 9/10 x 10/ 18 is the same as (9 x 10)/(10 x 18) by taking away the 10 in both halves you get 9/18 or 1/ 2.
Decimal calculations are done the same as fractions once you have converted the decimal into a fraction by adding a zero to the denominator for each position after the decimal dot. Thus 1 = 1/1 and 1.5 = 15/10 and 1.55 = 155/100 etc.
Percentages, usually written as 95%, are a specialised form of showing values as hundredths; 95% is really
95/100. If the percentage has a decimal like 12.5% the decimal
is removed by adding a zero to the denominator, 125/1000. A
number is converted to a percentage by multiplying it by 1/100.
A number can be shown as a percentage of another number by dividing it by the second and multiplying by 100 For
example 5 as a percentage of 25 is calculated 5 ÷ 25 x 100; is 5/25
x 100 or (5 x 100)/25 which is 20, thus 5 is 20% of 25.
The difficulty with percentages comes in calculating things with a percentage, such as commissions, taxes, interest. Especially with reverse and multiple level calculations. The confusion is due to not clearly identifying which is the whole or 100% value.
The three main types of percentage calculations are simple, reverse and accumulative.
This is the easiest and most common and used in calculating simple interest.
To calculate the percentage of a number you multiply it by the percentage as a fraction. Thus 15% of 450 is 450 x 15/100 or (450 x 15)/100 = 67.5. This gives you the value that the percentage is of the original figure.
To calculate the value with 15% added on, you add 100 to the percentage fraction; 450 with 15% (that is 450 plus 15%of 450 added on) is 450 x 115/100 = 517.5, which is the value of the original figure plus the percentage added to it.
This is used to take a figure back to its original value before a percentage was added. Often used to
calculate pre-tax values or pre-commission values.
To calculate the original value from a number that includes a percentage you divide the number by 100 + the percentage and multiply by 100. The number 600 includes 20% so the calculation is (600 ÷ 120) x 100 or 600/120 x 100 = 500. Most people try to take 20% off the 600 giving 480, having forgotten that the 600 is 120%.
This is where you have more than one percentage being applied, often used to calculate tax added
retail prices. You start with purchase price add profit margin, commission, then tax; all percentages of
the figure before it, not the original value. You start with one value add a percentage, then add a percentage
to the new value. The most common fault is thinking you can add the percentages together and apply them
to the original value, when it is really a series of separate calculations of price + profit, then that + commission,
and then that + tax. .
Example:..........800 with 25% profit, 15% commission, 10% tax is
.........................800 x 125/100 = 1,000 x 115/100 =1,115 x 110/100 = 1,226.50;
.........................25% + 15% + 10% = 50% then 800 x 150/100 = 1,200 is wrong
The perimeter is the outside edge of a shape. Most can be calculated by a simple formula. A parallelogram, square or rectangle perimeter is calculated by adding the length of the four sides together but easiest as (b + h) x 2 or (breadth + height) x 2.
The area is the space inside a 2 dimensional (flat) shape like a square or triangle.
The area of a parallelogram, square or rectangle (these are special parallelograms) is calculated by multiplying the breadth x height (b x h).
The area of a triangle is 1/2 b x h as any triangle when mirror imaged and added to itself will create a parallelogram.
Volume is the space inside a 3dimensional shape like a cube. This is calculated by using the area equations above to calculate the area of the base and then multiply it by the depth. Cones and pyramids are slightly different and not covered in this document.
To work out how much of a whole any one component is of a mix you add up the ratios and
divide the total mix by that value and then multiply that answer by the value of the relevant component.
Sometimes it is easier to calculate what fraction of the mix that component represents and then calculate
what that fraction is of the mix.
Example:..........sand / gravel / cement mixes in a ratio of 1/3/4 for 64 kilo mix
.........................1 + 3 + 4 = 8; 64 ÷ 8 = 8, values are 1 x 8 / 3 x 8 / 4 x 8, or 8 / 24 / 32;
.........................Same mix but you only want the amount of cement for 60 kilos
.........................1 + 3 + 4 = 8; 4 ÷ 8 = 1/2; 60 x 1/2 = 30
The probability, or chance, that a particular event will occur is calculated as the number of times that
event is possible, divided by the total number of event options available, and then expressed as a fraction.
The most common demonstrations involve a six-sided die. Each number has one chance of being rolled,
and the total number of options is 6, thus a probability of 1/6
for each number.
Probabilities are more complex when you have two or more probability events occurring together, as in throwing two or more dice together. Then the total number of options is calculated by multiplying the total number available on the first event by the total number available on the second event, etc. The chances of a specific result are calculated by working out all the possible answers; then counting the number of times that event can occur. With two six-sided dice the total options are 6 x 6 = 36 (with three dice the probability is 6 x 6 x 6 = 216) whilst the chances of rolling 12 (6 + 6) is 1 (1/36 ), the chances of rolling 11 (5 + 6 & 6 + 5) is 2 ( 2/36 , 1/18).
To calculate the various event values you draw a matrix, or grid with the number of events for one factor on the top and the other factor on the side. Then calculate the different event option in the matrix boxes. For two six-sided dice it would be:
Missing digit calculations are where you have all but one number in a figure and
know another aspect of the figure (usually that it is evenly divided by a certain value)
and you have to work out what the value of the missing digit is.
This is calculated by assuming that the missing digit is 0 (zero), then dividing the number that this makes by the divisor supplied. You then see what is left over and then check which of the available values will give you an even multiple of the divisor if the left over is added to the various values given. The trick here is to remember the positional value of the missing digit. If the missing digit is in the 10 value position it is a multiple of 10 plus the left over, etc.
Example:..........4568x is a number that can be evenly divided by 3.
Divide 3 into 45680 this gives you 15226 with two left over, thus x is a number that is a multiple of 3 plus 1. The only possible options are 1, 4, or 7.
Example:..........456x1 is a number that can be evenly divided by 3.
Divide 3 into 45601 this gives you 15200 with one left over, thus x is a number that is a multiple 10 and a multiple of 3 plus 2. In this case it is easier to look at values with the left over added eg 11, 21, 31, 41, 51, 61, 71, 81, 91 the only ones divisible by 3 are 21, 51 and 81; so x must be 2, 5, or 8. NB different values to the solution above.
Today the majority of countries use metric measuring scales. These all have a
decimal base and a common set of words to express the value relevance of the scale
in use. The whole unit is expressed by the relevant scale point without a value reference,
eg metre, gram, litre. Also some scales are multiple unit values, eg in weights you have
grams and tonnes; and some possible mixes are not used, eg centigram, decimetre, or
decametre. The main words and their values (with some common examples) are below;
some other examples are megahertz and, kilowatts.
Triangles are enclosed shapes that have three straight sides. There are three main types of triangles.
The value of the three angles in any triangle will always add up to 180o, the value of a straight line.
This can be proven by cutting a triangle out of paper, and tearing it into three parts; then placing the
angle points together with the sides touching but not overlapping. Knowing this we can always work
out the value of an angle in a triangle when we know the value of the other two angles. Main types are:
Equilateral..........all three sides are the same length, and so are the angles.....
Isosceles............has two sides of equal length............................................. .....
Scalene..............no two sides are the same length......................................... .....
Right angled.......it has a 900 angle which is just ‘right’ for construction use..... .....
Triangles can be duplicated and then put together to make up other shapes, like a parallelogram, square, rectangle, rhombus, octagon, hexagon, pentagon. With these last three they would make a ‘regular’ shape; that is one where all the angles are the same, the sides usually are too.
There is a special triangle called a ‘right angle’ triangle because it has an angle that is 900,
just right for building and ensuring things are square. Right angle triangles can be scalene or isosceles
triangles but not equilateral triangles.
In a right angle triangle the side opposite the 900 angle is called the hypotenuse and its length can be calculated from the other two sides as the ‘square of the hypotenuse will always equal the sum of the squares of the other two sides’. Thus you calculate the square for the other two sides, add them together and get the square root of that figure
One of the most common is called a ‘3/4/5’ triangle as the side lengths are kept to those ratios. 3 squared is 9, 4 squared is 16, that adds to 25 the square of 5. This is a scalene right angle triangle. The other angles are usually 300 and 600.
An isosceles right angle triangle would have the two sides from the 900 the same length and the other two angles are 450 each, as the angles is the same for each. In this case the value of the other two angles will always be 450
With an equilateral triangle the value of the angles will always be 600 as that is
the only way you can get the length of all sides to be the same.
NB: All these relationships above can be used to help calculate the value of angles and areas in other shapes.
From what we know about triangles we can use them to calculate the angles of multi-sided
shapes by simply changing the shape into a group of triangles. In all cases we can calculate
the total value of the angles in the shape, and in a regular shape we can calculate the
value of each angle. A regular shape is one where all the angles are the same, thus the sides
are the same and it has a uniform appearance.
Regardless of what the shape is you need only place a dot in the centre of the shape and then draw a line to the middle of each angle, thus dividing the shape up into a number of triangles; as per below. And then imagine a circle around the joining point.
hexagon... ........octagon... ........regular pentagon...
........pentagon... ........trapezium... ........parallelogram...
In these examples we can see that each shape is now a number of triangles with a circle in the middle. The sum of all the angles in any triangle is 1800 and all the triangles join to make a circle of 3600 in the centre. Thus we need only add up the number of triangles, multiply that by 1800 and subtract 3600 to have the sum of the outside angles; this will work in all the shapes.
Hexagon is 6 x 180 = 1080 – 360 = 7200 .........Octagon is 8 x 180 = 1440 – 360 = 10800
Pentagon is 5 x 180 = 900 – 360 = 5400 .........Parallelogram / Trapezium is 4 x 180 = 720 – 360 = 3600
The bolded answers above are the sum of the angles in each shape. Both pentagon calculations are the same, the sum of all the angles will remain the same but the individual angle values will be different between the regular and irregular pentagons. If the shape is a regular one then the value for each shape will be the bolded answer divided by the number of angles. Thus the octagon angles are 1080 ÷ 8 = 1350, the hexagon angles are 720 ÷ 6 = 1200, the regular pentagon angles are 540 ÷ 5 = 1080.
Tessellation is a fancy word that means patterned or patterning (depending upon usage). Some
shapes can tessellate, some cannot. To be able to tessellate the shape MUST have angles that can
divide evenly into 3600, or be able to be grouped into a set of 3600.
In the shapes above the regular octagon and regular pentagon cannot tessellate as their angles will
not go evenly into 3600, whereas the regular hexagon and irregular pentagon will;
as will all squares and rectangles. Some tessellation examples are:
.... .... ....
Most mathematical processes are simple calculations using only one function as either a
single process or a repeated process; such as adding up a column of figures. When the calculation
used two or more processes it becomes a complex calculation, most complex calculations involve
only a process and its opposite, such as addition and subtraction in a column of figures. When the
calculation involves mixed processes you must process them in a set order to ensure a correct answer.
The order of processing functions is remembered by the term BoDMAS. This is made
up from the letters of the processes in their processing order, Brackets opened, Division,
Multiplication, Addition, Subtraction. By having everybody following the same
order you all get the same results.
You perform one set of calculations at a time and start with the one in the deepest set of brackets and then repeat for each set of brackets until there are no brackets, then complete the other calculations. When processing within each set of brackets you process in the DMAS order. The example below shows the value of correct order.
Example:.......... 4 x 5 + 2 = ................Processed as if it had brackets:
A.......................(4 x 5) + 2 = 22...........20 + 2 = 22
B........................4 x (5 + 2) = 28............4 x 7 = 28
BODMAS tells us to process the calculations as process method A.
A formula is a way of expressing a constant mathematical relationship without using specific
numbers or values. Letters or symbols are used to express certain values and concepts. In
calculating the area of a rectangle you always multiply the length of one side by the length of the
other side that joins it at a corner; these are referred to as the height and breadth. Thus H x B = A
(where H is height, B is breadth, A is area). To calculate the actual area you just replace the H and B
with the relevant real values. This formula, like all formulas, will always work.
This is useful in helping to understanding mathematical relationships, and in finding mathematical relationships where we don’t know all the values. The formula allows us to play with the equation components and view various mathematical relationships without being distracted by the values. Change H x B = A into A ÷ B = H; and you create a formula to find the height where the total area and breadth are known. This process of manipulating the formula is known as Algebra.
When you can get a component of the formula or equation by itself on one side of the equals sign it is said to be isolated. By isolating a component we can then use the values of the other components to calculate the value of the isolated component. Thus algebra is used to isolate unknown component from the known components to permit its calculation.
In using algebra to manipulate formulas and equations we must always remember the Basic
Principle to keep both sides of the equals sign (=) balanced and BODMAS. In order to keep
both sides of the equation balanced, what is done to one side must be applied to the other side
in the exact same manner. Divide one side by a value you must divide the other side by the same
value, the same with all other functions. To manipulate the equation you don’t need to know the
values or what they relate to.
S = T x D........we know S and D and must find T so we divide both sides by D
.......................S ÷ D = T x D ÷ D.........Simplifies to.........T = S ÷ D
D ÷ C = L.......we know C and L and must find D so we multiply both sides by C
.......................D ÷ C x C = L x C.........Simplifies to.........D = L x C
A + B = C.......we know B and C and must find A so we subtract B from both sides
.......................A + B - B = C - B.........Simplifies to.........A = C - B
A - B = C.......we know B and C and must find A so we add B to both sides
.......................A - B + B = C + B.........Simplifies to.........A = C + B
With very complex formulas we can do this a number of times to isolate a variable.
D x G x M + L - C = H x K........we know all except M and want to know its value
.......................We can divide both sides by D, then G, then add C and minus L
.......................G x M + L - C = H x K ÷ D.........M + L - C = H x K ÷ D ÷ G
.......................M + L = H x K ÷ D ÷ G + C........M = H x K ÷ D ÷ G + C - L
In algebra or a formula multiplication and division signs are usually omitted. You show multiplication by putting the symbols together without any function, A x B is shown AB; and division is done with a divisor, D ÷ G is shown as D/G. The equations above with division and multiplication would normally be written as:
S = TD..............................................converts to.........T = S/D
D/C = L..............................................converts to.........D = LC
DGM + L - C = HK..........................converts to.........M = HK/DG + C - L
NB: The two division values can be placed together as a multiplying divisor.
A grid has a lot of lines running up and down and left to right, the darker line in each direction is the centreline for each axis. The horizontal or ‘x’ axis is positive above the line and negative below. The vertical or ‘y’ axis is negative on the left and positive on the right. A set of co-ordinates will have a number for each axis, eg 3,4 means 3 positive lines horizontally and then up, whilst 3, -4 means 3 across and 4 down. Always remember to walk along the aisle before climbing the ladder. A common use for grids is in locating things on maps and street directories.
Copyright by Ernest E Bywater