How to Calculate Quickly Without a Calculator
It is well known, that the easier the method you use to solve a problem, the faster you will solve it with less chance of making a mistake. It doesn't have much to do with intelligence or having a "mathematical brain". The difference between high achievers and low achievers is the best strategies the first use. The methods given it this article will amaze you by their simplicity and clarity. Enjoy your new math skills!
Multiplication
Multiplying Numbers up to 10
You don't need to memorize the multiplication table, just use this way at any time!
We will start by learning how to multiply numbers up to 10. Let's look how it works:
We'll take 7 × 8 as an example.
Write this example down in your notebook and draw a circle below each number to be multiplied.
7 × 8 =
( ) ( )
Now go to the first number (7) to be multiplied. How many more do you need to make 10? The answer is 3. Write 3 in the circle below the 7. Now go to the 8. How many more to make 10? The answer is 2. Write this number in the circle below the 8.
It should look like this:
7 × 8 =
(3) (2)
Now you have to subtract diagonally. Take either one of the circled numbers (3 or 2) away from the number, not directly above, but diagonally above. In other words, you either take 3 from 8 or 2 from 7. You only subtract one time, so choose the subtraction you find easier. Either way, the answer will be the same 5. This is the first digit of your answer.
8 − 3 = 5 or 7 − 2 = 5
Now multiply the numbers in the circles. Three times 2 is 6. This is the last digit of your answer. The answer is 56.
Tip!
Reference Number  is the number we take our multipliers away from. Write it left of the problem. We then ask ourselves, are the numbers we are multiplying above or below the reference number.
Multiplying Numbers in the Teens
Let's see how to apply this method to multiplying numbers in the teens. We will use 10 as our reference number and the following example:
(10) 13 × 14=
Both 13 and 14 are above our reference number, 10, so we put the circles above the multipliers. How much above? 3 and 4. So we write 3 and 4 in the circles above 13 and 14. Thirteen equals 10 plus 3 so we write a plus sign in front of the 3; 14 is 10 plus 4 so we write a plus sign in front of the 4.
+(3) +(4)
(10) 13 × 14=
As in the previous example, we work diagonally. 13+4 or 14+3 is 17. Write this number after the equals sign. Multiply the 17 by the reference number 10 and get 170. This number is our subtotal, so write 170 after the equals sign.
In the last step, we should multiply the numbers in the circles. 3 × 4=12. Add 12 to 170 and we get our finished answer 182.
+(3) +(4)
(10) 13 × 14=170+12=182
Tip!
If the circled numbers are above we ADD diagonally, if the numbers are below we SUBTRACT diagonally.
Multiplying Numbers Greater Than 10
This method is also working in the case of large numbers.
96 × 97=
What do we take these numbers up to? How many more to make what? 100. So write 4 under 96 and 3 under 97.
96 × 97=
(4) (3)
Then subtract diagonally. 963 or 974 is 93. This is the first part of your answer. Now, multiply the numbers in the circles. 4 × 3=12. This is the last part of the answer. The finished answer is 9,312.
96 × 97=9,312
(4) (3)
This method is certainly easier than the method you learned in school! We believe that everything genial is simple, and maintaining simplicity is a hard work.
Multiplying Numbers Above 100
Here, method is the same. We would use 100 as our reference number.
(100) 106 × 104=
The multipliers are higher than the reference number 100. So we draw circles above 106 and 104. How much more than 100? 6 and 4. Write these numbers in the circles. They are positive (plus) numbers because 106 is 100 plus 6 and 104 is 100 plus 4.
+(6) +(4)
(100) 106 × 104=
Add diagonally. 106+4=110. Then, write 110 after the equals sign. Multiply 110 by the reference number 100. How do we multiply by 100? By adding two zeros to the end of the number. That makes our subtotal 11,000.
Now multiply the numbers in the circles 6 × 4=24. Add the result to 11,000 to get 11,024.
God used beautiful mathematics in creating the world.
— Paul DiracMultiplying Using Two Reference Numbers
Previous method for multiplication has worked well for numbers that are close to each other. When the numbers are not close, the method still works but the calculation become more difficult.
It's possible to multiply two numbers that are not close to each other by using two reference numbers.
8 × 27=
Eight is close to 10, so we will use 10 as our first reference number. 27 is close to 30, so we use 30 as our second reference number. From the two reference numbers, we choose the easiest number to multiply by. It is 10. This becomes our base reference number. The second reference number must be a multiple of the base reference number. 30 is 3 times the base reference number 10. Instead of using a circle, write the two reference numbers to the left of the problem in brackets.
(10 × 3) 8 × 27=
Both the numbers in the example are lower than their reference numbers, so draw the circles below.
How much are 8 and 27 lower than their reference numbers (remember the 3 represents 30)? 2 and 3. Write these numbers in the circles.
(10 × 3) 8 × 27=
(2) (3)
( )
Now multiply the 2 below the 8 by the multiplication factor 3 in the parentheses.
2 × 3=6
Write 6 in the bottom circle below the 2. Then take this bottom circled number 6, diagonally away from 27.
276=21
Multiply 21 by the base reference number 10.
21 × 10=210
210 is our subtotal. To get the last part of the answer, multiply two numbers in the top circles, 2 and 3, to get 6. Add 6 to our subtotal of 210 and get our finished answer of 216.
Multiplying Decimals
When we write prices, we use a decimal point to separate the dollars from the cents. For example, $1.25 represents one dollar, and 25 hundredths of a dollar. The first digit after the decimal point represents tenths of a dollar. The second digit after the decimal point represents hundredths of a dollar.
Multiplying decimals is no more complicated than multiplying any other numbers. Let's see an example:
1.3 × 1.4=
We write down the problem as it is, but ignore the decimal points.
+(3) +(4)
(10) 1.3 × 1.4=
Although we write 1.3 × 1.4, we treat the problem as:
13 × 14=
Ignore the decimal point in the calculation and say 13+4=17, 17 × 10= 170, 3 × 4=12, 170+12=182. Our work isn't finished yet, we have to place a decimal point in the answer. To find where we put the decimal point we look at the problem and count the number of digits after the decimal points, the 3 in 1.3 and the 4 in 1.4. Because there are two digits after the decimal points in the problem there must be two digits after the decimal point in the answer. We count two places backwards and put the decimal point between the 1 and the 8, leaving two digits after it. So, the answer is 1.82.
Let's try another problem.
9.6 × 97=
We write the problem down as it is, but call the numbers 96 and 97.
(100) 9.6 × 97=
(4) (3)
963=93
93 × 100 (reference number)=9,300
4 × 3=12
9300+12=9,312
Answer is 931.2
Calculating Square Roots
There is an easy method for calculating the exact answer for square roots. It involves a process called cross multiplication.
To cross multiply a single digit, you square it.
3²=3 × 3=9
If you have two digits in a number, you multiply them and double the answer. For example:
34=3 × 4=12
12 × 2=24
With three digits, multiply the first and third digits, double the answer, and add this to the square of the middle digit. For example, 345 cross multiplied is:
3 × 5=15
15 × 2=30
30+4²=46
Rule for cross multiplication of an even number of digits!
Multiply the first digit by the last digit, the second by the second last, the third by the third last and so on, until you have multiplied all of the digits. Add them together and double the total.
In practice, you would add them as you go and double your final answer.
Rule for cross multiplication of an odd number of digits!
Multiply the first digit by the last digit, the second by the second last, the third by the third last and so on, until you have multiplied all of the digits up to the middle digit. Add the answers and double the total. Then square the middle digit and add it to the total.
Using Cross Multiplication to Extract Square Roots.
For example:
√2,809=
Firstly, pair the digits back from the decimal. For clarity, we will use ♥ as a sign of separation of digit pairs. There will be one digit in the answer for each digit pair in the number.
√28♥09=
Secondly, estimate the square root of the first digit pair. The square root of 28 is 5 (5×5=25). So 5 is the first digit of the answer.
Double the first digit of the answer (2×5=10) and write it to the left of the number. This number will be our divisor. Write 5, the first digit of our answer, above the 8 in the first digit pair 28.
To find the second digit of the answer, square the first digit of your answer and subtract the answer from your first digit pair.
5²=25
2825=3
Three is our remainder. Carry the 3 remainder to the next digit of the number being squared. This gives us a new working number of 30.
Divide our new working number 30 by our divisor 10. This gives 3, the next digit of our answer. Ten divides evenly into 30, so there is no remainder to carry. Nine is our new working number.
(5) (3)
10 √28♥09=
25
Finally, cross multiply the last digit of the answer. We don't cross multiply the first digit of our answer. After the initial workings the first digit of the answer takes no further part in the calculation.
3²=9
Subtract this answer from our working number.
99=0
There is no remainder: 2,809 is a perfect square. The square root is 53.
10 √2,809=53
Squaring Numbers
It's hard to believe, but now squaring big numbers without a calculator is possible! Learn fast techniques of mental math below here that will help you to perform like a genious.
To square a number simply means to multiply it by yourself. A good way to visualize this is, if you have a square brick section in your garden and you want to know the total number of bricks making up the square, you count the bricks on one side and multiply the number by itself to get the answer.
13² = 13 × 13 =169
We can easily calculate this using some methods for multiplying numbers in the teens. In fact, the method of multiplication with circles is easy to apply to square numbers, because it is easiest to use when the numbers are close to each other. In fact, all of the strategies taught here make use of the general strategy for multiplication.
Method of Using a Reference Number
(10) 7×8 =
The 10 to the left of the problem is our reference number. It is a number we take our multipliers away from.
Write the reference number to the left of the problem and then ask yourself, are the numbers you are multiplying above (higher than) or below (lower than) the reference number? In this case the answer is lower (below) each time. So we put the circles below the multipliers. How much below? 3 and 2. We write 3 and 2 in the circles. Seven is 10 minus 3, so we put a minus sign in front of the 3. Eight is 10 minus 2, so we put a minus sign in front of the 2.
(10) 7×8 =
(3) (2)
We now work diagonally. Seven minus 2 or 8 minus 3 is 5. We write 5 after the equals sign. Now, multiply the 5 by the reference number, 10. Five times 10 is 50, so write a 0 after the 5. (To multiply any number by 10 we affix a zero.) 50 is our subtotal.
Now multiply the numbers in the circles. Three times 2 is 6. Add this to the subtotal of 50 for the final answer of 56.
(10) 7×8 = 50
(3) (2) +6
__
56.
Tip!
If the circled numbers are ABOVE we ADD diagonally, if the numbers are BELOW we SUBTRACT diagonally.
Squaring Numbers Ending in 5
The method for squaring numbers ending in 5 uses the same formula we have used for general multiplication. If you have to square a number ending in 5, separate the final 5 from the digit or digits that come before it. Add 1 to the number in front of the 5, then multiply these two numbers together. Write 25 at the end of the answer and the calculation is complete.
For example:
35² =
Separate the 5 from the digits in front. In this case there is only a 3 in front of the 5. Add 1 to the 3 to get 4:
3+1 = 4
Multiply these numbers together:
3×4 = 12
Write 25 (5 squared) after the 12 for our answer of 1,225.
35² = 1,225
Let's try another:
We can combine methods to get even more impressive answers.
135² =
Separate the 13 from the 5. Add 1 to 13 to get 14.
13×14 = 182
Write 25 at the end of 182 for our answer of 18,225. This can easily be calculated in your head.
135² = 18,225
One more example:
965² =
96+1 = 97
Multiply 96 by 97, which gives us 9,312. Now write 25 at the end for our answer of 931,225.
965² = 931,225
That is impressive, isn't it?
This shortcut also applies to numbers with decimals! For instance, with 6,5×6,5 you would ignore the decimal and place it at the end of the calculation.
6,5² =
65² = 4,225
There are two digits after the decimal when the problem is written in full, so there would be two digits after the decimal in the answer. Hence, the answer is 42.25.
6.5² = 42.25
It would also work for 6.5×65 = 422.5
Likewise, if you have to multiply 3 ½ × 3 ½ = 12¼.
There are many applications for this shortcut.
Squaring Numbers Near 50
The method for squaring numbers near 50 uses the same formula as for general multiplication but, again, there is an easy shortcut.
For example:
46² =
46² means 46×46. Rounding upwards, 50×50=2,500. We take 50 and 2,500 as our reference points.
46 is below 50 so we draw a circle below.
(50) 46² =
(4)
46 is 4 less than 50, so we write a 4 in the circle. It is a minus number.
We take 4 from the number of hundreds in 2,500.
254 = 21
That is the number of hundreds in the answer. Our subtotal is 2,100. To get the rest of the answer, we square the number in the circle.
4² = 16
2,100+16 = 2,116. This is the answer.
Here's another example:
56² =
56 is more than 50 so draw the circle above.
+(6)
(50) 56² =
We add 6 to the number of hundreds in 2,500.
25+6=31. Our subtotal is 3,100.
6² = 36
3,100+36 = 3,136. This is the answer.
Let's try one more:
62² =
(12)
(50) 62² =
25+12 = 37 (our subtotal is 3,700)
12² = 144
3,700+144 = 3,844. This is the answer.
With a little practice, you should be able to call the answer out without a pause.
Squaring Numbers Near 500
This is similar to our strategy for squaring numbers near 50.
500×500=250,000. We take 500 and 250,000 as our reference points. For example:
506² =
506 is greater than 500, so we draw the circle above. We write 6 in the circle.
+(6)
(500) 506² =
500² = 250,000
The number in the circle above is added to the thousands.
250+6 = 256 thousand
Square the number in the circle:
6² = 36
256,000+36 = 256,036. This is the answer.
Another example is:
512² =
+(12)
(500) 512² =
250+12=262
Subtotal=262,000
12²=144
262,000+144 = 262,144. This is the answer.
To square numbers just below 500, use the following strategy.
We'll take an example:
488² =
488 is below 500 so we draw the circle below. 488 is 12 less than 500 so we write 12 in the circle.
(500) 488² =
(12)
Two hundred and fifty thousand minus 12 thousand is 238 thousand. Plus 12 squared (12² = 144).
238,000+144 = 238,144. This is the answer.
We can make it even more impressive.
For example:
535² =
(35)
(500) 535² =
250,000+35,000 = 285,000
35² = 1,225
285,000+1,225 = 286,225. This is the answer.
This is easily calculated in your head. We used two shortcuts  the method for squaring numbers near 500 and the strategy for squaring numbers ending in 5.
What about 635² ?
(135)
(500) 635² =
250,000+135,000 = 385,000
135² = 18,225
To find 135² we use our shortcut for numbers ending in 5 and for multiplying numbers in the teens (13+1=14; 13×14=182). Put 25 on the end for 135² = 18,225.
We say, "Eighteen thousand, two two five."
To add 18,000, we add 20 and subtract 2:
385+20 = 405
4052 = 403
Add 225 to the end.
The answer is 403,225.
Numbers Ending in 1
This shortcut works well for squaring any number ending in 1. If you multiply the numbers the traditional way you will see why this works.
For example:
31² =
Firstly, subtract 1 from the number. The number now ends in zero and should be easy to square.
30² = 900 (3×3×10×10)
This is our subtotal.
Secondly, add together 30 and 31  the number we squared plus the number we want to square.
30+31 = 61
Add this to our subtotal, 900, to get 961.
900+61 = 961. This is the answer.
For the second step you can simply double the number we squared, 30×2, and then add 1.
Another example:
121² =
1211 = 120
120² = 14,400 (12×12×10×10)
120+121 = 241
14,400+241 = 14,641. This is the answer.
Let's try another:
351² =
350² = 122,500 (use shortcut for squaring numbers ending in 5)
350+351=701
122,500+701 = 123,201. This is the answer.
One more example:
86² =
We can also use the method for squaring numbers ending in 1 for those ending in 6. For instance, let's calculate 86². We treat the problem as being 1 more than 85.
85² = 7,225
85+86 = 171
7,225+171 = 7,396. This is the answer.
Numbers Ending in 9
An example is:
29² =
Firstly, add 1 to the number. The number now ends in zero and is easy to square.
30² = 900 (3×3×10×10)
This is our subtotal. Now add 30 plus 29 (the number we squared plus the number we want to square):
30+29=59
Subtract 59 from 900 to get the answer of 841. (I would double 30 to get 60, subtract 60 from 900, and then add the 1.)
90059 = 841. This is the answer.
Let's try another:
119² =
119+1=120
120²=14,400 (12×12×10×10)
120+119=239
14,400239=14,161
14,400240+1 = 14,161. This is the answer.
Another example is:
349² =
350²=122,500 (use shortcut for squaring numbers ending in 5)
350+349=699
(Subtract 1,000, then add 301 to get the answer.)
122,500699 = 121,801. This is the answer.
How would we calculate 84 squared?
We can also use this method for squaring numbers ending in 9 for those ending in 4. We treat the problem as being 1 less than 85.
84² =
85²=7,225
85+84=169
Now subtract 169 from 7,225:
7,225169=7,056. This is the answer.
(Subtract 200, then add 31 to get your answer.)
Practice these in your head until you can do them without effort.
Squares
Number (X)
 Square (X²)


1
 1

2
 4

3
 9

4
 16

5
 25

6
 36

7
 49

8
 64

9
 81

10
 100

11
 121

12
 144

13
 169

14
 196

15
 225

16
 256

17
 289

18
 324

19
 361

21
 441

22
 484

23
 529

24
 576

25
 625

30
 900

Mental calculation can help you improve concentration, develops memory and enhances the ability to retain several ideas at once. This skill boosts your confidence, selfesteem and make you believe in your intelligence.
Mathematics affects our everyday life. There are many practical uses of mental calculation. We all need to be able to make quick calculations.
Methods discussed here are easier than those you have learned in the past so you will solve problems more quickly and make fewer mistakes. People who use better methods are faster at getting the answer and make fewer mistakes, while those who use poor methods are slower at getting the answer and make more mistakes. It doesn't have much to do with intelligence or having a "mathematical brain".
Have these techniques of mental math helped you?
Sync up Left and Right Hemispheres of Your Brain to Think Innovatively!
© 2018 Rada Heger
Comments
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