# Dividing decimals by decimals

This is a complete lesson with a video, instruction, and exercises about dividing decimals by decimals, meant for 5th grade. The lesson concentrates on the idea that we can transform any division with a decimal divisor into a whole-number division by multiplying BOTH the dividend and the divisor by 10, 100, 1000, or some other power of ten.

So, why do we move the decimal point in both the dividend and the divisor the same number of steps? This is just a shortcut, and it comes from the idea above; actually the dividend and the divisor are MULTIPLIED by some same number. In the video, I aim to make sense of this "rule". The actual lesson continues below the video.

You can make worksheets for decimal division here.

You have learned:
• ...how to divide decimals by whole
numbers,
using either mental math
or long division.

2.04 ÷ 2 = ________

0.24 ÷ 6 = ________

5.2 ÷ 10 = ________

5.2 ÷ 100 = ________

 7 ) 1 7.2 2
• ...how to divide decimals by decimals
mentally
, thinking of how many times
it fits
:
Solve.      2.5 ÷ 0.5 = _______

0.021 ÷ 0.003 = _______

But how can we solve divisions where the divisor is a decimal, yet the divisor does not fit an even number of times into the dividend? For example: 4.6 ÷ 0.029  or  0.23 ÷ 0.07 ?
That is based on the following principle:
• We can transform any decimal division problem into a new problem with the same answer,
but with a whole-number divisor. This new problem can be solved with normal long division.

1. Solve, thinking how many times the divisor “fits into” the dividend. What can you notice?

 a.    60   ÷   20  = _______ b.     6    ÷    2   = _______c.    0.6  ÷  0.2  = _______d.  0.06 ÷ 0.02 = _______ e.  350  ÷   50  = _______ f.    35   ÷   5    = _______ g.   3.5  ÷  0.5  = _______h.  0.35 ÷ 0.05 = _______ i.  2,000 ÷  10   = _______ j.    200  ÷   1    = _______ k.     20   ÷  0.1  = _______ l.       2   ÷  0.01 = _______

 What did you notice? It is no wonder: 0.02 fits into 0.06 as many times as 2 fits into 6, as many times as 20 fits into 60, or as many times as 200 fits into 600, and so on.

2. Solve the easier of the two problems in each box. The answers to both are the same.

 a.  5 ÷ 0.2 = _______      50 ÷ 2 = ________ b.  7 ÷ 0.35 = ________      700 ÷ 35 = ________ c.  36.9 ÷ 3 = __________      0.369 ÷ 0.03 = _______

The way to transform a more difficult decimal division problem, such as 3.439 ÷ 5.6, into a problem with the same answer, but with a whole-number divisor, is this:

• Multiply both the dividend and the divisor by 10 repeatedly, until the divisor becomes a whole number. Each problem you make this way will have the same answer!

Example. Solve 0.6 ÷ 0.03.

We multiply both numbers in the
problem by 10 until the divisor
is a whole number →

 0.6 ÷ 0.03 6 ÷ 0.3 60 ÷ 3 (This is the original problem.)       (The divisor is not a whole number yet.) ← Now the divisor is a whole number!

The last problem, 60 ÷ 3, is easy to solve. The answer is 20. So, the answer to 0.6 ÷ 0.03 is also 20.

Check by multiplying:  20 × 0.03 is 20 times 3 hundredths = 60 hundredths = 0.60 = 0.6. It checks.

Example. Solve 2.104 ÷ 0.4.

We multiply both numbers in the
problem by 10 until the divisor
is a whole number →

 2.104 ÷ 0.4 21.04 ÷ 4 (This is the original problem.)← Now the divisor is a whole number!

We take the last problem, 21.04 ÷ 4, and solve it with long division →

Notice that the dividend does not have to be a whole number.

The answer is 5.26. So, the answer to the original problem, 2.104 ÷ 0.4,
is also 5.26. Check by multiplying (using the original problem):

 1  2    5.2 6 ×   0.4 2.1 0 4 ← two decimal digits ← one decimal digit ← three decimal digits

 0 5.2 6 4 ) 2 1.0 4 - 2 0 1 0 -   8 2  4

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 partially done for you.

a. 0.445 ÷ 0.05

4.45 ÷ 0.5

44.5 ÷ 5

 5 ) 4 4.5

b.  2.394 ÷ 0.7

4. Multiply both the dividend and the divisor by 10, repeatedly, until you get a whole-number divisor.
Then, divide using long division.

 a. 0.832 ÷ 0.4 b. 0.477 ÷ 0.09 c. 9.735 ÷ 0.003 d.  1.764 ÷ 0.006 e.  2.805 ÷ 0.11 f.  546.6 ÷ 1.2

You can make worksheets for decimal division here.

This lesson is taken from Maria Miller's book Math Mammoth Decimals 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

#### Math Mammoth Decimals 2

A self-teaching worktext for 5th-6th grade that covers the four operations with decimals up to three decimal digits, concentrating on decimal multiplication and division. The book also covers place value, comparing, rounding, addition and subtraction of decimals. There are a lot of mental math problems.