Multiply Whole Numbers And Decimals Homework 13.2

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 = ________

 
 

)

 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

)

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

   

)

 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.

Download ($6.25). Also available as a printed copy.

=> Learn more and see the free samples!




Problem 1: Do you see a pattern in each table below? (Mouse over the red text)

Table 1
0.2658 x 1 = 0.2658
0.2658 x 10 = 2.658
0.2658 x 100 = 26.58
0.2658 x 1,000 = 265.8
0.2658 x 10,000 = 2,658.
0.2658 x 100,000 = 26,580.
0.2658 x 1,000,000 = 265,800.
Table 2
265,800. ÷ 1 = 265,800.
265,800. ÷ 10 = 26,580.
265,800. ÷ 100 = 2,658.
265,800. ÷ 1,000 = 265.8
265,800. ÷ 10,000 = 26.58
265,800. ÷ 100,000 = 2.658
265,800. ÷ 1,000,000 = 0.2658

In Table 1, we are multiplying the decimal 0.2658 by powers of 10. Each time we multiply by a power of 10, the decimal point is moved one place to the right. In Table 2, we are dividing the whole number 265,800. by powers of 10. Each time we divide by a power of 10, the decimal point is moved one place to the left. These patterns occur because a decimal is any number in our base-ten number system, and decimal places change by a factor of 10.

Problem 2: Which is greater: 56.5 x 80 or 8 x 565? Explain your answer.

Analysis: 56.5 x 80 = 56.5 x (10 x 8) = (56.5 x 10) x 8 = 565 x 8

Answer: They are equal.

As you can see in the problem above, multiplying one factor by a power of 10 and dividing the other factor by the same power of 10 maintains the equality of the expression. Thus, since 56.5 x 10 = 565, and since 80 ÷ 10 = 8, the expressions 56.5 x 80 and 8 x 565 are equal. Now that we have seen these patterns, we can look at some more problems.

Example 1: An electrician earns $18.75 per hour. If he worked 200 hours this month, then how much did he earn?

Analysis: The electrician earns $18.75 for each hour worked. For 200 hours of work, he will earn $18.75, a total of 200 times. We can multiply to solve this problem.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare the estimate with the product to verify that your answer makes sense.

Our product of $3,750.00 makes sense since it is close to our estimate of $4,000.00

Answer: The electrician earned $3,750.00 for 200 hours of work.

When multiplying a decimal by a whole number, placement of the decimal point is very important. Since there are two decimal digits in the factor $18.75, there must be two decimal digits in the product $3,750.00. This is because hundredths x whole number = hundredths. Estimating the product lets us verify that the placement of the decimal point is correct, and that we have a reasonable answer. For example, if our estimate was $4,000.00 and our product was $375.00, then we would know that we made a multiplication error. Let's look at some more examples of multiplying a decimal by a whole number.

Example 2: Multiply: 22.6 x 38

Analysis: There is one decimal digit in the factor 22.6. The whole number 38 is not a multiple of 10.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare your estimate with your product to verify that your answer makes sense.

Our product of 858.8 makes sense since it is close to our estimate of 800.

Answer: The product of 22.6 and 38 is 858.8.

In Example1, the whole number 200 is a multiple of 10. However, In Example 2, the whole number 38 is not a multiple of 10, which led to partial products when we multiplied. Since there is one decimal digit in the factor 22.6, there must be one decimal digit in the product. Perhaps you are wondering why this is so. When we ignored the decimal point in Step 2, we really moved it one place to the right (22.6 x 10 = 226.). Since we multiplied 22.6 by a power of 10, we need to compensate to get the right answer. To do this, we must divide by that power of 10 when we place the decimal point in our answer: Start from the right of the last digit in the product, and move the decimal point one place to the left. Let's look at another example.

Example 3: Multiply: 427 x 0.037

Analysis: There are three decimal digits in the factor 0.037.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare your estimate with your product to verify that your answer makes sense.

Our product of 15.799 makes sense since it is close to our estimate of 16.

Answer: The product of 427 and 0.037 is 15.799.

Look at the example above. Since there are three decimal digits in the factor 0.037, there must be three decimal digits in the product. When we ignored the decimal point in Step 2, we really moved it three places to the right (0.037 x 1,000 = 37.) Since we multiplied 0.037 by a power of 10, we need to  compensate to get the right answer. To do this, we must divide by that power when we place the decimal point in our answer: Start from the right of the last digit in the product, and move the decimal point three places to the left. Let's look at another example.

Example 4: Multiply: 0.874 x 401

Analysis: There are three decimal digits in the factor 0.874.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare your estimate with your product to verify that your answer makes sense.

Our product of 350.474 makes sense since it is close to our estimate of 400.

Answer: The product of 401 and 0.874 is 350.474.

Note that in Example 4, there are three decimal digits in the factor 0.874 and three decimal digits in the product. Since there is a zero in the tens place of the number 401, the second partial product consisted of zeros. Let's look at some more examples.

Example 5: Multiply: 40 x 3.5

Analysis: There is one decimal digit in the factor 3.5.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare your estimate with your product to verify that your answer makes sense.

Our product of 140.0 makes sense since the product of 40 and 3.5 ranges from 120 to 160.

Answer: The product of 40 and 3.5 is 140.0


Example 6: Multiply: 0.96 x 91

Analysis: There are two decimal digits in the factor 0.96.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare your estimate with your product to verify that your answer makes sense.

Our product of 87.36 makes sense since it is close to our estimate of 90.

Answer: The product of 0.96 and 91 is 87.36.


Example 7: Multiply: 6,785 x 0.001

Analysis: There are three decimal digits in the factor 0.001.

Step 1: Estimate the product.

Step 2: Multiply to find the product.

Multiply these numbers as if they were both whole numbers. Ignore the decimal point.

Compensate by placing the decimal point in the product.

Step 3: Compare your estimate with your product to verify that your answer makes sense.

Our product of 6.785 makes sense since it is close to our estimate of 7.

Answer: The product of 6,785 and .001 is 6.785.


Example 8: Look for a pattern. Then find each product using mental arithmetic.

35 x 698 = 24,430.
3.5 x 698 = 
0.35 x 698 = 
35 x 0.698 = 

Answer:

35 x 698 = 24,430.
3.5 x 698 = 2,443.
0.35 x 698 = 244.3
35 x 0.698 = 24.43

Example 9: What do each of these numbers have in common: 40, 0.001, 200?

Analysis: 

40 = 4 x 10
0.001 = 
200 = 2 x 100

Answer: Each number can be written as the product of a single digit and a power of 10.


Example 10: Gold costs $802.70 per ounce. How much would 4 ounces cost?

Answer: Four ounces of gold would cost $3,210.80.


Summary: When multiplying a decimal by a whole number, we use the following procedure:

  1. Estimate the product.
  2. Multiply to find the product.
    1. Multiply these numbers as if they were both whole numbers. Ignore the decimal point.
    2. Compensate by placing the decimal point in your product.
  3. Compare your estimate with your product to verify that your answer makes sense.

Estimating the product before we multiply lets us verify that the placement of the decimal point is correct, and that we have a reasonable answer. When we ignore a decimal point, we have really moved it to the right. Since we multiplied by a power of 10, we need to compensate to get the right answer. To do this, we must divide by that same power of 10 when we place the decimal point in our product: Start from the right of the last digit in the product, and move the decimal point the same number of places to the left.


Exercises

Directions: Read each question below. You may use paper and pencil to help you multiply. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

1.Multiply:  3.3 x 9
2.Multiply:  928 x 0.17
3.Multiply: $6.45 x 96
4.Multiply: 0.356 x 93
5.Danica Patrick can travel at 154.67 miles per hour in her race car. How far can she travel in 3 hours?

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