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All About ‘Additions’

‘The only way to learn mathematics is to do mathematics.’

—Paul Halmos

You must have done addition since you were four or five years old. It is one
of the most understood topics that most young people like you excel in. I
remember when I was your age, I used to love doing addition problems, I
found them rather easy. My teacher at school used to give me A+ with five
stars and a sticker whenever I answered a set of problems on addition
correctly.
Apparently one day when I was given a set of problems on addition to do
mentally, I struggled. I was asked to add two digits to another two digits in
a problem—it was somehow difficult. And I found bigger digits, say a
three-digit number plus another three-digit number even more cumbersome
and challenging to do mentally. It was all right to solve it on paper but my
mind drew a blank when I was asked to do the same problem mentally. It
was scary!
In our schools, mental mathematics is given due importance but there
aren’t any fixed rules given about how to approach a problem orally—be it
addition or division, for that matter. The oral system is taught nowhere
because we give importance to step-by-step calculation. I don’t refute that
but I feel we must also be taught the oral way of maths, which comes in
handy while doing the four operations—addition, subtraction,
multiplication and division—mentally as you will see in this book.
Doing the four operations and more in less time gives one a sense of
accomplishment and a solid, robust mental framework in calculations. It
adds to your creativity in the way you do maths. Any problem can be solved
mentally in more than one way, as I will show you now.
So, let’s get started with addition and learn how to do it mentally.

Mental Additions
Sometimes, we find it difficult to add numbers which end in 6, 7, 8 and 9.
For example, if we have 16 + 9, that’s a difficult problem to do mentally.
But we can make it easy. We can use a method called ‘by addition and by
subtraction’.

Let us try 16 + 9.
Since adding 9 directly is difficult for most of us, we add 10 which is easy
to do—mentally. So, since 9 is 1 less than 10, we can add 10 and then
subtract 1 from our answer.
Our sum looks like this:
16 + 9
We do: 16 + 10 = 26
26 - 1 = 25 is our answer.
Here, I would like to draw your attention to the method which is called ‘by
addition and by subtraction’. So, we add first and then subtract. Hope this
bit is clear to you.

Let’s take another example. Say, we have:

68 + 9

We do 68 + 10 = 78.
78 - 1 = 77 our answer.

Now, let’s try another example on our own.

59 + 8

We do: 59 + 10 = 69
Since 8 is 2 less than 10, we do:
69 - 2 = 67. The answer is 67.

Now can you tell me what happens if one of the numbers being added was
closer to 20?
Say, we have 116 + 18
So, to add 18, we add 20 and then subtract the 2. That was quite simple!
So, we do:
116 + 20 = 136
136 - 2 = 134. This is our answer.

Now let’s try addition with a larger number

139 + 69
Here, to add 69, we add 70 and then subtract 1 from the total.
139 + 70 = 209
209 - 1 = 208 our answer.

Let’s try another problem.

166 + 88
To add 88, we added 90 and then subtract 2.
166 + 90 = 256
256 - 2=254 was the answer.

ACTIVITY 1

Find the sum.

Find the sum.
Left to Right Mental Additions

Let us now see another method of quicker addition. This method is called
the ‘left to right mental addition’ method. So far, traditionally in maths, we
have been doing additions and other operations from right to left. Now, in
this method, we will get our answers efficiently from left to right, hence
disrupting and sidestepping the traditional calculating system.

Say for example, we have 78 + 45.

Step 1
We first add the figures in the left column. So 7 + 4 = 11. We keep that
figure in our head.

Step 2
We then add the figures in the right-hand column 8 + 5 = 13. We keep that
in our head too. The sum looks like this.
Step 3
In our final step, we add (or combine) the middle digits.

So, the answer is 123. I hope this is clear.

Let’s take another example to understand it better.

Say we have 87 + 38.

Step 1
We add the figures in the left column 8 + 3 = 11. We get 11, we keep that
figure in our head.

Step 2
We then add the figures in the right-hand column 7 + 8 = 15. We get 15. We
keep that in our head too. The sum looks like this now:
Step 3
In our final step, we add (or combine) the middle digits.

So, the answer is 125.

I hope you have understood how to add two-digit numbers. But what if we
have to add two three-digit numbers? There is a solution for that as well.

Say we have to add 582 + 759

Step 1
Start by adding the columns from left to right. The first column is 5 + 7 =
12. The middle column is 8 + 5 = 13.
Step 2
We add (combine) the middle digits of the first two columns. So, we have
133 in our head.
Then we add the last column on the right. So, we have 2 + 9 = 11. Now the
sum looks like this:

Step 3
So, in our mind, we have 133, 11. We then combine or add the digits on
either side of the comma. In this case, we add 3 and 1 and get 4. The final
answer is 1341.

Let’s take another example. Say, we have 698 + 576

Step 1
We add the first two columns from left to right. We get 11, 16.
Step 2
We add (combine) the first two columns. So, we have 126 in our head. We
then total up the final column and get 8 + 6 = 14. Finally our example looks
like this:

Step 3
We combine the digits on either side of the comma. So, in this case we add
6 + 1 = 7.

Our final answer is 1274.

ACTIVITY 2
Find the sum.
Find the sum.
Find the sum.

‘Mathematics is a language plus reasoning; it is like a language plus logic.

Mathematics is a tool for reasoning.’
—Richard P. Feynman, 1965 Nobel laureate

So that was mental addition made easy for you. These methods and systems
work elegantly if learnt and practised well. I hope you now have a solid
grounding in addition. But to take things little further and to make addition
really cool, I would like you to dabble with puzzles on addition from the
land of the rising sun—Japan. These puzzles will make your logical
thinking and mathematical reasoning skills strong.
So, did you know that Japan has given the world giants like the Sony
Corporation, sumo wrestling, Godzilla and tsunamis? And did you know
that if you were to rank countries with their maths scores, Japan ranks
among the world’s best! (Source: The Trends in International Mathematics
and Science Study [TIMSS] 2015 Global Study).

So, if you dig deeper into this and scout for reasons behind Japan’s success
in maths, you can be sure that it can be attributed to the amazing logical
puzzles created by some amazing Japanese mathematicians. Take Kakuro,
for instance.
The name Kakuro is Japanese for kasan kurosu, meaning ‘addition
cross’. Kakuro puzzles have been featured daily in the Guardian, the Daily
Mail, the New York Times and many other publications. The Japanese love
their puzzles and I can say that Kakuro is the second most popular puzzle in
the country. And it is catching on in other countries too because of the way
it helps improve mathematical reasoning and logical thinking skills. So, let
me share with you a 3 × 3 Kakuro puzzle and the rules on how to solve it.
We will start small and slowly expand to bigger Kakuro puzzles.

Rules: Just like letters in crossword puzzles, Kakuro puzzles are to be filled
with figures from 1 to 9, each figure to be used only once in a particular
entry. You should ensure that no digit is duplicated in any entry. You may
have repetitions of a digit in a different entry—that’s absolutely fine and
you will encounter many repetitions in 4 × 4 or bigger Kakuro puzzles. You
need to ensure that these figures add up to the numbers mentioned on the
puzzle. You must also remember that each Kakuro puzzle will have a
singular solution. All the best and have an awesome number-crunching
experience!

Kakuro puzzles will contain many clue squares; these are squares which
help you to solve the puzzle. A clue square can have an ‘across’ clue or a
‘down’ clue, or both.

Step 1
Let’s see step number one. So, start from the lowest number 5. So, we see
that we have 5 across.
So, our options are either 1 + 4 or 2 + 3.
Now for example, if we use 2 and 3 across, it would be invalid. Because
then it would be vertically 2 + 11 and the puzzle will look like this.
Even though this fits, we can’t use 11 as we have to use digits between 1
to 9. So, we change it and put 3 and 2.

Again, we can’t use 3 as we then need 10+3 to make 13. We can’t use 10
as the digits need to be between 1 to 9. So, this solution below is invalid as
well.

Step 2
So what else adds to 5? How about 4+1?
 So, we take the other option 4 and 1 to add up to 5. Then for 13 down, we
would need 9 which is correct. And then for 8 down, we have 7 and 1. Since
1 is already there we can now say that we have our unique solution.
Our completed puzzle looks like this:

Easy enough? Hope you liked this. You will love it more when you solve a
Kakuro puzzle on your own. So, let’s go ahead and get you initiated. In
Activity 3 below, I have given six Kakuro Puzzles and also some hints in
each of them. Try them now!
ACTIVITY 3

Now Kakuro puzzles can be of 3 × 3 or even 4 × 4 or even more than 30 ×

30 as well. I am giving you a few 4 × 4 Kakuro puzzles below as Activity 4,
so that you get to understand the basic concept behind it and then are able to
do Kakuro puzzles of any size. And to make it simpler for you, I have given
hints too. So, let’s get started!
ACTIVITY 4

I hope you saw mental additions in a new light and found Kakuro riveting!
Just Google Kakuro and you will find plenty of websites offering these
puzzles for practice for free. You could also visit
www.kakuroconquest.com.
Before we go forward, I must share with you the inspiring story of Jakow
Trachtenberg from Russia.
Jakow Trachtenberg was an ingenious chief engineer from St Petersburg,
Russia. He quickly climbed up the ranks in a shipping company to become
the supervisor of over 11,000 men in the early twentieth century. Being a
Jew and because of his radical political views, he was sent to Hitler’s
concentration camps amidst despair and horror. With little hope and faith,
Jakow started devoting his time at camp to building a new system of mental
arithmetic.
He had no paper, pencil or pen. Simply sitting in the camp, he found new
ways to look at numbers—mentally. Later, Jakow escaped camp after
spending years in it and founded the Mathematical Institute in Zurich,
Switzerland. Jakow believed and concluded that people are born with
‘phenomenal calculation possibilities’.
And this is why I wanted to share the story with you to urge you to
believe in yourself—no matter what the circumstances!
‘We must accept finite disappointment, but never lose infinite hope.’
—Martin Luther King Jr
Now, let me share with you Jakow Trachtenberg’s method of columnar addition.

High Speed Columnar Addition

Let’s take a sum. Say we have: 1256 + 8563 + 4658 + 2387
By the conventional method, we would have added from the rightmost column,
going 6 plus 3 plus 8 and so on. But in this system, we can begin by working on
any column. So, let’s start from the leftmost column.

Step 1
Here the rule is that we ‘never count more than 11’. We do the addition and
whenever the running total becomes greater than 11, we reduce it by 11 and go
ahead with the reduced figure called the running total. And as we do so, we make a
small tick or check mark beside the number that made our total higher than 11.
So, we have, from the leftmost column 1 + 8 = 9 + 4 = 13.
Now this is more than 11. So, we subtract 11 from 13. Make a tick and start
adding again with 2. We go on and add 2 so we get a running total of 4.
We write this under the column next to running total.
And next to ticks we write 1. Our example now looks like this:
We now go on to the next column from left and complete it too.
So, we go 2 + 5 = 7 + 6 = 13.
Therefore 13 is more than 11, so we put a tick and our running total becomes 2.
Then we go on 2 + 3 = 5.
So, our running total becomes 5 and number of ticks is 1.
Our sum at the end of all the four columns looks like this:
Step 2
Now we arrive at the final result by adding together the running total and ticks in
this way. We add in the shape of the alphabet ‘L’.
So, we have 2 plus 2 plus 0 = 4 in the unit’s place.
Then we have, 2 + 2 + 2 = 6 in the ten’s place.
Then we have, 5 + 1 + 2 = 8 in the hundreds place.
Then we have, 4 + 1 + 1 = 6 in the thousands place.
Finally, we have, 0 + 0 + 1 = 1
So, our answer becomes 16864
Easy? This is an alternative method to do quick columnar additions. Let’s go ahead
and take another sum as our example.
Say we have 89621 + 46892 + 14780 + 45893 + 29003
Step 1
We start doing the sum from the leftmost columns. We never add more than 11 as a
thumb rule. We get our running total and ticks as follows:

Step 2
We now add in the shape of the alphabet ‘L’.
For the units place we get 9 + 0 + 0 = 9
For the tens place, we get 6 + 2 + 0 = 8
For the hundreds place, we have 7 + 2 + 2 = 11. We put 1 down and carry-over 1.
For the thousands place, we have 0 + 3 + 2 + 1 (carry-over) = 6
For the ten thousands place, we have 8 + 1 + 3 = 12. We put 2 down and carry-over
1.
For the lakh’s place, we have 0 + 0 + 1 + 1 (carry-over) = 2.

So, our answer becomes 226189.