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Multiplication with Decimal Numbers
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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.

Multiplication with Decimal Numbers

Video Source (05:37 mins) | Transcript

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

Multiplication with decimals (Why it works)

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 that we are multiplying and move the decimal of the answer to the left that many times.

Example: \(5.8 \times 12.62\), \(5.8\) 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

Practice Problems

Evaluate the following expression:
  1. \(6.9 × 5 = ?\) (
    Solution
    x
    Solution: \(34.5\)
    Details:
    The 6 is in the ones place and the 9 in the tenth place. The number five is also in the ones place.
    The number 6.9 is written above the number 5 and the number 5 is placed directly under the 9. Above the nine is written the place value “tenths” and above the 6 is written the place value “ones.” There is a horizontal line the width of all the numbers written below the 5 and an x written off to the left indicating multiplication between the numbers.

    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.
    This image is the same as the previous image except now there is a new number 5 below the 9 and the 5 in the tenths column. This new number 5 is also below the solid horizontal line. There is also a small red number 4 written above the 6 in the ones column.

    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.
    This image is the same as the previous image except now there is a 3 and 4 written below the horizontal line next to the 5. This means that now under the line is the number 345.

    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.
    This is the same as the previous image except this time there is a box around the 9 in the original top number 6.9.

    The answer will have one decimal place.
    This is the same as the previous image except now there is also a box around the number 5 in the solution below the horizontal line.

    Finally, place the decimal point to the left of the number 5, one decimal place over in our answer.
    In this image the boxes around the 9 and 5 are no longer there and there is now a decimal point between the 34 and the 5 in the solution under the horizontal line. This makes the number under the horizontal line 34.5. There is an arrow pointing to the decimal point to add emphasis that it is one space over from the right.

    The answer is \(34.5\)
    )
  2. \(3.8 × 90 = ?\) (
    Solution
    x
    Solution:
    342
    )
  3. \(4.9 × 1.2 = ?\) (
    Video Solution
    x
    Solution: \(5.88)
    Details:

    (Video Source | Transcript)
    )
  4. \(9.3 × 625 = ?\) (
    Video Solution
    x
    Solution: \(5812.5\)
    Details:

    (Video Source | Transcript)
    )
  5. \(5.5 × 8.63 = ?\) (
    Solution
    x
    Solution: \(47.465\)
    Details:
    Place the number with more digits on top. The number \(8.63\) has more digits than \(5.5\), so place \(8.63\) on top.
    There are two numbers stacked one on top of the other. The top number is 8.63. Under it is the number 5.5. The 5 in the tenths place is directly under the 3 and the 5 in the ones place is directly under the 6. There is a solid horizontal line below the 5.5 stretching the width of all then numbers and there is an x to the left indicating multiplication.

    Start by multiplying the numbers on the right \(5 × 3\)
    This is the same as the previous image except the 3 and the 5 in the far right column of numbers are a different color indicating that we are working with them.

    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.
    This is the same as the previous image except below the rightmost column, under the horizontal line, is a 5. Above the 6 is a 1. Also, this time the 6 in the top number and the 5 in the rightmost column are blue indicating we will work with them next.

    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.
    This is the same as the previous image except now there is a 1 below the horizontal line and under the middle column of numbers. There is also a 3 above the 8 in the top number. The rightmost 5 and the number 8 are highlighted as the numbers to work with next.

    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.
    This image is the same as the previous one except that below the horizontal line there is a 3 to the left of the 1 and a number 4 to the left of the 3. This makes the number below the horizontal line four-three-one-five reading it from left to right.

    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.
    This image is the same as the previous image except now there is a 0 below the 5 in the previous solution.

    Now, start multiplying 3 by 5 in the ones place.
    This image is the same as the previous one except this time the 3 in the original top number and the 5 in the middle column below it are highlighted to show we are working with them next.

    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.
    This image is the same as the previous one except now there is a 5 to the left of the 0 in the bottom level of solutions. There is also a new number 1 above the 6 in the middle column.

    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.
    This image is the same as the previous one except now the left 5 and the 6 from the original numbers are highlighted. Down in the solution area, in the bottom solution line there is a 1 to the left of the 5, directly below the 3 from the previous solution of 4315. There is also a new number 3 above the 8 in the leftmost column of the original numbers.

    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.
    This image is the same as the previous one except this time the leftmost 5 and the 8 of the original numbers are highlighted. Down in the bottom line of the solution area there is a 3 to the left of the 1 and a 4 to the left of this 3. This makes the bottom solution number four-three-one-five-zero.

    Now add the answers from the multiplication of \(8.63\) times .5 (tenths) and \(8.63\) times 5 (ones).
    This image is the same as the previous image except now there is a new horizontal line below the two solution lines of numbers. There is also an addition symbol to the left of the solution area and the numbers in the two solutions lines are a different color than the other numbers indicating we will be adding the two solution numbers together and putting our answer below the new horizontal line.

    Add the products (answers to the multiplication). The sum is 47465.
    This image is the same as the previous image except all the previous numbers are in black and there are numbers below the addition line. These numbers are four-seven-four-six-five going from left to right.

    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.
    All the numbers in this image are the same as the previous image except they are all the same color. In the original numbers 8.63 and 5.5, the numbers 6, 3 and the rightmost 5 have boxes around them because they are to the right of the decimal point. Down in the solution area, below the addition line the right three numbers 4, 6, and 5 are also in boxes. This shows that there were three digits to the right of the decimal point in the original numbers and so we are highlighting the three rightmost numbers in our final solution.

    Finally, place the decimal point in the answer between 7 and 4; this is three places to the left of the answer.
    This image contains all the same numbers as the previous image except now there aren’t any boxes and there is just a decimal point to the right of the 7 in the final solution. This makes the final solution forty-seven-point-four-six-five.

    The answer is \(47.465\)
    )
  6. \(4.71 × 3.84 = ?\) (
    Solution
    x
    Solution:
    \(18.0864\)
    )

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