Carrying and Regrouping Values in Multiplication
The values in multiplication get very large very quickly. You might remember from addition that you will need to carry a number whenever the sum of the addends is greater than nine. The same rule applies in multiplication, but almost every set of values you multiply will be greater than nine. When that happens, you need to carry the extra value to the next column to the left. Carrying/regrouping will happen on almost every problem with two digits.Carrying and regrouping are the same ideas. You are moving numbers around to create new values. When you wind up with a value greater than nine, you need to bring some extra to the column to the left. To be honest, we like the term carry, but you need to say the word your teacher wants to hear.
Examples:
2 x 3 = 6 (no carrying/regrouping)
11 x 5 = 55 (no carrying/regrouping)
12 x 5 = 60 (carrying/regrouping needed)
- or -
12 x 5 60 |
How did we get that answer? Why did you need to carry? Shouldn't the answer be 50-something? When you multiplied the numbers in the first column (2 x 5) the answer was ten. You got a two-digit product, but you can only write down one number in the final answer. We write down the value from the ones column and carry the "1" over to the tens column. We regroup the tens in the problem. The next step has you multiplying 5x1 and then adding the carried "1" to the product. 5 x 1 = 5... 5 + 1 = 6.
Here are the steps...
(1) Multiply the values in the ones column. In our example, you multiply 2x5. The answer is "10".
(2) Write down the "0". That will be the ones value in your product.
(3) Take the extra "1" and write it just above the 1 in the tens column. Moving that "1" is called carrying or regrouping.
(4) Multiply out the tens value 1x5. Your tens product will be 5.
(5) Add the 1 that you carried (5+1). Your new value will be 6.
(6) Write the six in the tens column of your product. Your final product is 60.
12 x 5 ? |
1 12 x 5 2x5=> 0 |
1 12 x 5 (1x5)+1=> 60 |
So... multiply the ones, carry, multiply the tens, and add the carried amount.
Always Moving to the Left
Just a little reminder before we continue. As you get products that are greater than 9, you will add values to the left. When you multiply multiple columns, you always start with the smallest values. If you have a five-digit number such as 12,345 you will start multiplying values in the ones column first. Then you will move to the tens, hundreds, thousands, and ten thousands columns. If carrying is involved, you will take the carried value and place it in the column to the left. So if you were multiplying numbers in the tens column and you needed to carry a "4", you would place that "4" in the hundreds column.Problem:
999 x 5 ???? |
Answer:
(1) Start with the ones column: 9x5=45
(2) Since that product is greater than nine, you need to carry the four (4) to the next column to the left.
(3) Multiply the value from the tens column: 9x5=45. Then add the amount you just carried: 45+4=49.
(4) You got a value greater than nine again. Write the 9 in the tens column of your answer and carry the four to the hundreds column.
(5) Multiply the value from the hundreds column: 9x5=45. Then add the amount you just carried: 45+4=49.
(6) Since that's the end of the problem, write your value in the answer.
Final Answer: 4,995
We built our final product like this… ---5, then -95, then 4995.
999 x 5 ? |
4 999 x 5 9x5=> 5 |
44 999 x 5 (9x5)+4=> 95 |
44 999 x 5 (9x5)+4=> 4995 |
Example:
58 x 4 = ?
Step 1: Multiply the ones column. 8 x 4 = 32
Step 2: Write down the "2" and carry the "3" to the tens column.
Step 3: Multiply the tens column. 4 x 5 = 20
Step 4: Add the number you carried. 20 + 3 = 23
Step 5: Write down the 23.
Answer: 58 x 4 = 232
Multiply Then Add
After a few examples you can see the pattern: Multiply Ones—Carry—Multiply Tens—Add Carried Value. That pattern works if you are working with a two-digit number or a ten-digit number. It's just a lot more steps when you have more digits.Example:
296 x 8 = ?
MULTIPLY ONES: 6 x 8 = 48
WRITE AND CARRY: Write the "8" and carry the "4"
MULTIPLY TENS: 8 x 9 = 72
ADD: Add the carried amount. 72 + 4 = 76
WRITE AND CARRY: Write the "6" and carry the "7"
MULTIPLY THE HUNDREDS: 8 x 2 = 16
ADD: Add the carried amount. 16 + 7 = 23
WRITE: Write down the "23"
Answer: 296 x 8 = 2,368
296 x 8 ? |
4 296 x 8 6x8=> 8 |
74 296 x 8 (8x9)+4=> 68 |
74 296 x 8 (8x2)+7=> 2368 |
Moving Values Larger Than One
We just wanted to point something out. In addition, you usually carry a "1" to the next column. With multiplication you will carry numbers between 1 and 9. In one column you might add a carried "3", but in the next you will carry a "9". When you start carrying the higher numbers, there is a good chance you will need to carry again. So keep your eyes open when you start the carrying work. All of those little numbers can be mixed up.Useful Reference Materials
Wikipedia:https://en.wikipedia.org/wiki/Arithmetic
Encyclopædia Britannica:
http://www.britannica.com/topic/arithmetic
Encyclopedia.com:
http://www.encyclopedia.com/topic/arithmetic.aspx