Magic Math tricks for Fast Calculation: The method I have created is one that is holistic, containing as few methods as possible, subliminal, so that you learn advanced maths as you are doing the fundamentals, algebraic, meaning you think along abstract lines without initially realizing, and joyful because you can do things you hitherto thought impossible.
Maths Shortcut Tricks 2018 – Best Magic Math tricks for Fast Calculation
While reading this book, one question will constantly return to your mind.
‘WHY DON’T THEY TEACH THIS AT SCHOOL?’
I’ve been asked this many, many times, and there are a variety of answers.
For now, for you, you have in your hand the guide and passport to unlocking the secrets of maths and becoming one of its better users. You will be able to do things that will impress your friends, family, teachers and most of all yourself.
Take it on, be inspired, use the method for you and be a success!
Multiplication of Fractions
YOU ALREADY KNOW HOW TO DO IT!
Because it isn’t the same as adding and subtracting!
In fact, when we did add and subtracting, we picked up the skill we need in order to multiply.
I didn’t mention it at the time, but if we look at the currency conversion process, we can see it in action.
I just took it for granted that you would see that, for the first conversion
1 x 3 = 3
2 x 3 = 6
and each would change with no problem.
Although we are multiplying by
which is effectively 1, this is still written in the format of a fraction, and this is how we would also multiply any two fractions.
So you already know how to do it.
1 x 1 = 1
2 x 3 = 6 That’s it!
It is ironic that in school they spend all year learning how to add and subtract fractions, then when they come to multiplication they announce
‘Sorry everyone, this is completely DIFFERENT!’
When it isn’t. You intuitively and subliminally pick up the skill when you learn to add and subtract.
Finally, another way of saying ‘times’ when we are doing fractions is
So we might say
A half times a third or A half of a third.
If we look at what a half of a third actually is
We can see that it is indeed
Remembering the REFLEX
So why are simplifying now when I told you not to bother earlier?!?
So if you had
it’s easier to simplify
and then simplify afterward, as you would have to for any type of calculation when you get the final answer.
In the question
however, they are already as simplified as they can be.
Percentages – In A Minute
The word percentage, for example, comes from Latin/French. Do you know what ‘cent’ means?
Think of centimeters, dollars/euros, and cents, or centipede!
It means ‘100’.
The ‘per’ part means ‘for’ or ‘out of’. So we have that a percentage means ‘out of 100’.
This is the first introduction to an important mathematical idea. We compare what we are measuring with the number 1.
This is technically known as ‘normalisation’ and although I avoid jargon in this series of books like the plague, this one (pun, haha) is so important that I will refer to it again and again now and in future books.
This idea of measurement compared to one crop up in a variety of contexts in maths and once we understand its importance and significance, it makes some advanced concepts easier to understand also.
So percentage means a measure out of 100. For example, let’s say you were doing a survey, asking 100 people if they liked chocolate. Let’s say that 80 say yes, and 20 say no.
So what percentage of people like chocolate?
people liked it, so 80%.
In other words, the percentage sign is replacing the to say, ‘out of 100’.
What about if you got up the next day and thought ‘I’ll only ask 50 people today’?
So you ask 50 people, this time 35 say yes.
So what percentage of people like chocolate?
This time, it’s
But this does not mean 35%!
Here we have to scale it up to imagine we had asked 100 people, so since we only asked 50 (half of 100), we need to double that to make it 100.
In this case then,
Since we have to double 50, we must do that for 35 also (why?).
This is the first abstract concept in mathematics. Even though you haven’t asked 100 people, you’re taking a measurement as if you had. So this is the normalization process which allows us to compare to yesterday’s survey. We can now compare like with like.
It is something that is used widely, for example, in recording the profits a company makes compared to another – 10% for one, is not the same as 10% for another, in terms of the actual profit they made, but they must be growing at the same rate if they make the same percentage profit.
What about 5% of 3?
What is this?
Or 12% of 8?
You’ll note that they are exactly the same answer because the multiplication doesn’t ‘know’ which one has the percentage sign after it. So it’s the same result via the key change effect.
So this means we can do some more complicated looking percentages with ease, by switching them round to something easier.
For example, 50% of anything is half of it, as we saw above.
So 50% of 22 = 11.
What about 22% of 50? Instead of multiplying these together, we can notice that if we switch these around, to 50% of 22, we’ll get the same answer! So it may seem a bit tricky, but of course 22% of 50 = 11.
What about 9.8% of 50? Easy, 4.9.
What about 12.26% of 50? Easy, 6.13.
We can also do this with 25%, 75%, 20%, 200%, any percentage that is also an easy fraction.
So, for 25% of something is a quarter of it.
So 8% of 25 will equal 2 (as we’ve switched them around!)
Or 4.8% of 25? 1.2 of course.
35% of 20? 20% would be one-fifth of it (i.e. divide by 5), so we’re looking at 35/5 = 7.
And so on…
The goal is to achieve fluency and self-assuredness in something that only minutes ago seemed impossible. Impress your friends!
Before we looked at 14% of 180.
What’s 180% of 14?
If you use the same method, you get…
Of course, the same answer. In the multiplication, it doesn’t know which is the percentage and which isn’t! So you get 2 for the price of 1. This is something we can really use to our advantage.
Let’s say we need to find 22.4% of 50.
Ouch, multiplying them together may seem a little tedious. Can we get the answer immediately, without multiplying them?
Remember 22.4% of 50 = 50% of 22.4 as the multiplication doesn’t know which is the percentage! Of course, 50% of something is just half of it, so we can just half 22.4…
Seemingly difficult percentages are made easy.
How about 21% of 21? Easy, we have a squaring technique for that one,
available in Squaring & Area – In A Minute!
Or you could just do
21 X 21 —–441
In the usual way.
Dividing by 100 to make it a percentage,
When we go shopping and there is a sale on, the percentages and the new prices are usually already calculated for us. However, what if we’re the ones deciding! So we need to be able to calculate the discounts on things we buy (or sell).
Here’s a simple example, let’s say that you want to buy some shoes, which are normally £30. Today there’s a 10% discount. So the new price will be 10% off, which is £3
30 x 10 = 300, 300/100 = 3
and we can subtract that from the Original Price
30 – 3 = £27
So we know the Sale Price.
But is there a faster and more intuitive way to achieve this result? What is the percentage is a bit more complicated (say, 19%)? What is the price is more complicated (say, £37)?
Is there a faster way?
Funnily enough, there is.
What we can do instead is realize that we can do the reverse. When we’re subtracting 10% off something, as in the first example, we are effectively saying, ‘What is 90% of the Original Price?’. In other words, since we’re subtracting 10% from 100% (the Original Price), we’re left with 90%.
So in fact, we could have found 90% of 30
90 x 30 —-2700
Divide by 100
And found the final price much more quickly.
This technique really comes into its own with more complicated situations.
For example, Let’s say you want to buy a £76 pair of shoes, but today there’s a 22% discount. If we do this the traditional way, it’s a bit complicated, as we’ll end up with a nasty subtraction.
22% of 78
22 x 78 = 1716
Divide by 100
THEN we have to subtract that from £76…
So we have
….which is unpleasant!
How about we reverse finding the percentage and then subtracting to:
Subtract, then find the percentage.
So subtraction wise, subtract 22 from 100. 100 – 22 = 78% (since we want 78% of the Original Price).
So we find 78% of 76 using the usual method
78 x 76 —–59₁₀2₄8
Divide by 100 and we get
Much more intuitive, much easier.
So instead of finding the percentage and then subtracting our answer, it is much easier to subtract the percentage from 100%, the whole cost, and then find our new percentage!
This way we avoid a nasty subtraction, every time.
Instead of multiplying by 22%, using 78% instead is known as using a MULTIPLIER.
Here are a few to try:
- A pair of shoes are £62 with a 34% discount, what is the Sale Price?
- An iPod is £180 with a 22% discount, what is the Sale Price?
- A car is £8050, with an 11% discount, what is the Sale Price?
- 1. £40.92
- 2. £140.40
- 3. £7,164.50
Fast Calculation of Compound Interest
If we want to calculate compound interest, we find it’s a bit repetitive. It’s essentially the same step over and over for as many years (or times) as we compound. This kind of calculation is more computational, and a software program or a calculator is better suited to this kind of repetition.
As long as we know what we are doing, it is ok for the computer to do it for us!
So if we want to find an answer quickly, we can use the compound interest formula.
Where A = final amount, P = principal, or starting amount, r = rate of interest, expressed as a decimal, and t = time in years
So for example, a loan or savings of £5000 at 12% over 3 years would be written
Here we have a calculation we’ve not seen before. Technically known as a ‘binomial expansion’, which in English, means 2 numbers in a bracket multiplied. Binomial expansion is an A-level topic in the UK and is not in the scope of this book. For this calculation only, however, you are able to add the 1 and r (0.12) together.
Following BIDMAS, we first calculate and then multiply by 5000.
In reality, the indices calculation is telling you the growth over that period,
which is similar to the idea of the rule of 72. The actual starting amount doesn’t really matter, all that matters is the growth rate.
In the earlier examples, we saw that you could get up to 32 times your starting investment. In other words, the large part of that
is interest. So it almost doesn’t matter what you start with. The ‘DeLorean’ example started off with a pound!
Finding a Percentage of Any Number
First of all, what are we talking about?
What do we mean by finding a percentage of a number?
What we mean by this is to find a proportion of it, some fraction of it, or even some multiple of that number.
For example, 50% of 80 means half of 80 = 40.
200% of 120 means double 120 = 240.
So the fact that we are trying to find a fraction or a multiple of a number by finding its percentage can be very useful to us.
However, let’s quickly look at the school method.
For example, let’s say you want to find 14% of 21.
The school method is to either say ‘divide 14 by 100 and multiply by 21’ or to break the calculation up into finding
10%, which equals
1% which equals
(from above, dividing by ten again) = 0.21
Therefore, 14% can be found by adding 10% and 4 x 1% 2.1 + (4 x 0.21) = 2.1 + 0.84 = 2.94
This is a bit slow. However, the method works.
My version is more suitable to someone who has read and learned from ‘Multiplication In A Minute’.
As above, I noted that we are trying to find a fraction or a multiple of the number. This is a useful idea because we realize that we can simply, to find a percentage of any number, utilize the ‘Key Change’ effect once again, and multiply the numbers together!
So (from Multiplication In A Minute)
14% of 21 =
14 x 21 — 294
Then DIVIDE by 100 (from Decimals In A Minute)
So we instantly (almost) get the answer!
So to find any percentage of any number, we need to multiply them together and simply divide by 100.