Fun with Figures

The following tutorials are based on examples and exercises given in the book 'Fun with figures' by Kenneth Williams, which is a fun introduction to some of the applications of the sutras for children.

Click here to visit the bookstore, for more details
of the original book this work is based on

If you are having problems using the tutorials then you could always read the instructions.

Tutorial 1
Tutorial 2
Tutorial 3
Tutorial 4
Tutorial 5
Tutorial 6
Tutorial 7
Tutorial 8
Tutorial 9
Tutorial 10
Tutorial 11
Tutorial 12

Tutorial 1

Use the formula ALL FROM 9 AND THE LAST FROM 10 to perform instant subtractions.

For example 1000 - 357 = 643
We simply take each figure in 357 from 9 and the last figure from 10.

So the answer is 1000 - 357 = 643

And thats all there is to it!

This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.
Similarly 10,000 - 1049 = 8951
For 1000 - 83, in which we have more zeros than figures in the numbers being subtracted, we simply suppose 83 is 083.
So 1000 - 83 becomes 1000 - 083 = 917

Try some yourself:

1) 1000 - 777	=
2) 1000 - 283	=
3) 1000 - 505	=
4) 10,000 - 2345	=
5) 10000 - 9876	=
6) 10,000 - 1101	=
7) 100 - 57	=
8) 1000 - 57	=
9) 10,000 - 321	=
10) 10,000 - 38	=
Total Correct	=

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Tutorial 2

Using VERTICALLY AND CROSSWISE you do not need to the multiplication tables beyond 5 X 5.

Suppose you need 8 x 7
8 is 2 below 10 and 7 is 3 below 10.
Think of it like this:

The answer is 56.
The diagram below shows how you get it.

You subtract crosswise 8-3 or 7 - 2 to get 5,
the first figure of the answer.
And you multiply vertically: 2 x 3 to get 6,
the last figure of the answer.

That's all you do:

See how far the numbers are below 10, subtract one
number's deficiency from the other number, and
multiply the deficiencies together.
7 x 6 = 42

Here there is a carry: the 1 in the 12 goes over to make 3 into 4.

Multply These:

1) 8 8 x	2) 9 7 x	3) 8 9 x	4) 7 7 x	5) 9 9 x	6) 6 6 x
Total Correct =

Here's how to use VERTICALLY AND CROSSWISE for multiplying numbers close to 100.

Suppose you want to multiply 88 by 98.

Not easy,you might think. But with
VERTICALLY AND CROSSWISE you can give
the answer immediately, using the same method
as above.

Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:

As before the 86 comes from
subtracting crosswise: 88 - 2 = 86
(or 98 - 12 = 86: you can subtract
either way, you will always get
the same answer).
And the 24 in the answer is
just 12 x 2: you multiply vertically.
So 88 x 98 = 8624

This is so easy it is just mental arithmetic.

Try some:

1) 87 98 x	2) 88 97 x	3) 77 98 x	4) 93 96 x	5) 94 92 x	6) 64 99	7) 98 97 x
Total Correct =

Multiplying numbers just over 100.

103 x 104 = 10712
The answer is in two parts: 107 and 12,
107 is just 103 + 4 (or 104 + 3),
and 12 is just 3 x 4.
Similarly 107 x 106 = 11342
107 + 6 = 113 and 7 x 6 = 42

Again, just for mental arithmetic

Try a few:

1) 102 x 107 =

1) 106 x 103 =

1) 104 x 104 =

4) 109 x 108 =

5) 101 x123 =

6) 103 x102 =

Total Correct =

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Tutorial 3

The easy way to add and subtract fractions.

Use VERTICALLY AND CROSSWISE to write the answer straight down!

Multiply crosswise and add to get the top of the answer:
2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13.
The bottom of the fraction is just 3 x 5 = 15.
You multiply the bottom number together.

Subtracting is just as easy: multiply crosswise as before, but the subtract:

Try a few:

Total Correct =

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Tutorial 4

A quick way to square numbers that end in 5 using the formula BY ONE MORE THAN THE ONE BEFORE.

75² = 5625
75² means 75 x 75.
The answer is in two parts: 56 and 25.
The last part is always 25.
The first part is the first number, 7, multiplied by the number "one more", which is 8:
so 7 x 8 = 56
Similarly 85² = 7225 because 8 x 9 = 72.

Method for multiplying numbers where the first figures are the same and the last figures add up to 10.

32 x 38 = 1216

Both numbers here start with 3 and the last
figures (2 and 8) add up to 10.

So we just multiply 3 by 4 (the next number up)
to get 12 for the first part of the answer.

And we multiply the last figures: 2 x 8 = 16 to
get the last part of the answer.

Diagrammatically:

And 81 x 89 = 7209

We put 09 since we need two figures as in all the other examples.

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Tutorial 5

An elegant way of multiplying numbers using a simple pattern.

21 x 23 = 483

This is normally called long multiplication but
actually the answer can be written straight down
using the VERTICALLY AND CROSSWISE
formula.

We first put, or imagine, 23 below 21:

There are 3 steps:

a) Multiply vertically on the left: 2 x 2 = 4.
    This gives the first figure of the answer.
b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8
    This gives the middle figure.
c) Multiply vertically on the right: 1 x 3 = 3
    This gives the last figure of the answer.

And thats all there is to it.

Similarly 61 x 31 = 1891

6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1

Try these, just write down the answer:

1) 14 21 x	2) 22 31 x	3) 21 31 x	4) 21 22 x	5) 32 21 x
Total Correct =

Multiply any 2-figure numbers together by mere mental arithmetic!

If you want 21 stamps at 26 pence each you can
easily find the total price in your head.

There were no carries in the method given above.
However, there only involve one small extra step.

21 x 26 = 546

The method is the same as above
except that we get a 2-figure number, 14, in the
middle step, so the 1 is carried over to the left
(4 becomes 5).

So 21 stamps cost £5.46.

33 x 44 = 1452

There may be more than one carry in a sum:

Vertically on the left we get 12.
Crosswise gives us 24, so we carry 2 to the left
and mentally get 144.

Then vertically on the right we get 12 and the 1
here is carried over to the 144 to make 1452.

Any two numbers, no matter how big, can be
multiplied in one line by this method.

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Tutorial 6

Multiplying a number by 11.

To multiply any 2-figure number by 11 we just put
the total of the two figures between the 2 figures.

26 x 11 = 286
Notice that the outer figures in 286 are the 26
being multiplied.

And the middle figure is just 2 and 6 added up.
So 72 x 11 = 792

77 x 11 = 847
This involves a carry figure because 7 + 7 = 14
we get 77 x 11 = 7₁47 = 847.

234 x 11 = 2574
We put the 2 and the 4 at the ends.
We add the first pair 2 + 3 = 5.
and we add the last pair: 3 + 4 = 7.

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Tutorial 7

Method for diving by 9.

23 / 9 = 2 remainder 5
The first figure of 23 is 2, and this is the answer.
The remainder is just 2 and 3 added up!
43 / 9 = 4 remainder 7
The first figure 4 is the answer
and 4 + 3 = 7 is the remainder - could it be easier?

Divide by 9:

1) 61 = remainder

2) 33 = remainder

3) 44 = remainder

4) 53 = remainder

5) 80 = remainder

Total Correct =

134 / 9 = 14 remainder 8
The answer consists of 1,4 and 8.
1 is just the first figure of 134.
4 is the total of the first two figures 1+ 3 = 4,
and 8 is the total of all three figures 1+ 3 + 4 = 8.

Divide by 9:

6) 232 = remainder

7) 151 = remainder

8) 303 = remainder

9) 212 = remainder

10) 2121 = remainder

Total Correct =

842 / 9 = 8₁2 remainder 14 = 92 remainder 14
Actually a remainder of 9 or more is not usually
permitted because we are trying to find how
many 9's there are in 842.

Since the remainder, 14 has one more 9 with 5
left over the final answer will be 93 remainder 5

Divide these by 9:

1) 771 = remainder

2) 942 = remainder

3) 565 = remainder

4) 555 = remainder

5) 777 = remainder

6) 2382 = remainder

7) 7070 = remainder

Total Correct =

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Tutorial 8

Finding the digit sum of a number.

Any number of any size can always be reduced to a single figure by adding its digits.

For example 42 has two digits which add up to 6.
We say "the digit sum of 42 is 6".
The digit sum of 413 is 8 because 4+1+3=8.
For 20511 the digit sum is 9.

For 417 we get 12 when we add the digits.
But as 12 is not a single figure number we add its digits: 1+2 = 3.
So the digit sum of 417 is 3.

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Tutorial 9

Adding from left to right.

We can find 34 + 52 by starting at the left:
Adding the tens figure first: 3+5=8. Then 4+2=6.
So 34 + 52 = 86.
For 76 + 86 we get 15₁2
So we carry the 1 to the 15 to get 162 as the answer.

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Tutorial 10

Using digit sums to check answers.

Suppose you want to check that the simple
addition sum below is correct.

     5  3
     2  4  +
     ___
     7  7

We find the digit sums of 43, 32 and 75 and check
that the first two digit sums add up to the third digit sum.

     5  3         8
     2  4  +     6 +
     ___          _
     7  7          5

The digit sums are shown on the right and
8 + 6 = 5 is correct in digit sums because
8 + 6 = 14 = 5 (1 + 4 = 5)

This confirms the answer.

All sums, even the most complex, can be checked in this way.

Check these using digit sums and find out which are wrong:

1) 56 88 + ___ 144 ?=

2) 83 38 + ___ 121 ?=

3) 77 69 + ___ 156 ?=

4) 545 273 + ___ 818 ?=

5) 357 753 + ____ 1010 ?=

Total Correct =

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Tutorial 11

Multiplying left to right.

If you want to find 73 x 6 from left to right, multiply the 7 by 6 first, to get 42.
Now multiply the 3 by 6 to get 18.
So you have 73 x 6 = 42₁8,and the 1 is carried to the 42 to give 438.

For 234 x 7 you get 14₂1 first which you mentally simplfy to 161.
The multiplying 4 by 7 you have: 161₂8, which becomes 1638.

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Tutorial 12

Bigger divisions.

369 ÷ 72 = 5 remainder 9
Divide the 36 at the beginning of 369 by the first figure of 72: 36÷7=5 remainder 1.

This gives: 36₁9 ÷ 72 = 5

The remainder, 1, is placed as shown and makes 19 with the 9 following it.
From this 19 we subtract 2×5 (the answer figure multiplied by last figure of 72):
19-10=9, the remainder.

To sum up: for 369 ÷ 72:
36 ÷ 7 = 5 remainder 1 gives 36₁9 ÷ 72 = 5
and 19-2×5=9 the remainder, so 36₁9 ÷ 72 = 5 rem 9

1) 456 ÷ 87 = remainder

2) 468 ÷ 73 = remainder

3) 369 ÷ 84 = remainder

4) 543 ÷ 76 = remainder

5) 357 ÷ 61 = remainder

6) 131 ÷ 43 = remainder

Total Correct =

Similarly squares, squares roots are easily tackled (in one line) by the Vedic method. We can also solve equations and geometrical and trigonometrical problems. The Vedic system covers all areas of mathematics.

It is not possible to show all variations of the methods here: all the techniques shown can be extended in various ways.

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Instructions
for using the tutorials

Each tutorial has test sections comprising of several questions each. Next to each question is a box (field) into which you can enter the answer to the question. Select the first question in each test with the mouse to start a test. Enter the answer for the question using the numeric keys on the keyboard. To move to the answer field of the next question in the test, press the 'TAB' key. Moving to the next question, will cause the answer you entered to be checked, the following will be displayed depending on how you answered the question :-

Correct
Wrong
Answer has more than one part (such as fractions and those answers with remainders). Answering remaining parts of the question, will determine whether you answered the question correctly or not.

Some browsers will update the answer on 'RETURN' being pressed, others do not. Any problems stick to the 'TAB' key. Pressing 'SHIFT TAB' will move the cursor back to the answer field for the previous question.
The button will clear all answers from the test and set the count of correct answers back to zero.

1) 21 47 x	2) 23 43 x	3) 32 53 x	4) 42 32 x	5) 71 72 x
Total Correct =

6) 32 56 x	7) 32 54 x	8) 31 72 x	9) 44 53 x	10) 54 64 x
Total Correct =