Multiplication With Decimals

Multiplying a single digit number by a single digit number isn't to hard, just add the multiplicand the number of times the multiplier tells you to. For example 5 X 6 = 30. 5 Is the multiplier and 6 is the multiplicand. 6+6+6+6+6=30. We can reverse the order of the multiplier and the multiplicand and the answer or product will still be 30. Multiplying larger numbers like 56 X 31 = becomes tedious adding 31, 56 times. Thankfully there is an easier way to find the product. Lets examine 3 X 154 =? We could add 154, 3 times.

Or we can multiply 3 times each place value in 154 then add the values.

3X4=12

3X50=150

3X100=300

Remember that 4 is in the one's place so 3x4=12 5 is in the tens place and so it has a value of 50. 1 is in the hundreds place and has a value of 100. The expanded form of 154 is (1x100)+(5x10)+(4x1), which is 100+50+4.

You may find that multiplying each place value and adding is an easy way to multiply 2 or 3 digit numbers in your head without using paper.

Above are the steps in multiplying a single digit number by a number with multiple digits. Take a close look at the similarities in the method above used to multiply 3x154 and in the method of adding 154, 3 times and the carries. When we multiply 3x4 in the one's place resulting in 12 it is accomplishing the same thing as adding 4, 3 times in the one's place when adding 154, 3 times and in both cases we have a carry of 1 to the ten's place. Next when we multiply 3x5 in the ten's place we are accomplishing the same thing as when we added 5, 3 times when adding 154, 3 times. In both cases we get 15 + the carry 1 or 16 and then have a carry of 1 to the hundreds place. Last when we multiply 3x1 in the hundreds place getting 3 and then adding the carry of one which results in 4, we accomplish the same thing as adding the hundreds column of 1s resulting in 3 and then adding the carry of 1 for a result of 4 in the hundreds place of the product, when we added 154, 3 times.

Now look at multiplying a 2 digit by a 3 digit number.

258 X 35

First do it by multiplying all the place values and adding.

OK. Now the way it is taught in school.

First multiply the ones place of 35 times the the ones place of 258. 5x8=40, Put the 0 under the one's place and carry the 4. Next multiply the ones place of 35 times the tens place in 258. 5x5=25 Plus the carry of 4 is 29. Place the 9 under the ten's place and carry the 2. Now multiply the ones place in 35 times the hundreds place in 258. 5x2=10 Plus the carry of 2 is 12. Place the 2 in the hundreds place. Since there are no more digits in the top number or multiplicand, bring the 1 down in the thousands place. Cross out your carries so you don't accidentally add them in again and move to the 3 in 35.

We get the same answer in a few less steps. Looking at 35x258 closer. When we multiply the 5 in the one's place of 35 times 258 that is the same as adding 258, 5 times. Next when we multiply the 3 in the tens place of 35 by 258, it's the same as adding 258, 30 times. 3x258 is 774. We moved all the place values 1 place to the left so the 4 is in the ten's place, 7 in the hundreds and 7 in the thousands. There is a 0 not shown but understood to be there, in front of the 4. At the end of the problem we are adding 1290 to 7740. To multiply any number by 10 all you have to do is add a 0 to the end of it in front of the one's place but before the decimal point if it is a decimal number. 1x10=10, 2x10=20, 9x10=90, 7.12x10=70.12 and 15.613x10=150.613. The same pattern goes for multiplying any number by 100, only you add 2 zeros to multiply by 100. That is why we shift the product 1 place to the left each time we move left in the bottom number and multiply it by the digits in the top number. To multiply any number by 1000, just add 3 zeros in front of the one's place.

 
Here is a 3 digit number times a 3 digit number. 612x235
  3 1  |The carries
   612
   235
  X
  -----
  3060 |5x2=10, 0 Goes in the one's place. Carry the 1. 5x1=5 Plus the
       |carry is 6. 6 Goes in the ten's place. 5x6=30, 0 Goes in the
       |hundred's place. 3 Could be carried but since there are no more
       |digits we just bring it down into the thousand's place. This is
       |the same as adding 612, 5 times. Cross out those
       |carries before moving left to the 3 in 235.

    612
    235
   X
   -----
   3060 
  1836  |Now we move left to the 3 in 235 and multiply it by 2 in 612.
 1224   |Make sure and shift the product 1 place to the left so that
+       |the results are the same as adding 612, 30 times.
--------|3x2=6, 3x1=3 and 3x6=18. Now we move left again in the bottom
 143820 |number to 2 which is in the hundreds place and has a value of
        |200. All we do though is say 2x2=4 and put the 4 one more place
        |to the left and continue multiplying by the digits in the top
        |number. 2x1=2 And 2x6=12.

When we multiply numbers that have decimals in them, we do them the same way and ignore the decimal point until the very end. At the end, count up how many digits are in front of the decimal points in both the multiplier and multiplicand. Start at the far right digit in the answer and count left that many places and insert the decimal point. Since we just count up the places in front of the decimal point in the multiplier and in the multiplicand and place the decimal point that many places to the left in the product, you don't have to line the decimal points up in multiplication problems. It may seem strange to multiply a number like .5 X 6.44 and get a product of 3.220. When we multiply a whole number by a whole number the absolute value of the product is always greater than the multiplier and the multiplicand. Absolute value means we don't worry about the sign weather it is positive or negative, we just look at the value. Whole number means there is no fraction or decimal part of the number, just regular counting numbers like 1, 2, 3 and so on. How do you add 6.44, .5 times? If we were to multiply 1 times 6.44, we would add 6.44, 1 time and the product would be 6.44. If we take 1 divide it up in 10 equal parts, then take 5 of those parts that is .5. .5 Is the multiplier and tells us how many times we are to add 6.44. What we need to do is take 6.44 and chop it up in 10 pieces and then take 5 of those pieces to make .5x6.44.

Above the 0 in 0.5 is represented by ten blocks which have 10 blocks in each of them. The blocks with black are empty and the blocks with red in them are filled. Each column of ten blocks containing ten little blocks would be equal to 1 if filled. Each of the blocks that has ten small blocks in it is worth .1 or 1/10 if all the little blocks in it are filled. Each of the smallest blocks is worth .01 or 1/100 if filled. .5 Is represented by 5 of the .1 blocks fully filled with red. 6 In 6.44 is represented by 6 columns of fully colored in boxes. Looking at the rows of boxes that represent 6 we see that there are 10 rows of 6. To add 6, .5 times we can just circle 5 of those rows. We are circling 5/10 or .5 rows. Each 10 of those boxes is equal to 1. We end up with 3 sets of 10 for a value of 3. I rearranged them back into columns of 10 to show each 1 value more easily. The .4 in 6.44 is represented by 4 of the boxes that have 10 tiny boxes in them all filled up. If we unfold the tenth boxes like in the picture past the orange divider and put 4 of them together we get 4 columns of 10. We can circle 5 of these columns to graphically add .4, .5 times. Each time we have ten tiny full boxes that is equal to .1. We end up with 20 tiny boxes or .2. Next to add .04, .5 times graphically we just circle 2 of the 4 tiny dots as dividing these 4 little dots into 10 would be very hard to see. We would end up with 20 of them and each 10 would be worth .01. If you look at the representation of .5 you can see that .5 is 1/2 of one so we can just circle half the hundredth's values to add .04, .5 times. Next we add up the digits that we have graphically multiplied for a product of 3.22. Multiplying decimals is a lot like multiplying fractions only with decimals you are always using a denominators that are multiples of 10. If you were multiplying a fraction such as 3/4 times 100, divide the 100 up into 4 equal amounts and take 3 of those amounts. This is adding 100, 3/4 times. If you multiple .5X6.44 the way it is taught in school you will end up with 3.220 for a product. You can just drop the ending 0 off your answer if you want to and it will still be correct. 3.22 is the same value as 3.220. Also you may end up with leading zeros sometimes like in the number 0.6. You can just leave the leading 0 off your answer if you want to. .6 Is the same value as 0.6.

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