One of the famous tricks that you have probably known is how to multiply numbers by 11. If we multiply a 2-digit number by 11, we copy the ones digit to the ones digit of the product, add the ones and the tens digit and place the sum in the tens digit of the product, and then copy the tens digit to the hundreds digit of the product. For example, if we multiply 34 ×11.
(1) We copy 4 and place it in the ones digit of the product.
(2) Add 3 and 4 = 7 and place it in the tens digit of the product.
(3) Copy 3 and place it in the hundreds digit of the product.
Therefore, 34 ×11 = 374.
In case the sum is more than 10, we carry over to the hundreds digit of the product. For example 87 ×11 gives a product of the following digits.
ones digit: 7
tens digit: 8 + 7 = 15
We carry 1 to the hundreds digit.
hundreds digit: 8 + 1 = 9
We can generalize this to numbers with larger number of digits. For example, in multiplying 251 by 11, we can copy the first and last digits and add the adjacent digits. Below are the digits of the product.
ones digit: 1
tens digit: 5 + 1 = 6
hundreds digit: 2 + 5 = 7
thousands digit: 2
So, the final answer is 2761.
Why does the method work?
We can see why the method works by examining the standard algorithm of multiplication. Multiplying numbers by 11 is the same as multiplying it by 10 and then adding to the number. If we have a number abcd where a, b, c, and d are digits, for example, and we multiply it by 11, the we can multiply it by 10 and then add abcd.
That is abcd× 10 = abcd0
And the placement of the numbers will look like the one below.
Notice that the ones digit d and the digit with the highest place value a are copied in the sum. Then the other digits are the sum of the adjacent digits. In the case above, the digits x = a + b, y = b + c, and z = c + d.
This shortcut algorithm will work for all positive numbers. This is because multiplying any number by 10 will move all the numbers one place value to the left as 0 is added on the right. Adding the original number will result to the alignment as shown above.
As a seasoned mathematics enthusiast with a deep understanding of arithmetic concepts, particularly in multiplication, I'll delve into the fascinating trick of multiplying numbers by 11 and elucidate the underlying principles that make this method effective.
The method involves multiplying a 2-digit number by 11, where the ones digit is copied to the ones digit of the product, the sum of the ones and tens digit is placed in the tens digit of the product, and the tens digit is copied to the hundreds digit of the product. For instance, multiplying 34 by 11 proceeds as follows: (1) Copy 4 to the ones digit, (2) Add 3 and 4 to get 7, placing it in the tens digit, and (3) Copy 3 to the hundreds digit, resulting in 34 × 11 = 374.
This technique extends to cases where the sum exceeds 10, requiring a carryover to the hundreds digit. An example is multiplying 87 by 11, yielding a product with the following digits: ones digit (7), tens digit (8 + 7 = 15, carrying 1 to the hundreds digit), and hundreds digit (8 + 1 = 9).
Generalizing to numbers with a larger number of digits, like multiplying 251 by 11, involves copying the first and last digits and adding the adjacent digits. The digits of the product are then distributed accordingly: ones digit (1), tens digit (5 + 1 = 6), hundreds digit (2 + 5 = 7), and thousands digit (2), yielding the final answer of 2761.
The effectiveness of this method can be understood by examining the standard multiplication algorithm. Multiplying a number by 11 is akin to multiplying it by 10 and then adding the original number. If a number, abcd, is multiplied by 11, the result can be expressed as abcd × 10 = abcd0. Aligning the numbers in the standard multiplication grid, it's evident that the ones digit (d) and the digit with the highest place value (a) are copied, while the other digits are sums of adjacent digits (x = a + b, y = b + c, z = c + d).
This shortcut algorithm works for all positive numbers because multiplying any number by 10 shifts all digits one place value to the left, and adding the original number aligns them accordingly. The method provides a quick and efficient way to mentally calculate the product of numbers multiplied by 11.
