We use math in our daily life every day. Grocery shopping, Work, Home. Many times the average citizen depends on a calculator to accomplish tasks that may require simple calculation. Because of this, less and less people understand mathematics and let the calculator think for them instead of think with them. This is undesirable. If you make a mistake in typing the calculation, you may get an incorrect answer and not realize it.
Let me illustrate my point. Ask a group of students to do 2 + 2 / 2. Some students will get an answer of 2. Others will get an answer of 3. Which is correct? If we only used calculators, no one would know, since both came from inputting the same string of numbers on the keypad. The correct answer is 3, because of the order of operations.
However, if you understand the workings of mathematics, you’ll be able to tell if the answer is feasible. You may not even need to use a calculator at all.
By the end of this article, you will be able to multiply and divide large numbers in your head. No one will question your intelligence. What will they think when you’ll be able to estimate square roots, even cube roots, in your head, while they cannot even do that with a pencil and paper?
When someone is considered good at math, they’re usually perceived as “smarter” or “more intelligent”. This is not necessarily the case. These students may have a better problem-solving approach to mathematics: an understanding. The easier a method is, the more likely the problem will be solved correctly. The methods I am going to teach you are known as the “Vedic Mathematics”, a system based on the sixteen “Sutras”, or principles. It is what I consider the easiest method to learn mental calculation.
Without further ado, I present to you: How to Master Speed Mathematics and Never Need a Calculator.
Multiplication
For the sake of practicality, I’m going to assume that you already know your times tables. So we’ll start with something rather promising. We’ll start off with multiplications in the 90’s.
Take this problem: 92 X 97
Can you solve this problem mentally? Probably not. In fact, I doubt most people can. But you’ll be able to.
How do most people go about solving this?
That’s what most people are taught to do. This is very difficult to do mentally, however. Too many processes going on at one time. There is an easier, faster method that can easily be done mentally. That is the circles method.
From 92 X 97, you find a number both are near. In this case, 100 is a good case. 97 is 3 below 100 and 92 is 8 below 100. You write that like this.
From here, you subtract diagonally.
92 – 3 = 89.
97 – 8 = 89.
Both of these numbers will have the same answer, so choose whichever one is easier to do mentally. You take your running total of 89 and multiply it by the reference number, which in this case is 100. Your running total is now 8900.
Now you take the two circled numbers, 3 and 8, and multiply them together, to get 24. You then add this to your running total to get 8924. Do this on a calculator. It is the correct answer, and you’ve done very little work.
This method will work in nearly every senario of two relatively numbers in the same group. Take 45 X 49. It also works if you use a reference number of 50.
There is, however, an improvement. Say we want to multiply 103 by 105. You could use 110 and do as we did before, but you could also use 100 and do the problem even faster. The only difference is that you add diaganolly, instead of subtract. Set up the problem as you normally would, but in the circles place the difference to 100.
You do the problem as you normally would, and you simply add diagonally instead of subtract. There is no other difference.
However, when one number is above the reference number and one is below, we run into a small variation. Say you want to multiply 149 by 98. You would set it up as normal, choosing either to add or subtract diagonally…
However when you multiply 49 by 2, you’re really multiplying 49 by negative 2. So what you need to do is instead of adding the product of the numbers to your running total, you subtract it.
The answer is correct.
Direct Multiplication
While the previous method had the distinction of being very easy, it has a very serious flaw: If the two numbers have no immediate reference number, it is difficult to do. There are a few ways around that, but one of the most effective is this method: Direct Multiplication. Take 36 x 72. No immediate reference number leaps to mind, which poses a problem. Set the problem up as shown.
The first step is to multiply the tens digits, 7 and 3, and multiply them by 100 (Essentially, you’re multiplying 70 by 30. Its simplified into [7 x 10] x [3 x 10])
Then multiply crossways, as shown in the diagram, add them together, and multiply that by ten. (This works similarly to the last example). Add that to the previous total.
Finally, you multiply the ones digits together. After adding that to the running total, you’ll have the correct answer. A good thing about this method is that each step gives you a more accurate answer. If you need a rough estimate, than you just need to do the first step. A more refined estimate? The second step, and so on and so forth until you have the correct answer.
Multiplication: Odds and Ends
Another problem you may have may come from multiplying larger numbers by smaller numbers. This is actually quite simple and easy to understand. Take 43 X 6. The easiest way to do this is to just rewrite 43.
Solve this and add all the numbers together.
and you have the correct answer.
Addition
This is one of the most simple concepts, so I won’t dwell on this too much. The basic idea behind it is you use easier numbers in your calculations. Its easier to add 100 and take away 3 than it is to add 97. conundrums
Your rule of thumb is that if the number’s one digit is 5 or lower, you add the tens then add the ones. If the number is 6-9, you add an extra ten and subtract the difference.
Subtraction
This uses the same exact concept as addition. If you can master one, chances are you’ll master the other.
Squaring Numbers
Squaring numbers usually requires simply multiplying the number using the circles method. There are, however, a few tricks you can use to get certain answers faster.
If a number ends in 5, the number becomes easier to deal with. Say you have 35^2. You remove the five, add one to the remaining digit, and multiply those. place a 25 in front of the answer and you’re good to go.
If a number ends in 1, subtract one and square the answer. Add the original number to the result after you subtracted 1 and add the numbers together.
If a number ends in 9, you take a similar path. You add one and square the number, then add the two numbers, but subtract the result instead of add it.
Short Division
You can skip this area if you are relatively comfortable with short division.
What you must do is to draw a reference number nearby the far side of the equation. This number is usually 10. Then you draw circles below the divisor. Find a number that adds to the divisor to get the reference number. Then you add the number to the first digit to get the first part of your answer. Make another circle and Multiply the numbers in the circles. subtract the number from the second digit. This should give you the remainder. Add zeroes to the end of the dividend if you want a more detailed answer.
Long Division
Division by non-prime numbers is easy. You simply divide by factors. Find two relatively small numbers that multiply to get the number. Set up the equation like this.
Divide the number by the first number and then divide the second number and the quotient.
And you get your answer.
However, if you end up with a remainder, you’re better off setting up the equation a different way.
Estimate and Divide by factors. Multiply and continue normally, and Repeat estimation until you get your desired answer
Obviously, you could get a more exact answer by adding zeroes after the decimal point, but for many situations you won’t really need to.
Estimating Square Roots
Estimating square roots is a process broken down into six simple steps. We will use both 3145 and 70 as examples.
1: Pair the digits into groups of two
2: Find the two numbers that the answer is between, and choose the lower number. If you’re number is a perfect square, place that number. If there are more pairs, add a zero for every pair remaining.
3: Divide the number by the guess and subtract the number
4: Split the result
5: add result to original guess
6: Round a little bit down
Estimating Cube Roots
If you manage to learn this, no one will question your intelligence. The first step is to memorize the cube roots of the numbers 1-10.
Then you divide the digits of your problem into pairs of 3.
Like with square roots, find the two numbers and choose the lower number. Repeat this until you run out of pairs.
Then you divide the number by the estimate twice
Next you subtract the estimate from the result.
divide that by 3…
…and add it to your original estimate. You can round down here for more accurate results.
So there you have it! You can now do large calculations in your head, mentally, and have accurate ways of doing equations faster than you ever thought possible. There is a very simple way to check nearly any answer, but it is fairly lengthy, so I’ll hold off from posting unless I get a demand for such a thing. For more information, do a google search on Vedic Mathematics, or read a book on the subject. All my information was gotten by reading a book called “Speed Mathematics”. Good luck!
