Mathematics lesson: Multiplication
Teacher: Mona, Delhi.
Mathematics is a wonderful, elegant, and exceedingly useful language. It has its own vocabulary and syntax, its own verbs, nouns, and modifiers, and its own dialects and patois. It is used brilliantly by some, poorly by others. Some of us fear to pursue its more esoteric uses, while a few of us wield it like a sword to attack and conquer income tax forms or masses of data that resist the less courageous. This article does not guarantee to turn you into a Leibniz, or put you on stage as Professor Algebra, but it will, I hope, bring you a new, exciting, and even entertaining view on Multiplication.
In this article, you will learn to do multiplication in your head faster than you ever thought possible. After practicing the methods in this article for just a little while, your ability to work with multiplication will increase dramatically. With even more practice, you will be able to perform many calculations faster than someone using a calculator.
Multiplying Number by Eleven:
Let’s begin with how to multiply any two-digit number by eleven in your head. Consider the problem:
45 X 11
To solve this problem, simply add the digits, 4+5 = 9, put the 9 between the 4 and the 5, and there is your answer, 495. What could be easier? Now you try:
53 X 11
Since 5 + 3 = 8, your answer is simply, 583.
Now before you get too excited, suppose the problem is
67 X 11
Although 6+7= 13, the answer is NOT 6137, as before, the 3 goes in between the numbers, but the 1 needs to be added to the 6 to get the correct answer: 737
Now, if you tell a friend or teacher that you can multiply, in your head, any two-digit number by eleven. You will be amazed at the reaction you get.
Now, question is, Can we use this method for multiplying three-digit numbers (or larger) by eleven?
Answer is yes, you can, for instance, for the problem 314 X 11; the answer still begins with 3 and ends with 4. Since 3+1=4, and 1+4= 5, the answer is 3454. More you practice more you comfortable with multiplying any number with eleven.
Simple Methods for 2-BY-2 Multiplication:
When squaring two-digit numbers, the method is always the same. When multiplying two-digit numbers, however, you can use lots of different methods to arrive at the same answer. For me, this is where the fun begins. I will tell you about “addition method,” and similarly one can use “subtraction method” to solve all 2-by-2 multiplication problems.
The Addition Method:
To use the addition method to multiply any two two-digit numbers, all you need to do is perform two 2-by-1 multiplication problems and add the results together. For example to multiple 42 by 46 one can use following steps:
46
X 42 (40+2)
40X46 = 1840
2X46 = + 92
——–
1932
Here you break up 42 into 40 and 2, two numbers that are easy to multiply. Then you multiply 40X46, which is just 4X46 with a 0 attached, or 1840. Then you multiply 2X46= 92. Finally, you add 1840+92= 1932.
Here’s another way to do the same problem:
46 (40+6)
X 42
40X42 = 1680
6X42 = +252
——–
1932
The catch here is that multiplying 6X42 is harder to do than multiplying 2X46, as in the first problem. Moreover, adding 1680+252 is more difficult than adding 1840+92. So how do you decide which number to break up? I try to choose the number that will produce the easier addition problem. In most cases—but not all—you will want to break up the number with the smaller last digit because it usually produces a smaller second number for you to add.
Following are some of the tips on breaking up the number:
• If both numbers end in the same digit, you should break up the larger number
• Break the number which ends with 1
• If one number is much larger than the other, it often pays to break up the larger number, even if it has a larger last digit
• When you multiply a number in the fifties by an even number, you’ll want to break up the number in the fifties
Subtraction Method:
Subtraction method is more or like similar to addition method. Difference is that instead of using addition one use subtraction to simplify the number. So, I will not go in too much of details. The subtraction method really comes in handy when one of the numbers you want to multiply ends in 8 or 9.
The Factoring Method:
The factoring method is my favorite method of multiplying two digit numbers since it involves no addition or subtraction at all. You use it when one of the numbers in a two-digit multiplication problem can be factored into one-digit numbers.
To factor a number means to break it down into one-digit numbers that, when multiplied together, give the original number. For example, the number 24 can be factored into 8 X 3 or 6 X 4. It can also be factored into 12 X 2, but we prefer to use only single-digit factors.
To see how factoring makes multiplication easier, consider the problem of multiplying 46X42.
Previously we solved this problem by multiplying 46 X 40 and 46 X 2 and adding the products together. To use the factoring method, treat 42 as 7 X 6 and begin by multiplying 46 X 7, which is 322. Then multiply 322 X 6 for the final answer of 1932. You already know how to do 2-by-1 and 3-by-1 multiplication problems, so this should not be too hard.
SQUARE:
Now, I will tell you some basics of squaring
Here is another quick trick. As you probably know, the square of a number is a number multiplied by itself. For example, the square of 6 is 6X6= 36. Now, I will teach you a simple method that will enable you to easily calculate the square of any two-digit or three-digit (or higher) number. That method is especially simple when the number ends in 5, so let’s do that trick now.
To square a two-digit number that ends in 5, you need to remember only two things.
• The answer begins by multiplying the first digit by the next higher digit.
• The answer ends in 25.
For example, to square the number 45, we simply multiply the first digit (4) by the next higher digit (5), then attach 25. Since 4X5= 20, the answer is 2025. Therefore, 45X45 is 2025.
How about the square of 85? Since 8X9= 72, we immediately get 85X85= 7225.
We can use a similar trick when multiplying two-digit numbers with the same first digit, and second digits that sum to 10. The answer begins the same way that it did before (the first digit multiplied by the next higher digit), followed by the product of the second digits. For example, let’s try 63X67. (Both numbers begin with 6, and the last digits sum to 3+7=10.) Since 6X7=42, and 3X7=21, the answer is 4221.
Remember that to use this method, the first digits have to be the same, and the last digits must sum to 10. Thus, we can use this method to instantly determine that
41X49= 2009
42X48= 2016
43X47= 2021
44X46= 2024
45X45= 2025
Come back to squaring, I will tell you how squaring of two numbers can be made easy. You need to keep in mind following two rules:
1. Simplify given number by adding or subtracting with one digit number to make it in the term of tens, say 67 can be rounded off to 70 by adding 3.
2. You need to subtract/add (if in above step addition is used then here use subtraction and vice–versa) the given number by same one digit number used above.
You have two numbers from the above two steps. Now you need to multiply both the resultant numbers and finally adds the square of one digit number used in the above steps.
Let’s take an example to simplify the problem. You have to square the number 61:
Step-1, subtract 61 by 1 i.e. 61-1= 60
Step-2, add 61 by 1 i.e. 61+1= 62
Multiply 60 by 62= 3720
Add square of 1 to the answer i.e. 3720 +1=3721
So, the square of 61 is 3720+1=3721.
Take another example, square of 76:
76X76 => 80 (76+4) X 72 (76-4) = 5760
4X4 = 16
So, square of 76 is 5760+16 = 5776
CUBING:
I would like to end this article with a new method for cubing two-digit numbers. Recall that the cube of a number is that number multiplied by itself twice. For example, 5 cubed is equal to 5 X 5 X 5 = 125. As you will see, this is not much harder than multiplying two-digit numbers. The method is based on the algebraic observation that
Cube of A = (A – d) X A X (A + d) + sq (d) X A
Where d is any number. Just like with squaring two-digit numbers, I choose d to be the distance to the nearest multiple of ten. For example, when cubing 13, we let d = 3, resulting in:
Cube of 13 = (10 X 13 X 16) + (3 X 3) X 13
Since 13 X 16 = 13 X 4 X 4 = 52 X 4 = 208, and 9 X 13= 117, we have
Cube of 13 = 2080 + 117 = 2197
How about the cube of 35? Letting d = 5, we get
Cube of 35 = (30 X 35 X 40) + (52 X 35)
Since 30 X 35 X 40 = 30 X 1,400 = 42,000 and 35 X 5 X 5 = 175 X 5 = 875, we get
Cube of 35 = 42,000 + 875 = 42,875
With some practice you will able to solve lots of multiplication problem in your head without using any paper pen or calculator. This will be a big boost for your personalities and solving various kinds of competitive exams.
Teacher: Mona, Delhi.
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