What number is 100 less than 563
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Do you use the calculator for every invoice? It doesn't have to be. With our tips for mental arithmetic, you can multiply multi-digit numbers yourself and add and subtract fractions.
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Mental arithmetic: what is it and what do you need it for?
Mental arithmetic is the name given to solving a math problem without aids such as pen, paper or a pocket calculator. If you can manage a task with just a few notes, this is called half-written arithmetic.
Mental arithmetic is a central part of math lessons in elementary school. The pupils learn the one plus one and the multiplication tables there. You may be wondering what that should be good for (Finally there are calculators). But to learn mental arithmetic helps you throughout your school days and in later life.
The advantages of mental arithmetic
Those who are good at mental arithmetic not only have it easier in the supermarket, with baking recipes and tax returns. If you have mastered arithmetic without aids, then you also understand more difficult math problems more easily and can follow better in class.
Because mental arithmetic
- trains the brain
- helps to assess the plausibility of results
- makes it easier to solve more complex math problems
- and helps to develop a feeling for numbers.
Many of the arithmetic rules listed here belong to "Vedic mathematics". It is not clear whether the term refers to “Veda”, that is, to texts of Hinduism. The fact is, however, that with the rules of Vedic mathematics can be calculated very quickly and easily can.
Written arithmetic or quick mental arithmetic?
The arithmetic methods that you learn in school are very effective for written arithmetic, but unfortunately sometimes too complicated for fast mental arithmetic. The "Vedic calculation method" however, you should only use it in your head. Keep using the arithmetic methods you learned in school for your written math problems. Especially in math work is next to the result also the written calculation method graded.
Mental arithmetic: tricks for multiplying multi-digit numbers
Rule 1: If the first digits are the same and the last two digits add up to 10
With such a calculation, the Vedic rule “one more than the one before” comes into play.
And here oneStep-by-step instructions:
47 x 43 =?
|Step 1||Add the 1st digit of the first multi-digit number by 1 and multiply the result by the 1st digit of the second multi-digit number.||47 x 43|
(4 + 1) x 4 = 20 (1st partial result)
|2nd step||Multiply the two last digits together||47 x 43|
3 x 7 = 21 (2nd partial result)
|3rd step||Put the partial results together||20 and 21 put together results in 2021|
|solution||47 x 43 = 2021|
Rule 2: When the numbers are close to a tens base
If you want to multiply multi-digit numbers that are close to one Power of ten (i.e. 10, 100, 1000 etc.), the Vedic rule “vertically and crosswise” comes into play. Here you have to pay attention to whether the numbers you want to multiply, under (2a) or above (2b) a power of ten.
2a) If both numbers are less than a power of ten
|sequence||method||Example 95 x 82 =?|
|Step 1||Find the difference between the base of ten and the two numbers (e.g. 10 minus 6 or 100 minus 77).||100 – 95 = 5 and 100 - 82 = 18|
|2nd step||Cross-subtract the differences from the numbers (1st partial result).||95 – 18 = 77 and 82 - 5 = 77|
|3rd step||Multiply the differences together (2nd partial result).||5 x 18 = 90|
|4th step||Put the partial results together||77 and 90 become 7.790|
|solution||95 x 82 = 7,790|
Danger: If that 2. Partial result has more than two digits for an invoice with 100 and more than three digits for an invoice with 100, you have to transfer add to the 1st partial result!
2nd example (near 1000): 998 x 899 =?
- 1000 – 998 = 2 and 1000 - 899 = 101
- 998 - 101 = 897 and 899 - 2 = 897 (1st partial result)
- 2 x 101 = 202 (2nd partial result)
- Solution: 998 x 899 = 897.202
3rd example (with carryover): 95 x 74 =?
- 100 – 95 = 5 and 100 - 74 = 26
- 95 - 26 = 69 and 74 - 5 = 69 (1st partial result)
- 5 x 26 = 130< (2.="">
- Solution with carryover: 7030 (Carryover: 69 +1 = 70)
2b) When both numbers are above a power of ten
|sequence||method||Example 106 x 110 =?|
|Step 1||Find the difference between the base of ten and the two numbers.||106 – 100 = 6 and 110-100 = 10|
|2nd step||Add the differences from the numbers crosswise (1st partial result).||106 + 10 = 116 and 110 + 6 = 116 (1st partial result)|
|3rd step||Multiply the differences together (2nd partial result).||6 x 10 = 60 (2nd partial result)|
|4th step||Put the partial results together||116 and 60 become 11.660|
|solution||106 x 110 = 11,660|
3. If they are any two-digit numbers
Here, too, the Vedic rule “vertically and crosswise” applies.
|sequence||method||Example 64 x 32 =?|
|Step 1||Multiply the 1st digit of the two numbers (1st partial result).||6 x 3 = 18 (1st partial result)|
|2nd step||Multiply the 1st digit of the 1st number by the 2nd digit of the 2nd number and the 2nd digit of the 1st number by the 1st digit of the 2nd number.||6 x 2 = 12 and 4 x 3 = 12|
|3rd step||Add up both results (2nd partial result).||12 + 12 = 24 (2nd partial result)|
|4th step||Multiply the 2nd digit of both numbers together (3rd partial result).||4 x 2 = 8 (3rd partial result)|
|solution||64 x 32 = 2048|
Mental arithmetic: tricks for fractions
If you want to add or subtract fractions quickly, you can also use the “vertically and crosswise” rule.
|sequence||method||Example ¾ + ⅕ =?|
|Step 1||Multiply the numerator (upper number) of the 1st fraction by the denominator (lower number) of the 2nd fraction and the numerator of the 2nd fraction by the denominator of the 1st fraction.||3 x 5 = 15 and 4 x 1 = 4|
|2nd step||Add both results together (1st partial result = numerator).||15 + 4 = 19|
|3rd step||Multiply the two denominators together (2nd partial result = denominator).||4 x 5 = 20|
|solution||¾ + ⅕ = 19/20|
|sequence||method||Example ¾ - ⅕ =?|
|Step 1||Multiply the numerator of the 1st fraction by the denominator of the 2nd and the numerator of the 2nd fraction by the denominator of the 1st.||3 x 5 = 15 and 4 x 1 = 4|
|2nd step||Subtract both results together (1st partial result = numerator).||15 – 4 = 11|
|3rd step||Multiply the two denominators together (2nd partial result = denominator).||4 x 5 = 20|
|solution||¾ – ⅕ = 11/20|
Add up in your head
In addition to Vedic mathematics, there is also other mental arithmetic trickswith which you can add, subtract or multiply quickly and easily without the help of a calculator. When adding up in your head, it helps if you split the bill.
Example: 117 + 242 =?
- Add 7 + 2 = 9
- 11 + 24 = 35
- The solution: 117 + 242 = 359
Multiply by 11
Multiplying by 10 is fun as it is very easy. Just put a zero at the end of the multiplicand (i.e. the number that should be multiplied). But multiplying by 11 also works very well for all two-digit numbers without a calculator. The trick for this task: Add the two digits of the multiplicand and put it in the middle of the starting number.
An example: 43 x 11 =?
- 4 + 3 = 7
- Put the 7 in the middle of 43
- The solution: 43 x 11 = 473
The Trachtenberg method
The Trachtenberg method was invented in the 1940s by the Russian engineer Jakow Trachtenberg and enables fast mental arithmetic, or at least at least half-written arithmetic, even with large numbers. However, you have to remember a separate rule for each multiplier. We provide you with the calculation rules here as an example for 5, 6 and 11 in front.
Multiply by 5 using the Trachtenberg method
Take from each digit of the starting number respectively the right neighbor and cut it in half. Think of a zero to the right and left of the starting number, so z. B. 872 you think of as 08720. If the starting number is odd, always add the number five.
An example: 872 x 5 =?
- Neighbor of 2 is 0; 0: 2 = 0
- Neighbor of 7 is 2; 2: 2 = 1 but + 5 (because the 7 is odd) = 6
- The neighbor of 8 is 7; 7: 2 = 3.5 ≈ 3 (half numbers are rounded down)
- Neighbor of 0 is 8; 8: 2 = 4
- The solution: 872 x 5 = 4360
Multiply by 6 using the Trachtenberg method
The rule here is: Take the starting digit and add half of the right neighbor. Here, too, if the starting number is odd, you still have to add the number five.
Example: 741 x 6 =?
- 1 + ½ neighbor (0: 2 = 0) = 1 + 0 + 5 (because 1 is odd) = 6
- 4 + ½ neighbor (1: 2 = 0.5≈0) = 4 + 0 = 4
- 7 + ½ neighbor (4: 2 = 2) = 7 + 2 = 9; 9 + 5 (7 is odd) = 14 (Carry forward, because 14 is a two-digit number! That means the one is added to the next higher digit)
- 0 + ½ neighbor (7: 2 = 3.5≈3) = 0 + 3 = 3
- The solution: 4.446
Multiply three-digit numbers by 11
If you want to multiply a three-digit number by 11, you can also use a rule from the Trachtenberg method. It is: Add each digit of the multiplicand to its right neighbor.
An example: 542 x 11 =?
- 2 + 0 = 2
- 4 + 2 = 6
- 5 + 4 = 9
- 0 + 5 = 5
- The solution: 542 x 11 = 5962
Mental arithmetic: practice makes perfect
At first, this may all sound complicated to you. But with a little practice, you will soon be able to multiply multi-digit numbers and fractions while you sleep. Try our quiz right now.
Quiz Maker - a service from Riddle
Cover picture: © Evgeny Atamanenko /Shutterstock.com
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