Mental Math Techniques
The following are most popular mental math techniques.
- Multiplying with numbers ending in zero
- Multiply any number with 11
- Multiply 5 with an even number
- Find the Percentage of a number
- Square of a two-digit number whose last digit is 5
- Adding large numbers
- Multiply with 4 or 8
- Multiply by splitting
- The Advanced Star technique
- Tests of divisibility
1. Multiplying with numbers ending in zero
It is so simple. Multiply the two numbers without their zeroes. Count the number of zeros in both the numbers. Add as many zeros at the end of the product.
70x40 = 7x4 with 2 zeros = 28 and 00 = 2800
150x200 = 15x2 and 000 = 30 and 000 = 30000
7000x900 = 7x9 and 5 zeroes = 63,00,000
2. Multiply any number with 11
Add the neighboring digits and put the result in the middle of those digits. Carry forward the tens, if any, to the left digit.
11x23 = 2,2+3,3 = 2,5,3 = 253
11x79 = 7,7+9,9 = 7,16,9 = 7+1,6,9 = 869
3. Multiply 5 with an even number
It works only for even numbers. Reduce the even number to half by dividing it by 2. Then add a zero
5x8 = 8/2,0 = 4,0 = 40
5x28 = 28/2,0 = 14,0 = 140
5x488 = 484/2,0 = 244,0 = 2440
4. Find the Percentage of a number
First of all, the basic concept to find 10% of a number is to shift one digit.
It is easy for a number ending with 0. Ignore the last digit.
4280 x 10% = 428.0 = 428
10% of 10000 is 1000.0 = 1000
10% 1 crore is = Ignore the last zero, so only 6 zeros are left = 1 follwed by 000000 = 10 lakh
If the last number is non-zero, the process is similar. Don’t ignore the last digit. Shift decimal by one digit to the left.
10% of 25 = 2.5
10% of 105 = 10.5
10% of 5377 = 537.7
To calculate 1%, we shift two places left. If there are no places on the left add extra zeroes
1% of 786 = 7.86
1% of 780 = 7.80
1% 54 = 0.54 (note 0 was added)
1% of 5 = 0.05 (note two zeroes were added)
20% or 5% of a number – Find 10% as above and multiply or divide by 2
3% of a number – Find 1% and multiply by 3
5. Square of a two-digit number whose last digit is 5
Last two digits will be 25. Find the initial digit(s) as follows. Take the first digit. Multiply it by the next higher number.
65x65 = 6x(6+1) followed by 25 = 6x7, 25 = 42,25 = 4225
95x95 = 9x10,25 = 9025
You can also do it for three-digit numbers, but multiplication will be a little difficult
115x115 = 11x12,25 = 132,25 =13225
6. Adding large numbers
Bring the two numbers to their nearest 10s. Add or reduce the differences.
716+277 = 720+280-4(770-766)-3(280-277) = 720+280-4-3
= 1000-7 = 993
994+243 = 990+240+4+3 = 1230+7 = 1237
There is another way to add large numbers. But it comes naturally only with practice. Start from the left side. Add hundred, then tens and ones.
716+277 = 700+200+10+70+6+7 = 900+80+13 = 993
7. Multiply with 4 or 8
Double the numbers two times or three times.
4x45 = 45x2x2 = 90x2 = 180
4x125 = 125x2x2 = 250x2 = 500
8x45 = 45x2x2x2 = 90x2x2 = 180x2 = 360
8. Multiply by splitting
Split the tens and ones. E.g. 3x87 = 3x(80+7) = 3 is multiplied with both numbers and then added. 3x80 + 3x7 = 240+21 = 261
If you have learned tables up to 20, then you can even multiply bigger numbers. E.g. 13x473 = 13x400+13x70+13x3 = 5200+910+39. Now here, you need to use other techniques of addition already taught to you. So It becomes 5200+900+30+9 = 6100+10+39 = 6149
9. The Advanced Star technique
To multiply two-digit numbers, use the technique of I, X, I, where X stands for a star.
Here last 'I' means the product of the last two digits. X means you have to cross-multiply and add, and first, 'I' means the product of the first two digits. If one of the numbers is a single-digit number, we should add a zero before that number. Remember to take any carryforwards to the left side.
E.g. 23x45 = the last 'I' = 3x5 = 15, so the last digit is five and carries forward one.
X = cross multiply and add, i.e. 2x5+3x4 = 10+12 =22. So the second last digit is 2 plus 1 brought forward from the previous step= 3. We carry forward its 2 to the next step
The first 'I' = 2 x 4 = 8 and add 2 brought forward from the last step= 10
So, we got 10,3,5. So the answer is 1035.
You can write the answer digit by digit so that you do not need to memorize. But all other calculations of multiplication and addition are to be done mentally.
We can extend the same technique to three digits and four-digit multiplications also. I use the star method easily for bigger calculations with similar mental math formulas. But it is not possible to explain those techniques here without a visual aid. E.g. for three digits, the formula is I, X, XI, X, I. You can try to figure out how it works after you master the two-digit technique.
10. Divisibility Rules
Divisibily by 1Every number is divisible by 1.
Divisibily by 2If a number is even then it is divisible by 2.
Divisibily by 3If sum of all the digits in a number is divisible by 3 then it is divisible by 3.
Divisibily by 4If the last two digits of a number are divisible by 4, then that number is a multiple of 4 and is divisible by 4 completely.
Divisibily by 5Any number ending with 0 or 5 is divisible by 5.
Divisibily by 6If a number is divisible by both 2 and 3 then it is divisible by 6.
Divisibily by 8If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.
Divisibily by 9If sum of all the digits in a number is divisible by 9 then it is divisible by 9.
Divisibily by 10Any number ending with 0 divisible by 10.
Divisibily by 11If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11.
The mental math techniques are handy for solving simple to the complex calculations of addition, subtraction and multiplication, and even division. If you develop an interest in these techniques, you should determine why they always give correct results.
You will appreciate that mathematics follows a set pattern. After identifying the set pattern in these methods, you can try to find new techniques or extend those techniques to more complex calculations involving three or four digits. It will sharpen your brain further.
Your habit of making mental calculations will go a long way in solving day to day mathematical problems quickly, which may look daunting to beginners.