# Useful Formulas

Which one of these is it again…?

What is a problem? A problem = a fact + a judgment. That is a simple formula that tells us something about the way the world works. Maths is full of formulas, and that can intimidate some people if they don’t understand them or can’t remember the right one to use.

However, formulas should be our friends, as they help us to do sometimes complex calculations accurately and repeatably in a consistent and straightforward way. The following is a list of the most useful ones I’ve come across while teaching Maths to a variety of students at a variety of ages and at a variety of stages in their education.

## Averages

• The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
• The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
• The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.
• The range is the highest value minus the lowest, eg the range of the numbers 1-10 = 10 – 1 = 9.

## Geometry

• Angles around a point add up to 360º
• Angles on a straight line add up to 180º
• Opposite angles are equal, ie the two pairs of angles opposite each other when two straight lines bisect (or cross) each other
• Alternate angles are equal, ie the angles under the arms of a ‘Z’ formed by a line (or ‘transversal’) bisecting two parallel lines
• Corresponding angles are equal, ie the angles under the arms of an ‘F’ formed by a line (or ‘transversal’) bisecting two parallel lines
• Complementary angles add up to 90º
• Any straight line can be drawn using y = mx + c, where m is the gradient and c is the point where the line crosses the y-axis (the ‘y-intercept’)
• The gradient of a straight line is shown by δy/δx (ie the difference in the y-values divided by the difference in the x-values of any two points on the line, usually found by drawing a triangle underneath it)

## Polygons

• Number of diagonals in a polygon = (n-3)(n÷2) where n is the number of sides
• The sum of the internal angles of a polygon = (n-2)180º, where n is the number of sides
• Any internal angle of a regular polygon = (n-2)180º ÷ n, where n is the number of sides

## Rectangles

• Perimeter of a rectangle = 2(l + w), where l = length and w = width
Note that this is the same formula for the perimeter of an L-shape, too.
• Area of a rectangle = lw, where l = length and w = width

## Trapeziums

• Area of a trapezium = lw, where l = average length and w = width (in other words, you have to add both lengths together and divide by two in order to find the average length)

## Triangles (Trigonometry)

• Area of a triangle = ½bh, where b = base and h = height
• Angles in a triangle add up to 180º
• Pythagoras’s Theorem: in a right-angled triangle, a² + b² = c², ie the area of a square on the hypotenuse (or longest side) is equal to the sum of the areas of squares on the other two sides

Circles

• Circumference of a circle = 2πr, where r = radius
• Area of a circle = πr², where r = radius
• π = 3.14 to two decimal places and is sometimes given as 22/7

## Spheres

• Volume of a sphere = 4/3πr³, where r = radius
• Surface area of a sphere = 4πr², where r = radius

## Cuboids

• Volume of a cuboid = lwh, where l is length, w is width and h is height
• Surface area of a cuboid = 2(lw + lh + wh), where l is length, w is width and h is height

## Number Sequences

• An arithmetic sequence (with regular intervals) = xn ± k, where x is the interval (or difference) between the values, n is the value’s place in the sequence and k is a constant that is added or subtracted to make sure the sequence starts at the right number, eg the formula for 5, 8, 11, 14…etc is 3n + 2
• The sum of n consecutive numbers is n(n + 1)/2, eg the sum of the numbers 1-10 is 10(10 + 1)/2 = 110/2 = 55

## Other

• Speed = distance ÷ time
• Profit = sales – cost of goods sold
• Profit margin = profit ÷ sales
• Mark-up = profit ÷ cost of goods sold

# Shortcuts

It’s just through here…

There is always more than one way of solving a Maths problem. That can be confusing, but it can also be an opportunity – if only you can find the right trade-off between speed and accuracy.

I’ve taught a lot of QTS numeracy candidates recently, and the Maths itself isn’t particularly difficult, particularly in the mental arithmetic section.

The trick is to be familiar with all the possible short cuts and capable of using the right one at the right time. It may mean having to do more sums, but it will be much simpler and quicker in the long run. You don’t have to use all of these all the time, but it is useful to know what they are just in case you need them.

• Multiplying and dividing by 5
The most useful short cut I’ve come across is very simple. To multiply by 5, try multiplying by 10 and then dividing by 2 (or vice versa), eg
13 x 5
= 13 x 10 ÷ 2
= 130 ÷ 2
= 65
You have to do two sums rather than one, but the point is that you should be able to save time and improve the chances of getting the answer right by doing both in your head rather than having to work out a more difficult sum on paper.
You can do divide by 5 in a similar way by multiplying by 2 and dividing by 10 (or vice versa), eg
65 ÷ 5
= 65 x 2 ÷ 10
= 130 ÷ 10
= 13
You can do a similar trick with 50, 500 etc simply by multiplying or dividing by a higher power of ten.
• Chunking
If you have to multiply by a two-digit number outside your times tables, chunking is an easy way to do the sum in your head. Instead of writing it down on paper and using long multiplication (which would take a long time and is easy to get wrong!), try multiplying by the tens and the units separately and adding up the results, eg 16 x 15 = 10 x 15 + 6 x 15 = 150 + 90 = 240. The numbers might still be too tricky to do it comfortably, but it’s often worth a try.
• Rounding
To avoid sums with ‘tricky’ numbers, try rounding them up to the nearest ‘easy’ figure and adjusting at the end. This is particularly useful when working out start and end times, eg
‘The morning session in a school began at 09:25. There were three lessons of 50
minutes each and one break of 20 minutes. At what time did the morning session end? Give your answer using the 24-hour clock.’
If you assume the lessons last an hour, you can add three hours to 09:25 to get 12:25. You would normally then knock off 3 x 10 = 30 minutes, but the 20-minute break means you only need to subtract 10 minutes, which means the session ended at 12:15.
• Money problems
There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it – 1 x 2 x 3 is just the same as 3 x 2 x 1. The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning or turn multiplication by a fraction of a pound into a division sum, eg
‘All 30 pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get an average of 18 spellings correct each. The average amount of sponsorship was 20 pence for each correct spelling. How many pounds would the class expect to raise for charity?’
The basic sum is 30 x 18 x 20p, and there are a couple of ways you could do this:
1) Knock off the zeroes in two of the numbers, change the order of the numbers to make it easier and double and halve the last pair to give yourself a sum in your times tables, ie
30 x 18 x 20p
= 3 x 18 x 2
= 3 x 2 x 18
= 6 x 18
= 12 x 9
= £108
2) Convert pence into pounds, turn it into a fraction, change the order of the numbers, divide by the denominator and, again, double and halve the last pair to give yourself a sum in your times tables, ie
30 x 18 x 20p
= 30 x 18 x £0.20
= 30 x 18 x ⅕
= 30 x 18 ÷ 5
= 30 ÷ 5 x 18
= 6 x 18
= 12 x 9
= £108
• Percentages
Many students get intimidated by percentages, fractions and decimals, but they are all just different ways of showing what share you have of something. You will often by asked to add or subtract a certain percentage. The percentage will usually end in zero (eg 20%, 30% or 40%), so the easiest way is probably to find 10% first. That just means dividing by 10, which means moving the decimal point one place to the left or, if you can, knocking off a zero. Once you know what 10% is, you can simply multiply by 2, 3 or 4 etc and add or subtract that number to find the answer, eg
‘As part of the numeracy work in a lesson, pupils were asked to stretch a spring to extend its length by 40 per cent. The original length of the spring was 45 centimetres. What should be the length of the extended spring? Give your answer in centimetres.’
You need to find 40% of 45cm, so you can start by finding 10%, which is 45 ÷ 10 or 4.5cm. You can then multiply it by 4 to find 40%, which is best done by doubling twice, ie 4.5 x 2 x 2 = 9 x 2 = 18. Finally, you just add 18cm to the original length of the spring to find the answer, which is 45 + 18 = 63cm.
• Common fractions
An awful lot of questions involve converting between fractions, percentages and decimals. There is a proper technique for doing any of those, but it’s very useful if you learn the most common fractions and their decimal and percentage equivalents by heart, eg
½ = 0.5 = 50%
¼ = 0.25 = 25%
¾ = 0.75 = 75%
⅕ = 0.2 = 20%
⅖ = 0.4 = 40%
⅗ = 0.6 = 60%
⅘ = 0.8 = 80%
⅛ = 0.125 = 12.5%
⅜ = 0.375 = 37.5%
⅝ = 0.625 = 62.5%
⅞ = 0.875 = 87.5%
• Times tables
3 x 24
= (3 x 2) x (24 ÷ 2)
= 6 x 12
= 72
Alternatively, you can halve just one of the numbers and double the result, eg
24 x 9
= 12 x 9 x 2
= 108 x 2
= 216
• Multiplying by 4
If you have to multiply by 4 and the number is not in your times tables, a simple way to do it is to double it twice, eg
26 x 4
= 26 x 2 x 2
= 52 x 2
= 104
• Multiplying by a multiple of 10
If you have to multiply by a multiple of 10 such as 20 or 30, try knocking the zero off and adding it in again afterwards. That way, you don’t have to do any long multiplication and, with any luck, the sum will be in your times tables, eg
12 x 30
= 12 x 3 x 10
= 36 x 10
= 360
• Multiplying decimals
This can be a bit confusing, so the best way of doing it is probably to ignore any decimal points, multiply the numbers together and then add back the decimal point to the answer so that you end up with the same number of decimal places as you had in the beginning, eg
0.5 x 0.5
= 5 x 5 ÷ 100
= 25 ÷ 100
= 0.25
• Using the online calculator
The second section of the QTS numeracy test consists of on-screen questions that can be answered using an online calculator. This obviously makes working out the answer a lot easier, and short cuts are therefore less useful. However, just because the calculator’s there doesn’t mean you have to use it, particularly for multiple-choice questions. If you have to add up a column of cash values, for example, and compare it with a number of options, you could simply tot up the number of pence and pick the option with the right amount. Alternatively, the level of accuracy needed in the answer may give you a helping hand if it rules out all but one of the possible answers, eg 6 ÷ 21 to one decimal place is always going to be 0.3. Why? Well, it’s a bit less than 7 ÷ 21, which would be a third or 0.3 recurring. An answer of 0.4 would be more than that, and 0.2 would be a fifth, which is far too small, so it must be 0.3.
• Don’t do more than you have to!
There are several types of question that could tempt you into doing more work than you need to do. If you’re trying to work out how many tables you need at a wedding reception for a given number of guests, the answer is always going to need rounding up to the next whole number, so you don’t need to spend any time working out the exact answer to one or two decimal places. Equally, some numbers are so close to being an ‘easy’ number that you don’t need to add or subtract anything after rounding up or down to make the basic sum easier, eg
‘For a science experiment a teacher needed 95 cubic centimetres of vinegar for each pupil. There were 20 pupils in the class. Vinegar comes in 1000 cubic centimetre
bottles. How many bottles of vinegar were needed?
If you round 95cc to 100cc, the answer is 20 x 100 ÷ 1000 or 2 bottles, and the remainder consisting of 20 lots of 5cc of vinegar can safely be ignored.