Multiplication:

So what is multiplication all about and how is ti helpful?

Multiplication is a shortcut. It is a shorter and simpler way of expressing repeated addition and to a large extent, it is a matter of remembering the multiplication table. What do we mean by repeated addition?

Let us use the following example: What are we being asked to do when we are asked to calculate

Multiplying 6 by 7 is actually the same as adding six seven times:

Here is how it works:

We have added six seven times to get the result of 42. (If we were to add seven six times, we would get the same result.) If you remember your multiplication table, you will know that the result of 6 x 7 is 42. How is this helpful? Well, assume that you are asked to calculate the annual salary of an employee who is paid an hourly wage of $15.00/hour? How would you go about it? Well, there are 52 weeks in a year and most people are paid for 40 hours per week. Therefore we would multiply 52 x 40 x 15. If we would not use multiplication, we would have to add the number 52 forty times and then, once we would have the result, we would have to add that 15 times. It is certainly much faster to multiply the numbers together. So how do we multiply number of two or more digits, once we know our multiplication table?

We may write the multiplication task as 23 x 12, but in order to calculate the result, let us use the following format:

You will note that the number 2 is written in red. Why? Because it is the first number we multiply the first line by and we always start by multiplying the singles digit of the first line, progressing to the tens and if there are more digits, the hundreds, thousands, etc.

So we first multiply 3 in the first line by 2 and get the result of 6 and then we multiply the second number in the first line, 2 by 2 and get the result of 4 and we write it down as:

Then we go on to the next step of the calculation. We add a zero under 6. Why? Because now we are going to be calculating by the tens digit of the second number and we have to account for the fact that we are multiplying - in this case - not by 1, but by 10:

Notice that now we have written zero where it belongs and when we will now multiply our first line by the tens digit of the second line, which is now marked red, it will force us to write the result after the zero and thereby not make a mistake! We will first multiply 3 by 1 and then 2 by 1 and get the result of 23 (Obviously if you are multiplying any number by 1, the result is the same as the number you are multiplying).

And our result will be 276.

Now we have to learn one very important thing. In our above example, we have been multiplying 23 by first 2 and then by 1 and in both cases, our results did not exceed 10 What happens, when it does? Let us take a look at the following example::

According to our rule, we will first multiply the singles digit of the first line by the singles digit of the second line, that is we will multiply 6 by 8. The result is 48. How do we record and account for this? We will write down 8 and we have to "carry over" the 4 into our second multiplication of the tens digit of the first line by the singles digit of the second line, which will be 5 by 8, which will be 40 and we have to add the carryover of 4 to get 44:

Now let us return to our example of the employee who is paid $15.00 per hour. It is important to remember that whether we multiply 52 by 40 or 40 by 52, the result will be the same. When you multiply two numbers, whether you will multiply the first number by the second or the second by the first, you will get the same answer in both cases. Sometimes though, even though the result will be the same, it may be easier to multiply by one of the two numbers and the above is a good example:

Our first step is to multiply the first line by the singles digit of the second number. But look at our second pair: the singles digit of 40 is zero. What do we do when that happens? Just write zero under the zero and immediately right next to it, go on with multiplying the first line by the second, (tens) digit of the second number. We get the following for our two calculations:

For the first pair, we have only completed the first one of the two steps, multiplying first 0 and then 4 by 2 and we still have to add zero in our next line under zero and then to multiply the first line by the tens digit of the second line. However, in our second pair, our calculation is complete since multiplying the first line by zero gave us zero, which we have written down, but then continued immediately by multiplying first 2 and then 5 by the ten digit (4) number of the second line to get our answer, 2080.

So you see, it is always easier and faster by multiply by a two digit number than ends with zero. What about when we have multiplication of number with more than two digits, such as 365 by 24 (how many hours are there in a year?)

The most important part of multiplication (aside from getting the answers right) is make sure the numbers are aligned correctly! Don't forget to insert the zeros!

Examples:

Answers: