Calculation Tricks : Multiplication

I write this article today amidst the excitement of release of my first ever compiled ebook on parajumbles. Visit the FB page : https://www.facebook.com/mbadecoded for more on that. Today we look at the calculation tricks for multiplication. First we start with the specific tricks that pertain to multiplication by a specified number. Then eventually we look at how multiplication can be done faster in general.

Multiplication by 11

Step 1 : Put the right most digit of number being multiplied as the right most digit of your answer.
Step 2 : Each successive digit of number being multiplied is added to the digit next to it and put as the next digit moving right to left. If the sum is of two digits the ten's place digit is carried over to the left.


example : 5892 * 11
Step 1 : Answer : -----2 which is nothing but 2 + 0(as no right hand side neighbour)
Step 2 : 9 + 2 = 11. Carry over 1. Put 1 beside 2. Answer : ----12
Step 2(Contd) : 8 + 9 + 1(carry) = 18. Carry over 1. Put 8 beside 12. Answer : ----812
Step 2(Contd) : 5 + 8 + 1(carry) = 14. Carry over 1. Put 4 beside 812. Answer : ---4812
Step 2(Final) : 5 + 1(carry) = 6. Put 6 beside 4812 to get your final answer : 64812

How about multiplication by 12 or 13 or ..?
Before adding the right hand neighbour you double each digit first in each step.
5324 * 12 = Written as : 05324 * 12
4*2 = 8. Answer : ----8
2*2 + 4 = 8. Answer : ----88
3*2 + 2 = 8. Answer : ----888
5*2 + 3 = 13.Add 3 to the left and carry over 1. Answer : --3888
0*2 + 5 + 1(carry) = 6. Final answer : 63888
Try figuring out the logic and you will be able to generalize multiplication for more numbers..For now moving onto other special cases.


Multiplication by 25

As simple as it can get. 25 = 100/4. You add two zeroes to your number and divide it by 4. Just an alternative to your usual calculation.

example : 1761253 * 25 = 176125300/4 = 44031325

Multiplication by 111

To multiply a two-digit number by 111, add the two digits and if the sum is a single digit, write this digit twice  in between the original digits of the number.

example : 35x111= 3885 ; 63x111= 6993

The same idea works if the sum of the two digits is not a single digit, but you need to take the carry over to the left.

example : 57x111= 6(5 + 1) | 3(=2 + 1) | 2 | 7 = 6327
because 5+7=12

For 3 digit numbers,
Carry if any of these sums is more than one digit.

example :
123x111 = 1 | 3 (=1+2) | 6 (=1+2+3) | 5 (=2+3) | 3 = 13653
241x111 = 2 | 6(=2 + 4)| 7(=2 + 4 + 1) | 5(=4 + 1) | 1 = 26751

352x111=3 | 9 (=3+5 + 1) | 10 (=3+5+2)| 7 (=5+2)| 2 = 39072



Multiplication of numbers close to 100(called the "base")

Case 1 : 102*106 i.e both numbers greater than 100.

Left part of the answer will be : (102 + (106 - 100)) = 108 OR (106 + (102 - 100) = 108

Right part of the answer will be : A two digit multiplication of surplus from both numbers = 2*6 = 12. Carry the left most digits of this right part if the answer is more than two digits.

Answer : 10812


Case 2 : 95*97 i.e both numbers are less than 100
Similar approach. Just surplus replaced with deficits. Left part : (95 - 3) = (97 - 5) = 92
Right part : 3*5 = 15. Carry if required.

Answer : 9215


Case 3 : 104*95 i.e one number is less and other greater than 100

Again the similar approach. Left part will be : (104 + (-5)) = (95 + 4) = 99

Right part = 4*(-5) = -20.

Answer : 99 | -20 <- Does that look right to you? I hope so no. Let's look at how to make it look right.

Removing the negative sign
You borrow a 1 from the neighbouring 9 which by value is a 100. So your answer will become :
99 - 1 | 100 - 20 = 9880

Another example for such a kind where you need to borrow again but this time something different. Idea is to bring down the right hand side to a positive two digit number.

91*112 = (91 + 12) | -9*12 = 103|-108 = 103 - 2 | 200 - 108 = 101|92 = 10192

Let's end this one here today. Practice a lot on these techniques(Squares one earlier as well) if you want them to actually save your time in calculations. Solve at least 15-20 questions on each type by constructing various number combination. Like and share if you like the content on Lagom.
Next time we ll discuss the general multiplication techniques as well as when the base is changed from 100 i.e to say the numbers are close to say 50 or 400 or 1000 or anything like that. Keep following..Till then..



Cheers!
AS

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Calculation Tricks : Squares and Cubes

Calculation Tricks will be a series of articles to aid you in handling all kinds of calculations that can be done faster. The phase 1 of this learning involves understanding the method while phase 2 involves practicing it on basic problems on the concepts. Eventually, phase 3 on your end will involve imbibing the methods into all your calculations as the methods discussed will be specific to situations so taking out the right weapon at the right occasion is an art you will have to hone by practice.


The first article today will be focused upon calculating squares and cubes of any number with a detailed focus on squares as cubes are not a frequent occurrence in calculations.
Prerequisite : Must know squares from 1-25(actually till 30 is advisable) as presented in the image below for quick reference. Do not use calculators,pen or paper while going through these articles or while practicing these concepts as it would defeat the whole purpose. You need to master this series with only mental means and completely take the physical means out of running.




Concept 1 : Squares of numbers ending with the digit 5

Let the number be written as n5. Then the square is given as : n(n + 1) | 25 where | indicates just a separator to visualize the squared number in two parts.

examples : 15^2 = 1*(1 + 1) | 25 = 225 , 45^2 = 4*5 | 25 = 2025 , 135^2 = 13*14 | 25 = 18225

Practice yourself : 35^2 , 225^2 , 1005^2

Concept 2 : Finding squares 26 - 75

The numbers in this range can be seen as (50 - x) for 26-49 or (50 + x) 51 - 75.
The square in this scenario is given as : (25 -+ x) | x^2 where the right hand part(x^2) must be written as a two digit number only. Hence 3^2 = 9 will be written as 09 while in case of a three digit square for right hand part, the left most digit of the square is added as carry to left hand part.

examples : 44^2 = (25 - 6) | 6^2 = 1936 , 53^2 = (25 + 3) | 3^2 = 2809 , and
64^2 = (25 + 14) | 196 = (39 + 1) | 96 = 4096
Practice yourself : 37^2 , 62^2, 73^2


Concept 3 : Finding squares 75 - 130

The numbers in this range can be seen as (100 - x) or (100 + x). Right hand part must again be two digit number only while left hand part is calculated as (100 +- 2x).

examples : 93^2 = (100 - 2*7) | 7^2 = 8649 , 113^2 = (100 + 26) | 13^2 = (126 + 1) | 69 = 12769

Pick examples for yourself in this range and practice.


Concept 4 : Finding squares 175 - 230 , 275 - 330 and so on..


The concept primarily remains same as concept 3 with the only change in base of our square calculation. These are calculated : (200 +- 2x)*2 | x^2 where the right hand part again remains as a two digit number since we are dealing in hundreds.

examples : 202^2 = (200 + 2*2)*2 | 04 = 40804 , 193^2 = (200 - 2*7)*2 | 49 = 37249

Practice your own examples to get familiar with the concept.

For 275 - 300 range we take (300 +- 2x)*3 | x^2 and so on for 875 - 930 range..


Concept 5 : Find squares of n digit numbers where n >= 4 i.e thousands and beyond

The change for 1000s , 10000s and beyond is that right hand part is taken as a three digit number, four digit number and so on respectively..
examples :
1017^2 = (1000 + 2*17) | 17^2 = 1034 | 289 = 1034289 [ No carries this time ]
1994^2 = (2000 - 2*6)*2 | 6^2 = 3976 | 036 = 3976036  and so on..


Before moving onto cubes, did anyone think why are we able to do this ?
(50 +- x)^2 = 2500 + x^2 +- 100x = (25 +- x)*100 + x^2 = (25 + x)|x^2
(100 + -x)^2 = 10000 + x^2 +- 200x = (100 +- 2x)*100 + x^2 = (100 +-2x)|x^2
and  so on..
Whenever you learn shortcuts, do try to be curious about the concept behind it. Would give you a more generic outlook to the shortcut..


Concept 6 : Finding cubes

We ll discuss the method for finding two digit cubes as more than that is not useful for our purpose nor necessary. The prerequisite for this shortcut is you knowing the first 10 cubes by heart.
1^3 = 1 , 2^3 = 8 , 3^3 = 27 , 4^3 = 64 , 5^3 = 125 , 6^3 = 216 , 7^3 = 343 , 8^3 = 512 , 9^3 = 729 and 10^3 = 1000 [ Advisable to learn up to 15 if you have the time ]


Let's learn the calculation technique then :




That's about it you will ever need for calculating squares and cubes. Next article in this series would take up some other aspect of calculation part involved in any of your MBA exam papers. If you like the content here on Lagom, please help spread the word by sharing,liking and commenting on posts about the content.


Until next time..
Cheers !
AS

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