SECTION 1-WHOLE NUMBER ARITHMETIC

MULTIPLICATION-PART 2

At this point you can multiply any number by a single digit. Okay, you might say, "but how about if I'm multiplying by a double, or even a triple digit number?" And even if you don't ask this question, I'll ask for you. Isn't that nice of me?

Okay, welcome to the world of "long multiplication." As you will see, long multiplication is a combination of multiplying and adding. No, don't go running out of the room screaming--it's really not that hard. Simply follow these three steps.

Now if you are multiplying by a number that is larger than one digit, take it one step at a time.
- Think of a two digit multiplier as multiplying by two distinct numbers
  So if you were multiplying 235 by 12 think of it as multiplying 235 by 2 and multiplying 235 by 1.
- Think of a three digit multiplier as multiplying by three distinct numbers
  So if you were multiplying 324 by 738 think of it as multiplying 324 by 8, multiplying 324 by 3, and multiplying 324 by 7.
- Think of a four digit multiplier as multiplying by four distinct numbers
  So if you were multiplying 7892 by 1201 think of it as multiplying 7892 by 1, multiplying 7892 by 0, multiplying 7892 by 2, and multiplying 7892 by 1.
What do you think you would do for a 5 digit multiplier?
Now for the trick. Place some zeros onto the right hand side of the results you just obtained. How many zeros? It varies according to which digit of the multiplier you were using at the time. Here are the rules:
- Do not place any zeros on the end of the product that you obtained using the digit on the far right-hand side of the multiplier (the digit in the "one column").
- Moving to the left we will have the digit in the "tens column." Place one zero on the right hand side of the product you obtained when you multiplied by that digit.
- Moving to the left we will have the digit in the "hundreds column." Place two zeros on the right hand side of the product you obtained when you multiplied by that digit.
- Moving to the left we will have the digit in the "thousands column." Place three zeros on the right hand side of the product obtained when you multiplied by that digit.
What would you do when multiplying by the next digit (which would be in the "ten-thousand" column)?
Now that you have your "subproducts", you can achieve the final product, by adding them all together.

These concepts will be illustrated in the next three examples

Multiplication example 1

235 x 12

The multiplicand is 235, and the multiplier is 12.
We are interested in 12 groups of 235 (i.,e. the result of adding 235 to itself a total of 12 times).
We can figure out that 235 x 2 = 470 and 235 x 1 = 235.
The digit in the "ones column" of the multiplier is 2. When we multiplied 235 by 2 we recieved a product of 470. We are not going to use any zeros at this stage.

The digit in the "tens column" of the multiplier is 1. When we multiplied 235 by 1, we got a value of 235. However since the 1 is the "tens-value" of our multiplier, the rule tells us to place one zero on right hand side of 235. The number 235 is now transformed into 2350.

Now write the problem as follows

235 x 12 470 2350
The final step is to add 470 and 2350 together.
470 + 2350 = 2820.

235 x 12 470 2350 2820
235 x 12 = 2820

Some math books will suggest that you indent the "subproducts" rather than add zeros. If you used that procedure you would get the following:

235 x 12 470 235 2820

Multiplication example 2

324 x 738

The multiplicand is 324, and the multiplier is 738.
We are interested in 738 groups of 324 (i.,e. the result of adding 324 to itself a total of 738 times).
324 x 8 = 2592
324 x 3 = 972
324 x 7 = 2268
The digit in the "ones column" of the multiplier is 8. When we multiplied 324 by 8 we recieved a product of 2592. We will not use any zeros at this stage.

The digit in the "tens column" of the multiplier is 3. When we multiplied 324 by 3, we got a value of 972. However since this was the result of multiplying by the digit in the "tens column" of the multiplier, the rule tells us to place one zero on right hand side of 972. The number 972 is now transformed into 9720.

The digit in the "hundreds column" of the multiplier is 7. When we multiplied 324 by 7, we got a value of 2268. However since this was the result of multiplying by the digit in the "hundreds column" of the multiplier, the rule tells us to place two zeros on right hand side of 2268. The number 2268 is now transformed into 226800.

Now write the problem as follows

324 x 738 2592 9720 226800
The final step is to add 2592, 9720, and 226800 together.
2592 + 9720 + 226800 = 239112.

324 x 738 2592 9720 226800 239112

324 x 738 = 239,112

Some math books will suggest that you indent the "subproducts" rather than add zeros. If you used that procedure you would get the following:

324 x 738 2592 972 2268 239112

Assignment: In this problem I multiplied 324 by 738 and ended up with a result of 239,112. Your assignment is to multiply 738 by 324. You should come out with the same answer.

Multiplication example 3

7892 x 1201

The multiplicand is 7892, and the multiplier is 1201.
7892 x 1 = 7892
7892 x 0 = 0
7892 x 2 = 15784
The digit in the "ones column" of the multiplier is 1. When we multiplied 7892 by 1 we recieved a product of 7892. We do not use any zeros at this stage.

The digit in the "tens column" of the multiplier is 0. When we multiplied 7892 by 0, we got a value of 0. Ordinarily we would extend this with one zero, but in this case that would not change the value of this number (00 = 0). In fact, adding zero will not have any effect when we get to step 3 of this problem.

The digit in the "hundreds column" of the multiplier is 2. When we multiplied 7892 by 2, we got a value of 15784. However since this was the result of multiplying by the digit in the "hundreds column" of the multiplier, the rule tells us to place two zeros on the right hand side of 15784. The number 15784 is now transformed into 1578400.

The digit in the "thousands column" of the multiplier is 1. When we multiplied 7892 by 1, we got a value of 7892. However since this was the result of multiplying by the digit in the "thousands column" of the multiplier, the rule tells us to place three zeros on the right hand side of 7892. The number 7892 is now transformed into 7892000.

Now write the problem as follows

7892 x 1201 7892 1578400 7892000

Notice that I did not even bother to write the 0 in the second row of the addition section of this problem. After all adding zero will have no effect on the finished sum.
The final step is to add 7892, 1578400, and 7892000 together.
7892 + 1578400 + 7892000 = 9478292.

7892 x 1201 7892 1578400 7892000 9478292

7892 x 1201 = 9,478,292

Some math books will suggest that you indent the "subproducts" rather than add zeros. If you used that procedure you would get the following:

7892 x 1201 7892 15784 7892 9478292

Assignment: In this problem I multiplied 7892 by 1201 and ended up with a result of 9,478,292. Your assignment is to multiply 1201 by 7892. You should come out with the same answer.

There is one important fact that you may have realized. in multiplication it doesn't matter which number is the multiplicand and which is the multiplier. You'll get the same result either way.

3 x 2 = 2 x 3
6 x 4 = 4 x 6
324 x 738 = 738 x 324
7892 x 1201 = 1201 x 7892

So if I asked you to multiply 21 and 22, you could either multiply 21 by 22 (i.e. 21 is the multiplicand and 22 is the multiplier) or you could multiply 22 by 21 (i.e. 22 is the multiplicand and 21 is the multiplier). The choice is yours. Either way you should reach a product of 462.

See multiplication isn't so bad after all! Now you try it!

Click on the "generate a multiplication problem" button to see your exercise. Place your answer in the text box and then click the "check my answer" button. The computer will then tell you whether or not your answer is correct.

After you complete the problem, you can generate another problem by first clicking the "reset button" and then clicking the "generate a multiplication problem" button. Continue as many times as you wish, and then:

All text, images, and source code c2003 Martin Selditch.