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Section 5.5 Dividing Decimals
Objectives To successfully complete this section, In this section, you will learn to: you need to understand:
- Divide decimal numbers by whole numbers. • Dividing whole numbers (1.4)
- Divide decimal numbers by decimals. • Repeating decimals (5.1)
- Write fractions as decimal numbers. • Terminating decimals (5.1)
- Divide decimal numbers by powers of 10. • Writing decimal fractions as decimals (5.1)
- Multiplying decimals by powers of 10 (5.4)
INTRODUCTION
We know that 40 ÷ 4 = 10. What about 36 ÷ 4?
If we think about it, 36 is a little bit less than 40, so the quotient of 36 ÷ 4 should be a little bit less than 10, and it is: 36 ÷ 4 = 9.
Likewise, the quotient of 52 ÷ 4 should be a little bit more than 10 13 We can check this by because 52 is a little bit more than 40. In fact, 52 ÷ 4 = 13: 4 52 multiplying:
We also know that 4 ÷ 4 = 1. What could we say about 5.2 ÷ 4? Using the same reasoning, because 5.2 is a little bit more than 4, the quotient of 5.2 ÷ 4 should be a little bit more than 1, and it is: 5.2 ÷ 4 = 1.3.
The point is this: when we divide a decimal number by a whole 1.3 We can check this number, the quotient will also be a decimal number. 4 5.2 by multiplying:
DIVIDING DECIMAL NUMBERS BY WHOLE NUMBERS
Dividing Decimal Numbers by Whole Numbers Using Long Division
- Place a decimal point in the quotient directly above the decimal point in the dividend.
- Divide as if each were a whole number.
- Place zeros at the end of the dividend if needed.
quo.tient divisor div.idend!
Example 1: Divide using long division: 6.52 ÷ 4 Procedure: When the divisor (4) is a whole number and the dividend (6.52) has a decimal point in it: (1) we place a decimal point in the quotient directly above the decimal point in the dividend; and (2) we ignore the decimal point and divide as if each were a whole number.
1.63 Notice that the decimal point in the quotient is Answer: 4 6.52 directly above the decimal point in the dividend.
- 4 25 Check by multiplying:
- 24 6.52 ÷ 4 = 1.63 1. 12 x 4
- 12 6. 0
YTI #1 Divide using long division. Use Example 1 as a guide.
a) 6.92 ÷ 4 b) 10.482 ÷ 6 c) 50.4 ÷ 8
Recall from Section 1.4 that if the divisor doesn’t divide into the first digit, we can place a 0 above the first digit and then continue to divide into the first two digits, and so on.
Example 3: Divide using long division: 1.37 ÷ 2
Procedure: Extend the number of decimal places in the dividend by placing as many zeros at the end of the decimal as needed.
0.685 We need to place only one extra zero at the end of the dividend. Answer: 2 1.3700 However, it’s okay to place more (two extra zeros are shown
- 1 2 here). Usually, we don’t know ahead of time how many 17 zeros we will actually need. Sometimes, we may need to
- 16 add more than we originally thought. 10
- 10 Once we get a remainder of zero, we’re finished. 0 1.37 ÷ 2 = 0.
YTI #3 Divide using long division. Use Example 3 as a guide. (In each of these, you’ll need to place at least one zero at the end of the dividend.)
a) 3.7 ÷ 5 b) 0.5 4 c) 0.77 ÷ 8
Think about it #1: When dividing into a decimal, when is it necessary to add zeros to the end of the dividend?
Sometimes we can place as many zeros as we wish, but we never get a remainder of 0. When this happens, the quotient is a repeating decimal, as you will see in the next example. The repeating pattern will begin to become clear when you get a recurring remainder. Once you see the pattern, you can stop dividing and place a bar over the repeating digits (the repetend), just as you did in Section 5.1.
Example 4: Divide using long division: 2.5 ÷ 3 Procedure: You can probably see that—temporarily ignoring the decimal—3 will not divide evenly into 25. We will need to extend the number of decimal places in the dividend by placing some zeros at the end. This time, however, we’re going to get a repeating decimal. When you recognize that the pattern is repeating, write the quotient with the bar over the repeating part.
Answer: 3 2.5000 Place plenty of zeros at the end of 2.5.
- 2 4 10 The remainder is 1; bring down the 0.
- 9 10 The remainder is 1; bring down the 0.
- 9 1 The remainder of 1 is recurring, let’s stop.
2.5 ÷ 3 = 0.8 3
Sometimes the pattern takes a little longer to develop.
quo.tient divisor div.idend!
DIVIDING WHEN THE DIVISOR CONTAINS A DECIMAL
To this point, you have learned to divide a decimal number by a whole number, such as 6.52 ÷ 4 (as was demonstrated in Example 1). What if we need to divide 6.52 by the decimal number 0. instead of the whole number 4?
quotient divisor dividend dividend ÷ divisor = quotient dividend divisor = quotient
To answer this question, we will use these three ideas to help us develop a procedure for dividing by a decimal number:
(1) Any division can be written as a fraction: 6.52 ÷ 0.4 =
( 2 ) We can multiply any decimal number by a power of 10, moving the decimal point to the right:
6.52 x 10 = 65.
and 0.4 x 10 = 4
(3) We can multiply any fraction by a form of 1 without changing the value:
0.4 x^
10 =^
which is 65.2 ÷ 4.
Notice that we have changed 6.52 ÷ 0.4 into 65.2 ÷ 4 by adjusting the decimal point in each number, moving it one place to the right:
The point is this: If the divisor is a decimal number, then we can make it a whole number by moving the decimal point in both the dividend and the divisor by the same number of places.
In 6.52 ÷ 0.4, we can move each decimal point one place to the right in one of three settings:
6. 5 2 ÷! 0. 4
6 5. 2 ÷ 4
Standard division Place the decimal point above
0. 4. 6. 5. 2 the new location right away.
Long division form
Make the divisor a whole number.
Fractional form
Make the denominator a whole number.
Whichever setting you choose, the actual division should be done using long division.
Example 6: Divide. a) 0.741 ÷ 0.3 b)
Procedure: First adjust each number so that the divisor is a whole number:
a) Because the divisor has one decimal b) Because the divisor has two decimal place, move each decimal point places, move each decimal point one place to the right: two places to the right:
Now set up the long division using these adjusted numbers.
2.47 5. Answer: a) 3 7.41 b) 6 35.
- 6 – 30 14 5 4
- 12 – 5 4 21 0
- 21 0
0.741 ÷ 0.3 = 2.47 0.354 ÷ 0.06 = 5.
Here are two examples showing the moving of the decimal point directly:
Dividing by 10, which has only one zero, has the effect of moving the decimal point of 3.65 one place to the left. We
.^ can write in 0 as the whole number.
a) 3.65 10 = 0.
Dividing by 1,000 suggests that we need to move the decimal point in 45.2 three places to the left. However, there aren’t enough whole numbers in 45.2, so we must place some zeros at the beginning of it so that there is someplace to move the decimal point to.
b) 45.2 ÷ 1,000 = 0.
00045.2 ÷ 1,000 = 0.
Here is the procedure that results from all of this:
Dividing a Decimal by a Power of Ten (1) Count the number of zeros in the power of ten. (2) Move the decimal point that many places to the left.
YTI #6 Divide by moving the decimal point the appropriate number of places to the left.
a) 8.7 ÷ 10 b)
10 c)^
100 d)^ 20.9^ ÷^ 1,
WRITING FRACTIONS AS DECIMAL NUMBERS
Any fraction in which both the numerator and denominator are whole numbers can be written as a decimal number, either a terminating decimal or a repeating decimal.
Start by writing the fraction as division, then divide using long division. For example,
4 = 3^ ÷^ 4.
Because 4 won’t divide into 3 directly, it is necessary to write 3 as 3.000 (possibly with more or with fewer ending zeros) in order to divide.
Example 7: Find the decimal equivalent of each fraction. a)
4 b)^
Procedure: Write the fraction as division and then divide using long division. Write each numerator with a decimal point followed by some ending zeros.
Answer: a)
4 can be thought of as 3^ ÷^4 b)^
6 can be thought of as 5^ ÷^6
0.75 0. 4 3.000 6 5.
- 2 8 – 4 8 20 20
- 20 – 18 0 20
- 18 20 3 4 =^ 0.75, a terminating decimal^
6 = 0.8^3 , a repeating decimal
YTI #7 Find the decimal equivalent of each fraction. Use Example 7 as a guide.
a)
5 b)^
9 c)^
YTI #8 Divide. Use Example 8 as a guide.
a) 0.523 ÷ 8 b)
0.3 c)^ 0.079^ ÷^ 0.
d) 7 ÷ 0.004 e) 4.1062 ÷ 0.009 f)
Below is a chart of the equivalent decimal values of some common fractions.
Decimal Equivalents of Some Fractions
2 = 0.^
5 = 0.^
6 = 0.8333...^
3 = 0.333...^
5 = 0.^
8 = 0.^
3 = 0.666...^
5 = 0.^
8 = 0.^
4 = 0.^
5 = 0.^
8 = 0.^
4 = 0.^
6 = 0.1666...^
8 = 0.^
Think about it #2: Based on the chart above,
a) What is the decimal equivalent of
9? __________________
b) What is the decimal equivalent of
9? __________________
c) Does the decimal equivalent of 39 appear somewhere else in the chart? Explain your answer.
You Try It Answers: Section 5.
YTI #1: a) 1.73 b) 1.747 c) 6.
YTI #2: a) 0.157 b) 0.007 c) 0.
YTI #3: a) 0.74 b) 0.125 c) 0.
YTI #4: a) 0.6 7 b) 0.071 6 c) 0.47 09
YTI #5: a) 0.7 b) 97 c) 680
YTI #6: a) 0.87 b) 3.46 c) 0.764 d) 0.
YTI #7: a) 1.2 b) 0. 4 c) 0.
YTI #8: a) 0.065375 b) 27.7 3 c) 1.
d) 1,750 e) 456.2 4 f) 3.4 09
20. 0.676 ÷ 8 21.
3^22.^
23. 4.8 ÷ 9 24. 5.7 ÷ 9 25. 3.4 ÷ 11
11^27.^ 0.49^ ÷^15^28.^ 0.35^ ÷^6
Divide.
0.6^30.^ 3.42^ ÷^ 0.9^^31.^ 0.468^ ÷^ 0.
32. 0.736 ÷ 0.8 33. 0.198 ÷ 0.05 34. 0.267 ÷ 0.
35. 5.6 ÷ 0.08 36. 3.6 ÷ 0.09 37.
38. 125 ÷ 0.05 39. 0.24 ÷ 0.012 40. 0.96 ÷ 0.
41. 0. 126 ÷ 0.3 42.
0.5^43.^ 0.07128^ ÷^ 0.
44. 0.0579 ÷ 0.3 45. 0.0455 ÷ 0.09 46. 0.0316 ÷ 0.
0.11^48.^ 0.038^ ÷^ 0.11^^49.^ 0.0015^ ÷^ 0.
50. 0.0053 ÷ 0.004 51. 0.062 ÷ 0.25 52.
Divide by moving the decimal point the appropriate number of places to the left.
53. 25.8 ÷ 10 54. 90.3 ÷ 10 55.
10^57.^ 74.9^ ÷^100^58.^ 2.36^ ÷^100
100^60.^
100^61.^ 0.45^ ÷^100
62. 1.03 ÷ 100 63. 42.1 ÷ 1,000 64. 8.2 ÷ 1,
Find the decimal equivalent of each fraction.
2^66.^
2^67.^
4^68.^
5^70.^
5^71.^
8^72.^
73.^1320 74.^2920 75.^1725 76.^4325
6^78.^
6^79.^
9^80.^
11^82.^
11^83.^
15^84.^
37^86.^
37^87.^
7^86.^
Find the prime factorization of the denominator of each of these fractions. What does each prime factorization have in common? 2 3 · 3 ,^
2 · 7 ,^
2 · 11 ,^
2 · 2 · 3 ,^
5 · 7 ,^
5 · 11 ,^
2 · 3 · 11 ,^
2 · 2 · 3 · 7 ,^
Each denominator has prime factors other than 2 and 5.
103. Based on your answers to #97 and #98, how can you determine if a fraction will be a terminating decimal or a repeating decimal? If a fraction’s denominator has only 2 or 5 as prime factors, then the fraction can be written as a terminating decimal. If a fraction’s denominator has a prime factor other than 2 or 5, then the fraction can be written as a repeating decimal.
