One thing that fascinates me is performing mental math. Being
able to quickly perform additions, subtraction, multiplications etc is a
good way to impress your friends. The problem is, I’m not a math
genius, and I don’t know much behind simple arithmetic.
If you’re anything like me, but you’d still like to learn some basic math tricks, I hope you’ll find this list useful.
Example:
What happens if the sum of the two numbers is bigger than 9? In this case you add 1 to the first number, followed by the last digit of the addition of the two numbers, and then again you add the second number
Note that if the product of the second digits is below ten, you have to add a 0 in front of it.
Usually, you would first sum up 4 to 45, and then and 30 to the result. But by using the left to right approach, you first sum up 30 to 45, and then you add 4 to the result. Although this example is very simple, you’ll see the advantages of this method as you start to use it.
If you’re working with three digits numbers, the process is the same.
This example is a bit more complicated than the previous one, yet it’s very easy to solve using the left to right approach. You first start by adding 600 to 459, which results in 1059. Now the problem is simplified to 1049 + 37. You simplify it even further by adding 30 to 1049, and then you finally add 7 to the result.
In this case, you first start by subtracting 10 to 64, resulting in 54, and now you only have to subtract 7 to 54. You can, however, subtract 20 to 64 and add 3 to the result. This way you don’t have to worry about borrowings.
Instead of following the standard left to right approach, you could solve this problem by subtracting 400 to 674 and then add 42 back to the result. 42 is the difference from 100 and 58. A good question is: how do you find 42?
Note that there’s a simple pattern for calculating the second number. In particular, the sum of the first digits always sum up to 9, and the sum of the second digits always sum up to 10. The only exception is when the number ends with 0, which is simpler.
You can use this technique to solve any subtraction very easily.
As you may have already guessed, we’re going to use the left to right approach to solve simple multiplication very easily. Take the following example:
We can reduce it by first calculating 30 × 7 (which is like 3 × 7 plus a 0) and then add 6 × 7 on the result.
This approach can be used for even larger numbers. Note that you can also round up instead of rounding down:
To multiply 5 simply cut the # in half then multiply by 10.
eg. 17*5
1/2 of 17 = 8.5
8.5 * 10 = 85
Use this trick when multiplying numbers with multiple digits
let {a;b;c;d…} represent digits of a number
ab x cd = (axc), (axd + bxc), (bxd)
the commas represent separation of digits, so “axc” represents the digit in the hundreds place, etc.
eg) 23 × 14 = (2×1), (2×4 + 3×1), (3×4)
8 + 3
= 2,11,12
in the event of double digits in the same digit place, the number in the digit’s place (starting with the unit’s place) carries the ten’s place digit of the digit place to the following digit place [what a mouthful!]
like in this instance
= 2, 11, 12 = 2, 12, 2 = 3, 2, 2
the answer is 322
the theory behind this is the “distribution property” of numbers commonly used with equations like (x + 1)(x + 4)=0 to make x^2 + 5x + 4=0
the same principles can be applied with 3 digit numbers as well
abc x def = (axd),(axe+bxd),(axf+bxe+cxd),(bxf+cxe),(cxf)
for multiplying 2 digit with 3 digit numbers, just use the 3×3 digits method but use a zero in the hundreds place of the 2 digit number
Vedic mathematics provides lots of short cuts like shown here.
e.g.-
If you need to square a number close to 10^n, you can do so easily. Like if you want 92^2, lets take its answer as abcd.
Now, 92 is 8 before 100, so subtract 8 from 92, i.e. you get ab as 84. For finding cd, square 8 i.e. 64.
Hence the square of 92 comes as 8464.
For square of 87, let the answer is abcd again. Here 87 is 13 short of 100, so subtract 13 from 87 You get 74 as ab. For finding cd, square 13 i.e. 169. Since cd is only of two digits, add this extra 1 to ab.
So the answer becomes 7569.
To square 2 digit numbers ending with ‘5’ eg 75 × 75
1. The answer will end with ‘25’
2. Take the first digit ‘7’ multiply by the number after ‘7’ => 7 × 8 = 56
75 × 75 = 5625
Test it out with 95 × 95.
Did you get 8125?
take any number and find out how much to add to get it to the nearest tens subract and add that number to the orignal number multiply add the square
example:
(999+1) (999-1) + (1^2)
(998) (1000) + 1
999^2 + 998001
A math trick I noticed when I was young. If you are squaring a number it is always equal to the total of the number times 2 subtract one of the previous squared number. This is helpful if you dont want to write it out. For instance most people know that 10×10=100 or 11×11=121 even 12×12=144 so lets say you dont know 13×13. Its equal to (13×2)-1(plus the previous squared number which was 12×12)144=169
Suppose AB is the number,
Then arrange the number as follows,
A*A|2*A*B|B*B
( if A*A or A*B is one digit add 0 prior to that – eg: 4 should be written as 04, 5 should be 05 etc..)
Take a number : 35
09|30|25 ( 3*3 | double of 3*5 | 5*5 )
From right to left, keep the right most number as it is and add the number coming both side of | symbol.
ie. Keep 5 as it is, add 2+0, add 3+9
1225
Take another example 43
16|24|09 = 1849
Here’s a list of what you can expect to learn from the book:
If you’re anything like me, but you’d still like to learn some basic math tricks, I hope you’ll find this list useful.
Simple tricks
How to multiply any two digits number by 11
Let’s say that you want to find the product of 36 and 11. One way to find it would be to multiply 36 by 10 and then add 36 on the result. There is, however, a simple trick that’ll do the job for any two digits number. To find out the result, write the first digit followed by the addition of the first and second digit, followed by the second digit.Example:
What happens if the sum of the two numbers is bigger than 9? In this case you add 1 to the first number, followed by the last digit of the addition of the two numbers, and then again you add the second number
Square any two digits number that ends with 5
Calculating the square of a number below 100 is extremely simple. If you want to find the square of 25 for example, you simply have to take the first digit (2), multiply it for the next higher number (3), and then add 25 to the result.
Multiply any two digits numbers with the same first digit and the second digit that sums up to 10
Let’s say that you want to multiply 42 and 48 together. Notice that they both start with 4, and that the sum of their second digit is 10. In this case there’s a simple rule that you can use to find their product. Simply multiply the first digit (4) for the next higher number (5) and then append the product of their second digits.
Note that if the product of the second digits is below ten, you have to add a 0 in front of it.
Multiply by 9
To multiply by 9, simply multiply by 10 and then subtract the number itself.Quickly find percentages
- To find out the 15% of a number, divide it by 10 and the add half of it.
- To find out the 20% of a number, divide it by 10 and multiply the result by two.
- To find out the 5% of a number, divide it by 10 and the divide it by two.
Addition
When we were at school, we have been taught how to sum two or more numbers together by using the right to left approach. With this method, you first sum the decimal part of the number, then you move to the hundreds and so on. This works good on paper, but it’s a pain when you’re doing mental calculations. Fortunately, the solution is very easy.Left to right approach
Instead of using a right to left approach, we can start from the left and move to the right. Take the following example:Usually, you would first sum up 4 to 45, and then and 30 to the result. But by using the left to right approach, you first sum up 30 to 45, and then you add 4 to the result. Although this example is very simple, you’ll see the advantages of this method as you start to use it.
If you’re working with three digits numbers, the process is the same.
This example is a bit more complicated than the previous one, yet it’s very easy to solve using the left to right approach. You first start by adding 600 to 459, which results in 1059. Now the problem is simplified to 1049 + 37. You simplify it even further by adding 30 to 1049, and then you finally add 7 to the result.
Subtraction
Like with addition, you can use the left to right approach for subtracting to numbers together. This time, however, it may feel uncomfortable to keep track of borrowings (a borrowing occurs when you subtract a number to a bigger one, like 16 – 9). Let’s see an example of this.In this case, you first start by subtracting 10 to 64, resulting in 54, and now you only have to subtract 7 to 54. You can, however, subtract 20 to 64 and add 3 to the result. This way you don’t have to worry about borrowings.
Using complements to simplify subtractions even more
There is a way to easily calculate 3 or 4 digits subtractions very quickly in your head. This technique makes use of complements. For example. let’s say that you’re facing the following problem:Instead of following the standard left to right approach, you could solve this problem by subtracting 400 to 674 and then add 42 back to the result. 42 is the difference from 100 and 58. A good question is: how do you find 42?
Note that there’s a simple pattern for calculating the second number. In particular, the sum of the first digits always sum up to 9, and the sum of the second digits always sum up to 10. The only exception is when the number ends with 0, which is simpler.
You can use this technique to solve any subtraction very easily.
Multiplication
In order to solve simple multiplications, it’s helps a lot being comfortable with the multiplication table for numbers below 10.As you may have already guessed, we’re going to use the left to right approach to solve simple multiplication very easily. Take the following example:
We can reduce it by first calculating 30 × 7 (which is like 3 × 7 plus a 0) and then add 6 × 7 on the result.
This approach can be used for even larger numbers. Note that you can also round up instead of rounding down:
User contributions
the following are some math tricks contributed by the users.Multiply by 5
Contributed by Scott.To multiply 5 simply cut the # in half then multiply by 10.
eg. 17*5
1/2 of 17 = 8.5
8.5 * 10 = 85
Multiply numbers with multiple digits
Contributed by Tom PetersonUse this trick when multiplying numbers with multiple digits
let {a;b;c;d…} represent digits of a number
ab x cd = (axc), (axd + bxc), (bxd)
the commas represent separation of digits, so “axc” represents the digit in the hundreds place, etc.
eg) 23 × 14 = (2×1), (2×4 + 3×1), (3×4)
8 + 3
= 2,11,12
in the event of double digits in the same digit place, the number in the digit’s place (starting with the unit’s place) carries the ten’s place digit of the digit place to the following digit place [what a mouthful!]
like in this instance
= 2, 11, 12 = 2, 12, 2 = 3, 2, 2
the answer is 322
the theory behind this is the “distribution property” of numbers commonly used with equations like (x + 1)(x + 4)=0 to make x^2 + 5x + 4=0
the same principles can be applied with 3 digit numbers as well
abc x def = (axd),(axe+bxd),(axf+bxe+cxd),(bxf+cxe),(cxf)
for multiplying 2 digit with 3 digit numbers, just use the 3×3 digits method but use a zero in the hundreds place of the 2 digit number
Square a number close to 10^2
Vedic mathematics provides lots of short cuts like shown here.
e.g.-
If you need to square a number close to 10^n, you can do so easily. Like if you want 92^2, lets take its answer as abcd.
Now, 92 is 8 before 100, so subtract 8 from 92, i.e. you get ab as 84. For finding cd, square 8 i.e. 64.
Hence the square of 92 comes as 8464.
For square of 87, let the answer is abcd again. Here 87 is 13 short of 100, so subtract 13 from 87 You get 74 as ab. For finding cd, square 13 i.e. 169. Since cd is only of two digits, add this extra 1 to ab.
So the answer becomes 7569.
Square two digits ending with 5
To square 2 digit numbers ending with ‘5’ eg 75 × 75
1. The answer will end with ‘25’
2. Take the first digit ‘7’ multiply by the number after ‘7’ => 7 × 8 = 56
75 × 75 = 5625
Test it out with 95 × 95.
Did you get 8125?
Squaring any number
take any number and find out how much to add to get it to the nearest tens subract and add that number to the orignal number multiply add the square
example:
(999+1) (999-1) + (1^2)
(998) (1000) + 1
999^2 + 998001
Squaring a number
A math trick I noticed when I was young. If you are squaring a number it is always equal to the total of the number times 2 subtract one of the previous squared number. This is helpful if you dont want to write it out. For instance most people know that 10×10=100 or 11×11=121 even 12×12=144 so lets say you dont know 13×13. Its equal to (13×2)-1(plus the previous squared number which was 12×12)144=169
Squaring two digit numbers
Suppose AB is the number,
Then arrange the number as follows,
A*A|2*A*B|B*B
( if A*A or A*B is one digit add 0 prior to that – eg: 4 should be written as 04, 5 should be 05 etc..)
Take a number : 35
09|30|25 ( 3*3 | double of 3*5 | 5*5 )
From right to left, keep the right most number as it is and add the number coming both side of | symbol.
ie. Keep 5 as it is, add 2+0, add 3+9
1225
Take another example 43
16|24|09 = 1849
Want more tricks?
All these tricks I learned are from the fantastic book secrets of mental math. This is one of the few books (probably the only one) that I would carry with me all the time. It’s extremely cool to be able to perform mental calculations very quickly, and you can get around it without being a nerd.Here’s a list of what you can expect to learn from the book:
- Additions and subtractions.
- Basic and advanced multiplications.
- Divisions.
- Guessing a number (when it’s good enough).
- Pencil and paper math.
- How to memorize numbers.
- Many other tricks that will impress your friends.
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