Introduction

In this lesson, you will learn the multiplication algorithm, or pattern, used in this course, as well as why this method works.

Note: You are welcome to use a different system of multiplication if it works better for you.

Scripture Connection

In Matthew 18:22, Jesus taught Peter about forgiveness. Peter asks if he should forgive his brother seven times but Jesus replies, “I say not unto thee, Until seven times: but Until seventy times seven.” One might think this gives permission to stop forgiving someone after forgiving them 490 times, but Elder Bruce R. McConkie explained that this really means “there is no limit to the number of times men should forgive their brethren” (The Mortal Messiah: From Bethlehem to Calvary, 4 vols. [1979-81], 3:91).

These videos illustrate the lesson material below. Watching the videos is optional.

• Multiplying by One Digit (04:19 mins) | Transcript
• Multiplying with Multiple Digits (07:12 mins) | Transcript

Multiplying by One-Digit Numbers

While you can memorize single digit multiplication, you can’t memorize all multiplication problems. Instead, you can use single digit multiplication and some steps to solve large multiplication problems. The following instructions will teach these steps. If you have a different method of doing multiplication, you can use the method you are comfortable with.

Example 1
$32\times 8$

• Put the number with the most digits on top.
  
  • In this case, 32 has the most digits, so it goes on top. 8 will go on the bottom.

Figure 1

• Start with the bottom number furthest to the right. Multiply it to all the digits in the top number, one by one, starting at the right and going left.
• If the numbers in the first column multiplied together are greater than 9, then carry the first digit of their solution into the next column.
  
  • Example: $8\times 2=16$. Carry the 1 into the tens column and write the 6 below the ones column in the solution area.

Figure 2

• Continue by multiplying the bottom rightmost number to the number in the next place value to the left in the top number and add any number that was carried into that column to this solution.
  
  • Next, in this case, multiply the eight and the three. $8\times 3=24$, but since you carried the one over earlier, you need to add it to the total, which is 25.
• Continue following this pattern by multiplying the rightmost number on the bottom to all the individual digits in the top number and writing the solution in the solution area. Remember to carry anything greater than 9 to the next column.
  
  • Finishing this step gives us the total answer for this multiplication problem: 256.

Figure 3

Consider why this works. 32 is really the same as $30+2$, or three tens and two ones. Since multiplication is repeated addition, you’re adding thirty together eight times and two together eight times, which looks something like this:

$30+30+30+30+30+30+30+30+2+2+2+2+2+2+2+2$

30 added together eight times is 240, and two added together eight times is 16. Added together, this gives the total answer: $240+16=256$.

The multiplication problem shows this same principle. $2\times 8=16$, and sixteen is the same as one ten and six ones. Represent this by putting a one in the tens column.

Figure 4

Remember that the three in 32 represents 30, and $8\times 30=240$, so you’ll put a two in the hundreds column, a four in the tens column, and a zero in the ones column.

Figure 5

Now add 16 and 240, which equals 256. The algorithm saves us a few steps, but you are really doing the exact same thing, and it gives us the same answer.

Figure 6

Multiplying by Multiple-Digit Numbers

How do you go about solving a problem when both of the numbers in the multiplication problem have multiple digits?

Follow the same steps as when working with one digit, but include the following:

• After multiplying all the digits in the top number by the bottom right digit, move to the next digit to the left in the bottom number and repeat the process.  Be sure to start a new solution line and include a 0 in the rightmost column of it to show you have moved to the tens place.
  
  •  Each time you begin the steps again with a new digit in the bottom number, start a new solution line and add an additional 0 in the rightmost column to represent the place value of the number you are multiplying across the top number. 
•  When you have repeated this process for all the digits in the bottom number, add the lines in the solution together.

Example 2
$534\times 127$

Begin by putting the number with the most digits on top. When both numbers have the same number of digits, it’s often best to put the larger number on top. In this example, that means 534 will go on top.

Put the number with fewer digits on the bottom. In this case, 127 will be on the bottom.

Figure 7

Next, take the number on the bottom right and multiply it successively by each digit of the top number. In this example, start with the seven and multiply it by all the individual numbers in 534.

Since $7\times 4=28$, carry the 2 over to the tens place and leave the 8 in the ones column.

Figure 8

$7\times 3=21$, and when you add the 2 that you carried over, it makes 23. Leave the 3 in the tens column and carry the 2 over to the hundreds column.

Figure 9

Lastly, $7\times 5=35$, and when you add the extra 2, it becomes 37.

Figure 10

It is crucial to remember that when you start the next round of multiplication, you’ll begin in the tens column. When you start in the tens column, you must put a zero in the ones column because you’re multiplying everything by tens now. Then proceed as before.

Figure 11

Start with $2\times 4=8$. There’s nothing to carry here, so document 8 and go on to the next column.

Figure 12

$2\times 3=6$ , which again leaves nothing to carry. Simply put the 6 in the hundreds column of the solution area.

Figure 13

$2\times 5=10$ and gives a total of 10680 for this solution line.

Figure 14

Now you’ve reached the hundreds column of the bottom number. This time put two zeros at the end because you’re working with hundreds. Since you’re multiplying the top number by one, it simplifies the work; anything multiplied by one stays the same number.

Figure 15

This works because what you are really doing is multiplying each place value in the bottom number by each place value in the top number and adding them together.
At this stage, add the numbers together column by column.

• The numbers in the ones column add up to 8.
• The numbers in the tens column add up to 11, so put a 1 in the tens place and carry a 1 over to the hundreds column.
• The numbers in the hundreds column add up to 17. When you add the additional 1 that you carried over from the tens place, you get 18. Put the 8 in the hundreds place and carry the 1 over to the thousands column.
• The numbers in the thousands column add up to 6. The additional 1 that you carried over makes it a 7 in this place.
• Finally, the numbers in the ten thousands column add up to 6. 

The total is 67818.

Figure 16

How does this work? What you are really doing is multiplying each place value in the bottom number by each place value in the top number, and then adding them together. In the image below, the different colored arrows show how you multiply each of the top numbers by each of the bottom numbers, respectively.

Figure 17

In the example equation, you did the same thing. However, instead of writing down every step of the equation and only adding numbers together at the end, you did some of the addition as you went.

Things to Remember

• Keep an eye on the numbers you carry over from other columns. Be careful not to lose them or add them to the wrong section.
• Make sure that the columns are appropriately aligned before they are added together. For example, be careful not to add a number from the tens column to a number in the ones column.
• Remember to add zeros to the appropriate columns when multiplying.

Practice Problems

1. $22\times 3=?$ (
   Solution

   x

   Solution: 66
   Details:
   Analyze the multiplication of $22\times 3$ in parts.
   First, 22 can be broken into two parts: 2 tens (or 20), and 2 ones. Multiply $20\times 3$.
   $20\times 3=60$
   
   Next, multiply the second part, 2 by 3.
   $2\times 3=6$
   
   The answer to each separate part multiplied by 3 is 60 and 6.
   Add these answers together to get the final answer.
   $60+6=66$
   
   The 66 represents the answer to the original problem of $22\times 3$.
   The final answer is 66.
   )
2. $70\times 5=?$ (
   Solution

   x

   Solution: 350
   Details:
   Using what you already know about multiplication from previous lessons:
   $7\times 5=35$
   
   But you are actually multiplying $70\times 5$ in this problem and 70 is $7\times 10$.
   
   So for the final answer, multiply the $35\times 10$ which becomes 350.
   
   (See the lesson on Multiplying and Dividing by Powers of 10.)
   
   $70\times 5=350$
   )
3. $75\times 48=?$ (
   Video Solution

   x

   Solution: 3600
   Details:
   

   
   
   | Transcript)
4. $542\times 14=?$ (
   Solution

   x

   Solution:
   7,588
   )
5. $294\times 400=?$ (
   Solution

   x

   Solution:
   117,600
   )
6. $876\times 204=?$ (
   Video Solution

   x

   Solution: 178,704
   Details:
   

   
   
   | Transcript)
7. $807\times 655=?$ (
   Solution

   x

   Solution:
   528,585
   )