Multiplication: Base Method 1

Hello. Today I am going to teach a new two way trick on how to do the Base Method. The Base Method is a way to use bases to multiply numbers. Some people prefer the Rainbow Method over the Base Method, but that is up to you.

Example: 98 x 95

Step 1:  First, find the closest base of the two factors that is higher than the two factors. A base is a number that is either 10 or a power of 10. This includes 100, 1000, 10000, etc because they are all powers of 10. In this case, the closest base of the two factors that is higher than the two factors is 100.

Step 2: Pick one of the factors in the problem. Figure out that factor minus the base you chose in the previous step. Place that number on top of the factor you chose in the beginning of the step.

                                                                                             -5

                                                                     Example: 98 x 95

As you can see, 95 (the factor I chose) - 100 (the base) =-5. Therefore I put that number on top of 97.

Step 3: Repeat Step 2 for the other factor in the problem.

                                                                                     -2       -5

                                                                     Example: 98 x 95

Step 4: Now do something I like to call "cross-adding". It's when you pick any number and add it to whatever is diagonal from it. For example, if you choose 95, you would add -2, because -2 is diagonal of 95, as shown below:

                                                                                    -2     -5

                                                                     Example: 98 x 95

95 is diagonal of -2. 95 +-2 = 93.

That sum of those 2 numbers is the first 2 digits of your answer.

                                                                 

                                                                  -2     -5

                                                  Example: 98 x 95                Answer: 93

Step 5: Pay attention to your base. Count the number of zeros your base has. This tells you how many digits your answer has left.

                                                      Base: 100.                     Answer: 93_ _

Step 6: To figure out the last 2 digits of your answer. Simply multiply the 2 numbers at the top.

                                                                                     -2     -5

                                                                    Example: 98 x 95   

                                                                        -5 x -2 = 10

                                                                     Answer:       9310

The answer is 9310. Hope this helped! If you have any questions, please post a comment below about it. Thanks!

             

Multiplication: Rainbow Method

 Hello! Today I will teach you how to multiply quicker than you already can with a method known as the Rainbow Method. This is a shortcut on how to multiply any 2 digit number with any other two digit number.

Example: 23 x 55

Step 1: To calculate the first two digits of your answer, multiply the first digit of the first factor to the first digit of the second factor. Then leave a space after the answer.

23 x 55     Answer: 10_

Step 2: To calculate the last digit of your answer, multiply the last digit of the first factor to the last digit of the second factor. If its a two digit number, carry over the tens place digit to the space before.

                                                                                       1

23 x 55        Answer: 10_5

Step 3: Now, to calculate the final digit of the answer (the space) you will use the Rainbow Method. Simply multiply the first digit of the first factor to the last digit of the last factor, and add that to the product of the last digit of the first factor to the first digit of the last factor, as shown below:

                                              

This shows that you are adding "2 x 5"(as shown in the outer rainbow line on the picture) to "3 x 5"(as shown in the inner rainbow line on the picture). The sum is 25. This is what you put in the space in the answer. Since it is a 2 digit number, carry over the tens place digit of this number to the place before in the answer.

                                                                                      21

23 x 55          Answer: 1055 

Step 4: Now do your finishing touches on your answer by adding the digits you carried over to the digits below them, as shown below.

                                                                                                                   21

23 x 55          Answer: 1055 

                                                                        

                                                                          True Answer: 1265

The answer is therefore 1265. I hope you've learned how to use this method! If you have any questions, please post a comment below about it. Thanks!

Division: Dividing Fast!

 

This video will teach you how to divide any number by 9.

Multiplication: "Odd" Theory

Here's an interesting theory I expanded on. I hope you like it! In order to do this trick however, you need to know your perfect squares.

If you want to know the answer to a multiplication problem such as:

                                                            11 x 19
This is how you do it with my theory:

Step 1: Identify the factors. In this case, it would be 11 and 19.
Step 2: Find the average of the factors. In this case, it would be 15.
Step 3: Calculate that number squared:


                                                            15 x 15= 225
Step 4: Increase one of the of the factors in this problem and decrease the other factor by 1. Then calculate it. It should look like this:


                                                            16 x 14 = 224

Note: If you noticed, the product of the first problem happens to be higher than the product of the second problem by 1. 225 - 224 = 1!

Step 5: Repeat Step 4. It should look like this:


                                                            17 x 13 = 221


Note: If you noticed, the product of the second problem happens to be higher than the product of the third problem by 3. 224 - 221 = 3!


Do you notice a pattern in the products? Let me show you: 

As you can see, each time you increase one of the factors by one and decrease the other by 1. The difference between the product of these 2 numbers and the product of the previous two numbers, increases one odd number higher. Using this theory, you will be able to widen your range in number of math facts!

11 x 19 = 209

Hope this helped!

Addition: No Carrying!

How you ever thought in addition: "I don't want to carry over. Its so hard!" Well, I have a solution for that! Its called the no-carrying trick, and I'm to guide you step-by-step in the process on how to do it! Let me show you an example:

   129

+ 423

Now for those of you, who can do can't do mental math, you're all probably wondering: "How can I do this problem without carrying over?" So let me show you the step by step process:

Step 1: Look at the place values farthest on the left. In this case, it would be 4 and 1. 
Step 2: Find their true values. For example, since 4 and 1 are in the hundred's place, their true values are 400 and 100.
Step 3: Find out the sum of the 2 numbers. Then put the answer below. Make sure its aligned correctly:

                                                                                 129

                                                                            +   423

                                                                                 500

Step 4: Now move over one place value to the right (In this case, since we started with the hundreds place, moving one place value to the right will get us to the tens place).

Step 5: Look at the place value you are on now. Repeat Step 2 and Step 3. Make sure its aligned correctly:

                                                                               129

                                                                           +  423

                                                                               500

                                                                                 40


Step 6: Repeat Step 4 and Step 5 for the ones place (in this example). You will notice that the sum for the ones digits place is 12. So how can you fit it in? The answer is simple, simply move the tens digit of the number (in 12, 1 is the 10s digit) one place value to the left. The result should look like this:


                                                                               129
                                                                          +   423 
                                                                               500
                                                                                 40
                                                                                 12


Step 7: Add the "solution numbers" together for you to get the solution of the entire problem. It should look like this:

                                                                               129

                                                                           +  423

                                                                               500
                                                                                 40             
                                                                           +    12
                                                                               552



It may seem confusing now, but once you get it, you'll be able to use this method in your head for various problems!

                                                                          

                                                                          
                                                        

                                                                                  

Division: Divisibility Rules

Have you ever wondered if 198676 is divisible by 3, but you don't want to do the math to figure it out? Well, there is a way to do this, and it is by using the Divisibility Rules. The divisibility rules are rules that tell you if a number (x) can be divided by certain number or not. For example if you didn't know if 26 can be divided by 2 or not, just check the divisibility rule for 2:

 The last digit must be divisible by 2, or be even.

Using that divisibility rule, you can apply that rule to the number to see if its divisible by 2 or not (which it is).  Here are the list of divisibility rules from 2-12:

Divisibility Rules

Rule for 2: The last digit must be divisible by 2 or be even.

Rule for 3: The sum of the digits must be divisible by 3.

Rule for 4: The last two digits must be divisible by 4.

                   (Example: 536 is divisible by 4 because 36 is divisible by 4)

Rule for 5: The last digit must be 0 or 5.

Rule for 6: The number must be even and divisible by 3.

Rule for 7: The number must be broken down into numbers that are both divisible by 7.

Rule for 8: The last three digits must be divisible by 8.

Rule for 9: The sum of the digits must be a multiple of 9.

Rule for 10: The last digit must be 0.

Rule for 11: The difference of the sum of odd-placed digits and the sum of the even-placed digits of the number must be 11 or 0. Let me explain:


Example: 139271


The bolded digits are odd placed digits. This is because if you were to count from right to left, 1 is the first digit from the right side, 2 is the third digit from the right side, and 3 is the fifth digit from the right side. 


As you notice each of those numbers are an odd number of digits from the right side. That is why they're odd placed digits. This also goes with the evens. 


Rule for 12: The number must be divisible by 3 and 4.

I think you might be able to find the pattern in the divisibility rules and use them to make rules for all the numbers. I hope this helped! 

                         

Multiplication: Nine Times Table At Your Fingertips

Having trouble remembering the 9x table? The answers are literally at your fingertips! Using the Fingertip Method, I will teach you how to remember the 9x table with no trouble at all!