Multiplication and Division:

Multiplication with Decimal Numbers

While we can always use a calculator to do multiplication with multiple digits or with decimal numbers, it’s important to know how to do it, and practicing strengthens our brain and prepares us for the coming lessons. Multiplication with decimal numbers only adds one step to the multiplication we do with large whole numbers. The following video will show the steps of this process.

Video Source (05:37 mins) | Transcript

The following video explains why we can do multiplication with multiple digits as explained in the previous video.

Video Source (04:43 mins) | Transcript

Multiplication with decimals is the same as the process when there are no decimals. The only difference is that at the end we count how many decimal places are represented in the two original numbers we’re multiplying and move the decimal of the answer to the LEFT that many times.

Example: 5.8 × 12.62, 5.8 has 1 digit after the decimal, 12.62 has 2 digits after the decimal, so our answer will have 1 + 2 = 3 digits after the decimal.

Note: Having your multiplication facts memorized will help you do these problems faster and more accurately. These exercises also help reinforce the multiplication facts which will help you when we get into fractions and algebra.

Additional Resources

Khan Academy: Introduction to Multiplying Decimals (03:49 mins, Transcript)
Khan Academy: Multiplying Decimals (06:28 mins, Transcript)
Khan Academy: Multiplying Decimals Examples (02:52 mins, Transcript)

Practice Problems

Evaluate the following expression:

6.9 × 5 = ?
3.8 × 90 = ?
4.9 × 1.2 = ?
9.3 × 625 = ?
5.5 × 8.63 = ?
4.71 × 3.84 = ?

Solutions

34.5 (Written Solution)

The 6 is in the ones place and the 9 in the tenth place. The number five is also in the ones place.

Multiply 5 times 9. The answer is 45. Place the 5 to the far right below the line. Place the 4 above the 6 in the ones place.

Next, multiply 6 × 5. The answer of 6 × 5 = 30, plus the 4 that was brought over from the previous multiplication, is 34. Since there aren’t any more place values to the left to multiply, place the 34 in the answer next to the 5.

The last step is to put a decimal point in the answer. To determine where to place the decimal point, consider the decimal places in the factors (the numbers being multiplied together). There is only one decimal place .9 or 9 tenths.

The answer will have one decimal place.

Finally, place the decimal point to the left of the number 5, one decimal place over in our answer.

The answer is 34.5
342
5.88 (Solution Video | Transcript)
5812.5 (Solution Video | Transcript)
47.465 (Written Solution)

Place the number with more digits on top. The number 8.63 has more digits than 5.5, so place 8.63 on top.

Start by multiplying the numbers on the right 5 × 3

Multiply 5 × 3 = 15. Place the 5 on the bottom right and place the 1 over the 6. This moves the 1 into the tenths place.

Multiply the 5 in the first column on the right with the 6.

This gives us 5 × 6 = 30, plus the 1 that was brought from the previous multiplication, for a total of 31.

Place the 1 at the bottom, to the left of the 5, and place 3 over the 8. This moves the 3 into the ones place.

Multiply 8 × 5 = 40, plus the 3 brought from the previous answer, makes 43. This time, place 43 below the line with the 4 next to the 3 because there aren’t any more columns to multiply.

We must now multiply the other 5 in the problem by all the numbers in the first row.

Before multiplying, place a zero below the first column on the right, under the 5, as a place-holder.

Now, start multiplying 3 by 5 in the ones place.

Multiply 3 × 5 = 15. Place the 5 to the left of 0 and place the 1 over the 6. This moves the 1 into the tenths place.

Multiply 5 × 6 = 30 and add the 1 that was brought from the previous multiplication to makes 31. Place the 1 at the bottom, to the left of the 5, and place 3 over the 8. This moves the 3 into the ones place.

Multiply 8 × 5 = 40 and add the 3 brought from the previous answer to make 43. This time, place 43 below the line with the 4 next to the 3.

Now add the answers from the multiplication of 8.63 times .5 (tenths) and 8.63 times 5 (ones).

Add the products (answers to the multiplication). The sum is 47465.

To place the decimal point, determine how many decimal places are in each factor. In this example, there are three decimal places in the numbers used to multiply, so count three places in the answer. Start on the left side going right.

Finally, place the decimal point in the answer between 7 and 4; this is three places to the left of the answer.

The answer is 47.465
18.0864