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!
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 =
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.
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.
(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.
(10) 13 × 14=170+12=182
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=
Then subtract diagonally. 96-3 or 97-4 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
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.
(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 Dirac
Multiplying 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=
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.
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.
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.
(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=
93 × 100 (reference number)=9,300
4 × 3=12
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
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.
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.
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.
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.
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.
Subtract this answer from our working number.
There is no remainder: 2,809 is a perfect square. The square root is 53.
Mental calculation can help you improve concentration, develops memory and enhances the ability to retain several ideas at once. This skill boosts your confidence, self-esteem 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.
Have these techniques of mental math helped you?
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