In the previous post, we learned a cool math trick on squaring numbers (positive integers) ending in 5. We did not only learn the math trick itself, but we also discussed why it works. In this post, we will explore another math trick which is multiplying 2 digit numbers by 11.
The Copy-Add-Copy Method
To multiply 2 digit numbers by 11, we will use the copy-add-copy method. We will copy and add the digits of the number multiplied by 11. Below are the steps of this method.
Steps in Multiplying 2-Digit Numbers by 11
- Copy the ones digit of the number multiplied by 11 to the ones digit of the product.
- Add the ones digit and the tens digit of the number and copy the sum (see *) to the tens digit of the product.
- Copy the tens digit of the number (see **) to the hundreds digit of the product.
* copy the ones digit of the sum if it is greater than 10
** add 1 if the sum in step 2 is greater than 10
Why the Math Trick Works
If we let mn be the number multiplied by 11, then, in the math trick, the digits of the product will have the following properties.
Ones Digit: n
(a) m + n if sum < 10
(b) m + n – 10 if sum >= 10
Hundreds Digit: m if (a) and m + 1 if (b)
Notice that we add 1 (in m + 1 ,he hundreds digit of the product) if the sum of the ones and tens digit is greater than or equal to 10. This is because we carry over 10 to the hundreds digit. Now let us examine what happens in the multiplication.
What Really Happened
Any two digit number can be written as 10t + u, where t is its tens digit and u is the ones digit. The number 25 for instance can be written as 10(2) + 5, while 83 can be written as 10(8) + 3. Therefore, the generalized number above can be written as 10m + n. Now, the multiplier 11 can be written as 10 + 1. Multiplying the two numbers, we have the following.
Notice that the ones digit of the product is the same as the ones digit of mn. Also, the tens digit is the sum of m + n and the hundreds digit is the m. Therefore, the copy-add-copy method above works.