Tag Archives: base multiplication

MSC 19: Base Multiplication: One Number Above and One Number Below the Base

When one number is above and the other number is below the base, we either

a) add the excess of the number above the base to the other number or
b) deduct the deficiency of the number below the base from the other number.

Example 19.1: Find 104 x 98
Our base is 100, 104 is 4 above the base, and 98 is 2 below the base.
104 x 98
= (100 + 4) *(100 – 2)
= (104 – 2) * 100 + 4 *(-2)

Note that 104 – 2 and 98 + 4 will both yield 102
= 10,200 – 08
= 10,192

Example 19.2: Find 102 x 93
102 x 93
= (100 + 2)(100 – 7)
= 93 + 2 | 2 * (-7) Easier than 102 – 7
= 95 | -14
= 9500 – 14
= 9,486

Example 19.3: Find 109 x 97
109 x 92
= 109 – 3 | 9*(-3)
= 106 | -27
= 10,573

Here we just deducted 1 from 106 (109 – 3) and affixed the ten’s complement of 27 = (9 x 3) at the end.

Example 19.4: Find 114 x 88
114 x 88
= (100 + 14) (100 – 12)
= 114 – 12 | 14(-12)
= 102 | -168
= 100 | 200 – 168
= 10,032

Here the right hand part is -168, so we deducted 2 from the left hand part.

Proof:

Let x = base; a, b = excess / deficiency from the base

(x + a)(x – b)
= x² + (a – b)x + a(-b)
= (x + a – b)x – ab
= [(x + a) – b]x – ab
or
= [(x – b) + a]x – ab

Exercise 19: Find the following products using the base method

) 12 x 9 =
) 103 x 98 =
) 102 x 97 =
) 102 x 98 =
) 103 x 97 =
) 105 x 93 =
) 75 x 103 =
) 112 x 89 =
) 1012 x 991 =
) 1125 x 995 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

MSC 17 – Base Multiplication: Multiplying “Teen” Numbers and Others

Most of us have learned by heart the multiplication table up to 10 x 10. A simple technique will enable us to extend our multiplication power up to 20 x 20.

Example 17.1: Compute 14 x 12

Cover one of the ten’s digit and add what remains to the other number: 14 + 2 or 4 + 12 will both give 16. This will be the first or left hand part of the product.
Cover both ten’s digit and multiply what remains: 2 x 4 = 8. This will be the second or right hand part of the answer.
Thus 14 x 12 = 16 | 8 or 168

What we actually did in the first step is to add the excess over the base (10) of a number to the other.In the second step we multiplied the excesses. The algebraic proof of this method, known as base multiplication, is shown below:
(x + a) * (x + b)
= x² + (a + b) * x + a * b
= (x + a + b) * x + a * b
= [ (x + a) + b] * x + a * b

In our example above, x = 10, a = 4 and b = 2. So we have
14 x 12
= [ (10 + 4) + 2 ] * 10 + 4 * 2
= 160 + 8
= 168

In this case both multiplicands are above the base.

Example 17.2: Compute 16 x 13

(10 + 6) x (10 + 3)

Add 3 (excess of 13 over 10) to 16 to get 19
= ( 10 + 6) + 3 |
= 19 |
Multiply the excesses 6 x 3 = 18.
= 19 | 6 x 3
= 19 | 18
Since there is only one zero in the base, only one place is allotted for the right hand part. Therefore the 1 of 18 must be carried or added to 19 to get a final answer of 208
= 20 | 8 = 208

Example 17.3: Find 107 x 104

The left part is 107 + 4 = 111, and the right part is 7 * 4 = 28
(100 + 7) * (100 + 4)
= 107 + 4 | 7 x 4
= 111 | 28

= 11,128Note that in this example we did not write the two zeroes after 111 but two places are reserved for the 28.

Example 17.4: Find 1,025 x 1,012

The left part is (1025 + 12 ) = 1037 and the right part is 25 * 12 = 300
1,025 x 1,012
= (1000 + 25) * (1000 + 12)
= 1025 + 12 | 25 * 12
= 1037 | 300
= 1,037,300

Example 17.5: Find 115 x 111

The left part is (115 + 11) and the right part 165
115 x 111
= (100 + 15) (100 + 11)
= 115 + 11 | 15 * 11
= 126 | 165
Since we have only two zeroes in our base, we can allot only two spaces for the right hand part of the answer, so we must “carry” the 1 of 165 into the left side.
= 12,765

Example 17.6: Find 102 x 104

This is pretty straightforward with the left 102 + 4 = 106 and the right is 2 * 4 = 8
102 * 104
= (100 + 2) (100 + 4)
= 102 + 4 | 2(4)
= 106 | 08
In this case, the product of the excesses is 8 but since the base is 100, two spaces are allotted to it. We write it as 08.
= 10,608

Example 17.7: 103 x 119

The left side is 103 + 19 = 3 + 119 = 122 and the right side is 3 * 19 = 57
(100 + 3) * (100 + 19)
= 119 + 3 | 3 * 19
= 12,257
Note that in this example we chose to add the smaller excess to the other number. This always leads to a simpler calculation.

Exercise 17: Find the following products using base multiplication

12 x 13 =
14 x 17 =
15 x 18 =
108 x 101=
116 x 102 =
108 x 112 =
112 x 114=
123 x 106 =
1021 x 1006 =
1432 x 1002 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

25 Math Short Cuts

By Ike Prudente

This collection of 25 Math Short Cuts is published in commemoration of the 25th Foundation Day of MSC Institute of Technology.

I have selected what I think will be most useful for everyone, whether you are students, teachers, professionals or plain folks who need to calculate every day. Much of the hard work in compiling this work is not in researching for materials because I know these short cuts by heart but in selecting what short cuts to include.

From our grade school years, many of us have learned many ways of simplifying calculations. I am sure that for most of us, “multiplying by 10” is one of the first math short cut we have learned. And we continue to use it, which is the reason why I did not include it here. Unfortunately, most of us have forgotten most of the short cuts we have been taught in the past. As for my case, I am somewhat “lazy” at calculations so that I always looked for easier ways of computing. From my grade four teacher, I learned “squaring numbers ending in 5’ To this was added some more techniques which I picked-up from various books. I developed more short cuts by applying what I learned in algebra to simplify arithmetic computations. Then there were also some “tricks” I learned when I began to study “Vedic Math” in 2003, when they became available in the Internet.

$Math Short Cuts$

Even very young children can learn math short cuts, even adding left-to-right ( versus the traditional right to left )

I observed that my students enjoyed the short cuts that I shared with them and many came to love Math and actually improved their grades in the subject. I am also sure that these short cuts helped countless of my students pass the Philippine Science High School, UP Rural High School and UP entrance examinations.

Realizing that most readers like me, don’t want to read explanations, I will try to present the techniques in the form of examples, avoiding long explanations as much as possible. But sometimes we will cite Sutras or “word formulas” used in Vedic Math, which are very useful in remembering the short cuts.

We hope that with these shortcuts you will eventually be able to do most calculations mentally. However, we also realize that many will have to write down the answer first before reciting it. So we will explain the procedures to arrive at a written answer using a mental solution. With a little practice, many can dispense with the written answer.

Math Short Cuts (MSC)

We will add one short cut every week until Foundation Day 2014

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Tag Archives: base multiplication

MSC 19: Base Multiplication: One Number Above and One Number Below the Base

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MSC 17 – Base Multiplication: Multiplying “Teen” Numbers and Others

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