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Dividing decimals by decimals - a video and an online lesson

Why do we move the decimal point in both the dividend and the divisor when dividing decimals by decimals? It is because both the dividend and the divisor are multiplied by 10, 100, 1000, or some other power of ten. In the video below, I aim to make sense of this "rule". Please see also the lesson that continues below the video. The lesson are taken from Math Mammoth Decimals 2 book ($4.00 download). The book has more problems than shown in this online lesson.

Divide decimals - why do we move the decimal point?

Divide Decimals by Decimals

The above problems illustrate a way to solve decimal division problems. You noticed that in each case, the quotients (answers) were the same! And it is no wonder. Think of it as, “How many times does the divisor fit into the dividend?” 0.02 fits into the 0.06 as many times as 2 fits into 6. Or, 0.007 fits into 0.35 as many times as 7 fits into 350.

If we have a more difficult decimal division problem, such as 3.439 ÷ 5.6, we can transform it into a problem with the same answer, but with a whole-number divisor, which can be solved with long division.

Look at the problems in #1 again, this time moving from the bottom up. In each step, the dividend increases by a factor of 10 (that is, it is multiplied by 10), and so does the divisor! When both the dividend and the divisor are increased by the same factor, the quotient remains the same!

The table on the right illustrates this idea again. Each line is one division problem. Each problem has the same answer, 28. Each problem’s divisors differ from each other by a factor of 10, and so do each problem’s dividends. This idea is VERY important! Let’s write some problems using the division line instead of the ÷ symbol. We can write the equal sign “=” between the problems because they all have the same answer. The last step,   340.2 7 , can be done with long division, and the answer is 48.6. If you still doubt that they’re all the same, then check each of the division problems on the right with a calculator.

× 10          × 10           × 10                    0.3402 0.007 = 3.402 0.07 = 34.02 0.7 = 340.2 7 = 48.6                            × 10           × 10           × 10

2. Continue the patterns, multiplying the dividend and divisor in each step by 10, so that
    the quotients (answers) remain the same.

3. Multiply both the dividend and the divisor by 10, repeatedly, until you get a whole-number divisor.
    Then, divide using long division. The first one is done for you.

a. 0.445 ÷ 0.05 0.445 0.05 = 4.45 0.5 = 44.5 5 = 8.9       0 8.9 5  )  4 4.5   - 4 0        4 5      - 4 5     0

(More problems available in the book.)

See also: Divide decimals by a whole number