Mastering Mental Math: Methods in Trachtenberg Speed System

Calculate at the Speed of Thought with Trachtenberg Mastery!

The Trachtenberg Speed System is a mental math system developed by Jakow Trachtenberg to perform calculations quickly and mentally. It comprises several methods or techniques designed to simplify arithmetic operations. The system includes techniques for addition, subtraction, multiplication, and division. Here's an overview of some of the key methods within the Trachtenberg Speed System:

• Multiplication
  

1. Multiplication by 11: This method involves adding the digits of the number and placing the sum in the middle while carrying over any tens. For example, to multiply 23 by 11, add 2 + 3 = 5, and the result is 253.
   

2. Multiplication by 5: For multiplying a number by 5, one can simply divide the number by 2 and then multiply by 10. For instance, 36 multiplied by 5 is (36 ÷ 2) × 10 = 18 × 10 = 180.
   

3. Multiplication by 9: To multiply a number by 9, you subtract 1 from the number and then subtract the result from 9. For example, 7 multiplied by 9 is (7 - 1) = 6, 9 - (7 -1 ie. 6) = 3 so the result is 63.
   

4. Multiplication by 4: This method involves doubling the number twice (which is equivalent to multiplying by 4). For example, 9 multiplied by 4 is (9 × 2) × 2 = 18 × 2 = 36. or, 9 + 9 = 18, 18 + 18 = 36.
   

5. Multiplication by 25: To multiply a number by 25, divide the number by 4 and then multiply by 100. For instance, 16 multiplied by 25 is (16 ÷ 4) × 100 = 4 × 100 = 400.
   

6. Multiplication by 99: This method involves subtracting the number from 100 and then subtracting the result from the original number. For example, 63 multiplied by 99 is (100 - 63 ie. 37, so The tens and units digits of the result are 37), (63 - 37= 26, so The difference, 26, will be the hundreds and thousands digits of the final result), Combine these results and the final result is 6237.

• Addition

1. Left-to-right addition: Instead of adding digits column by column, you add from left to right. This method helps in reducing errors and makes mental addition faster. For example, to add 348 + 275, you would add 300 + 200 = 500, then 40 + 70 = 110, and finally 8 + 5 = 13, giving you the result of 500 + 110 + 13 = 623.
   

2. Adding multiples of 10: When adding numbers that end with zeros (multiples of 10), you can simply ignore the zeros and add the remaining digits. For example, to add 370 + 430, you would add 37 + 43 = 80 and then add the zeros back to get 800.
   

• Subtraction
  

1. Left-to-right subtraction: Similar to left-to-right addition, this method involves subtracting digits from left to right. For example, to subtract 892 - 346, you would start with 800 - 300 = 500, then 90 - 40 = 50, and finally 2 - 6 (borrowing 1 from the 90) = 6 (after borrowing, it becomes 80 - 60 = 20), giving you the result of 500 + 50 + 20 = 570.
   

2. Subtracting multiples of 10: When subtracting numbers that end with zeros (multiples of 10), you can simply ignore the zeros and subtract the remaining digits. For example, to subtract 750 - 240, you would subtract 75 - 24 = 51 and then add the zeros back to get 510.
   

• Division

1. Division by 5: To divide a number by 5, you can multiply the number by 2 and then move the decimal point one place to the left. For example, 45 divided by 5 is (45 × 2) = 90, and moving the decimal point gives 9. Another example: to divide 180 by 5, you would multiply 180 by 2 to get 360 and then move the decimal point to get 36.
   

2. Division by 9: One technique for division by 9 involves writing the quotient as the tens digit and the remainder as the ones digit. For example, to divide 72 by 9, the quotient would be 8 and the remainder would be 0, giving you 8 as the result.
   

These are some of the techniques used in the Trachtenberg Speed System for addition, subtraction, and division. They are designed to simplify mental calculations and make arithmetic operations faster and more efficient.