# The Tier System – Multiply Any Number By Twelve

January 29th, 2017 by Math Tricks | Filed under Math Tricks.

By Ernesto Zuniga

First let me say that I am not an advanced mathematician. I find myself trying to see patterns in numbers and things around me. Call this the Tier System for multiplying any two digit number by 12 through 19. It is meant to be learned so that you can do fast calculations in your mind. Below you will see how to multiply any number by twelve. The steps outlined will apply to all tiers from 13 to 19 as well. This will make sense when you learn how to multiply any number by thirteen. So here are the steps do doing this. With practice doing this will be automatic. Share this with children who are learning how to multiply.

So here are the steps to doing this.

1. Determine what the last digit will be by multiplying the last digit of each number as in normal multiplication. Mentally set that number aside. If the product is a two digit number we are only concerned with the last digit of the product.  For example with 34 x 12 we multiply 4 by 2 and get 8. we mentally set the 8 to the side. If we multiply 78 by 12 we multiply 8 by 2 and get 16. We only mentally set aside the 6 and not the 1.
1. Multiply 2 by the first digit of the number being multiplied by 12. In the examples above 34 x 12 would multiply 2 by 3 and get six. And with 78 x 12 we multiply 2 by 7 and get 14 (we mentally keep thewhole number 14 unlike step 1 where we dropped the 1 from 16).
1. Add the product from step 2 to the number being multiplied by twelve. In the example of 34 x 12 we add 6 to 34 and get 40. And here is why I call this the tier system. If the number being multiplied by twelve ends with 5 or greater we add 1 to the product of step two. With 78 x 12 the 8 is greater than 5. Therefore we instead of adding 14 to 78 we add 15 and get 93. Can you add 78 and 15 in your mind?
1. Attach the result for Step 1 to the end of the sum from step three.

Let’s  try a couple.

51

We know it ends in 2.

Steps 2 and 3 give me a result of 61

Step 4 gives us 612.

51 x 12 = 612

Here’s another one.

95

We know it ends in 0.

Steps 2 and 3 gives us a result of 114

Step 4 gives us 1140

95 x 12 = 1140.

Below is a visual of the Tier System as it applies to multiplying two digit numbers by twelve.

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### One Response to “The Tier System – Multiply Any Number By Twelve”

I use something very similar with my students in Number Sense. When multiplying any 2 digit number by a number between 12-19, we only use the last digit of the number. For example, if we were to do 12 x 34. We will only focus on the digit “2”. I call it the “Multiply, add to the right” trick.

Example #1

12 x 34 =
Step 1: 2 x 4 = 8. This will be the last digit in the answer.

Step 2: 2 x 3 = 6. Add 4, and you get 10. The the “0” will be the next digit in the answer and the digit “1” will be added to 3 to get 4.

Example #2

13 x 37

3 x 7 = 21, “1” will be the last digit in the answer and “2” will carry over to the next step.

3 x 3 = 9, 9+7+2 = 18, “8” will be the next digit in the answer and “1” will carry over to the next step.

1 + 3 = 4. “4” is the final digit.