dividing decimals.docx

Estimating Decimal Quotients

Example 1: An electrician's apprentice earns $7.58 per hour. If he earned $316.99 last week, then about how many hours did he work?

Analysis: The phrase about how many indicates that we need to estimate. To solve this problem, we will estimate the quotient of these decimals. There are many strategies that we could use. We will use compatible numbers for this problem. The number 7.58 is close to 8 and the number 316.99 is close to 320.

Estimate:

Answer: The electrician's apprentice worked about 40 hours.

In the example above, we changed the divisor 7.58 to 8 and the dividend 316.99 to 320. The numbers 8 and 320 are compatible since they make it easy to do mental arithmetic. Let's look at some more examples of estimating decimal quotients.

Example 2: Estimate the quotient: 4,189 ÷ 6.3

Analysis: Since 42 ÷ 6 = 7, we will round 4,189 to 4,200 and 6.3 to 6 .

Estimate:

Answer: The estimated quotient of 4,189 and 6.3 is 700.

In Example 2, each factor was changed to a compatible number. The numbers 4,200 and 6 are compatible since they make it easy to do mental arithmetic. When estimating decimal quotients, it is easiest to use compatible numbers. This is the strategy we will use throughout this lesson.

Example 3: Estimate the quotient: 3,041.9 ÷ 23

Analysis: Since 30 ÷ 2 = 15, we will round 3,041.9 to 3,000 and 23 to 20.

Estimate:

Answer: The estimated quotient of 3,041.9 and 23 is 150.

Example 4: Estimate the quotient: 1,981.5 ÷ 401

Analysis: We will round 1,981.5 to 2,000 and 401 to 400.

Estimate:

Answer: The estimated quotient of 1,981.5 and 401 is 5.

Now that we have learned how to estimate decimal quotients, we can look at some word problems.

Example 5: If you bought 3 items for $219.75, then about how much does each item cost?

Analysis: To solve this problem we need to estimate the quotient of $219.75 and 3.

Estimate:

Answer: Each item costs about $70.

Example 6: The Lachance family drove on a 1,587.2 km trip in 7.5 days. Estimate the average number of km driven per day.

Analysis: To solve this problem, we need to estimate the quotient of 1,587.2 and 7.5.

Estimate:

Answer: The average number of km driven per day is about 200.

Example 7: Nine jars of candy cost $18.99. Derek estimated that he would have to pay $2.00 for one jar. If the actual cost is $2.11 for one jar, did he overestimate or underestimate? Explain your answer.

Answer: Derek underestimated because his estimated quotient of $2.00 was lower than the actual cost of $2.11.

Example 8: Twenty-four blank DVDs cost $70.80. Robin estimated that she would have to pay $3.00 for one DVD. If the actual cost is $2.95 for one DVD, did she overestimate or underestimate? Explain your answer.

Answer: Robin overestimated because her estimated quotient of $3.00 was higher than the actual cost of $2.95.

Example 9:

Taylor divided 8,326 by 9.2 and got a quotient of 905. Use estimation to determine whether his answer is reasonable or unreasonable. Explain your answer.

Answer: Taylor's answer of 905 is reasonable since 8,100 ÷ 9 = 900.

Dividing Decimalsby Whole Numbers

Example 1:

The Lachance family drove cross country on a 4,615.8 mile trip in 49 days. Find the average number of miles driven per day.

Analysis: We need to divide 4,615.8 by 49 to solve this problem.

Step 1: Estimate the quotient using compatible numbers.

Step 2: Use long division to find the quotient.

Decide where to place the first digit of the quotient.

Round to estimate the quotient digit.

Multiply, subtract and compare.

Bring down the next digit from the dividend. Continue dividing.

Bring down the 5.

4 x 49 = 196

Bring down the 8.

2 x 49 = 98

Place the decimal point in the quotient.

Check your answer: Multiply the divisor by the quotient to see if you get the dividend.

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

Our quotient of 94.2 makes sense since it is close to our estimate of 90.

Answer: The average number of miles driven per day was 94.2.

Estimating the quotient 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 90 and our quotient was 9.42, then we would know that we made a division error. Let's look at some more examples of dividing a decimal by a whole number.

Example 2:

Analysis:The divisor 8 is larger than the dividend 6.6. Therefore, we will need to place a zero in the quotient.




Round to estimate the next quotient digit.






Our quotient of 0.825 makes sense since it is close to our estimate of 0.7.

Answer: The quotient of 6.6 and 8 is 0.825

Example 3:

Analysis: The divisor 36 is larger than the dividend 18.288. Therefore, we will need to place a zero in the quotient.




Round to estimate the next quotient digit.



Write 0 as our quotient digit.

Bring down the second 8.

8 x 36 = 288.




Our quotient of 0.508 makes sense since it is close to our estimate of 0.5.

Answer: The quotient of 18.288 and 36 is 0.508.

In Examples 1 to 3, we used long division and showed each step. When dividing a decimal by a whole number, we use the following procedure:

1. Estimate the quotient.2. Perform the division.

a. Remember to place a zero in the quotient when the divisor is larger than the dividend.

b. Place the decimal point in your quotient.c. Check your answer: Multiply the divisor by the quotient to

see if you get the dividend.3. Compare your estimate with your quotient to verify that the

answer makes sense.

Example 4:

Analysis: 28 is the divisor and 355.6 is the dividend.




Round to estimate the quotient digit.



Bring down the 5.

2 x 28 = 56

Bring down the 6.

7 x 28 = 196






Example 5:

Analysis: 6 is the divisor and 6.036 is the dividend.






Bring down the 0.

0 does not divide 6

Bring down the 3.

3 does not divide 6

Bring down the 6.

6 x 6 = 36






Example 6: Look for a pattern. Then find each quotient using mental arithmetic.

36 ÷ 8 = 4.5

3.6 ÷ 8 =

0.36 ÷ 8 =

0.036 ÷ 8 =

Answer: 36 ÷ 8 = 4.5

3.6 ÷ 8 = 0.45

0.36 ÷ 8 = 0.045

0.036 ÷ 8 = 0.0045

Example 7: Sandy has 5.8 kg of coffee. If she puts the coffee into 8 bags, how much coffee will each bag contain?

Answer: Each bag will contain 0.725 kg of coffee.

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

1. Estimate the quotient.2. Perform the division.

a. Remember to place a zero in the quotient when the divisor is larger than the dividend.

b. Place the decimal point in your quotient.c. Check your answer: Multiply the divisor by the

quotient to see if you get the dividend.3. Compare your estimate with your quotient to verify

that the answer makes sense

Division of a Decimal by another Decimal

To divide a decimal by another decimal:

Move the decimal point in the divisor to the right until it is a whole number. Move the decimal point in the dividend to the right by the same number of places as the

decimal point was moved to make the divisor a whole number. Then divide the new dividend by the new divisor

Example 36

Solution:

Alternatively, we can divide as follows:

Note:

Example 37

Solution:

Alternatively, we can divide as follows:

Note:

Remember? When you multiply whole numbers by 10, 100, 1000, and so on (powers of ten),you can simply “tag” as many zeros on the product as there are in the factor 10, 100, 1000 etc.

There is a similar shortcut for multiplying decimal numbers by numbers such as 10, 100, and 1000:Move the decimal point to the right as many places as there are zeros in the factor.

10 × 0

. 4 9 = 04.9 =

4.9

Move the decimal point one step to the right(10 has one zero).

100 × 2

. 6 5 = 265. = 265

Move the decimal point twosteps to the right (100 hastwo zeros). The number 265.is 265 (as shown above).

1000 × 0 . 3 7 0 = 3

70. = 370

1000 means we move the point three steps. Write a zero at the end of 0.37 so that the decimal point can “jump over to” that place.

1. Multiply.

a. 10 × 0.04 = ________

b. 100 × 0.04 = ________

c. 1000 × 0.04 = ________

d. 10 × 0.56 = ________

e. 100 × 0.56 = ________

f. 1000 × 0.56 = ________

g. 10 × 0.048 = ________

h. 100 × 0.048 = ________

i. 1000 × 0.048 = ________

Another helpful shortcut! Since 100 × 2 = 200, obviously the answer to 100 × 2.105 will be a little more than 200. Hence, you can just write the digits2105 and put the decimal point so that the answer is 200-something: 210.5.

2. Let’s practice some more.

a. 100 × 5.439 = ________

b. 100 × 4.03 = ________

c. 1000 × 3.06 = ________

d. 100 × 30.54 = ________

e. 30.73 × 10 = ________

f. 93.103 × 100 = ________

105 × 0. 1 2 0 0 0

= 12000. =

12,000

105 = 100,000 has five zeros. Again, write additional zeros so that the decimal point can “jump over to” those places.

3. Now let’s practice using powers of ten.

a. 102 × 0.007 = _____________

103 × 2.01 = _____________

105 × 4.1 = ______________

b. 105 × 41.59 = _____________

3.06 × 104 = ______________

0.046 × 106 = _____________

The shortcut for division by 10, 100 and 1000 (powers of ten) is similar. Can you guess it?

Move the decimal point to the ( left / right ) for as many places (steps) as there are

____________________________ in the factor 10, 100, or 1000.

0 0 2 . 8 ÷ 100 =

0.028

Move the decimal point two steps to the ____________. You need towrite zeros in front of the number.

0 0 0 5 6. ÷ 104 = 0.0056

Move the decimal point four steps to the ____________. You need to write zeros in front of the number.

4. Divide.

a. 0.4 ÷ 10 = ________

0.4 ÷ 100 = ________

4.4 ÷ 100 = ________

b. 15.4 ÷ 100 = ________

21.03 ÷ 10 = ________

0.39 ÷ 10 = ________

c. 5.6 ÷ 10 = ________

34.9 ÷ 100 = ________

230 ÷ 1000 = ________

5. Now let’s practice using powers of ten.

a. 0.7 ÷ 102 = _____________

45.3 ÷ 103 = _____________

568 ÷ 105 = _____________

b. 2.1 ÷ 104 = _____________

4,500 ÷ 106 = _____________

9.13 ÷ 103 = _____________

Why does this SHORTCUT work?

When 0.01 (a hundredth) is multiplied by ten, we get ten hundredths, which is equal to one tenth. Or, 10 × 0.01 = 0.1.

The entire number moved one “slot” to the left on the place value chart. This looks like moving the decimal point in the number to the right.

O t h th

0 . 0 1

O t h th

0 . 1

A hundred times two tenths islike multiplying each tenth by10, and by 10 again. Ten timestwo-tenths gives us two, and

T O t h th

0 0 . 2

T O t h th

When 3.915 is multiplied by100, we get 391.5. Each partof the number (3, 9 tenths,1 hundredth, 5 thousandths)

H T O t h th

3 . 9 1 5

H T O t h th

ten times that gives us 20.

Again, it is like moving the

2 0 .

number over two “slots” to the left in the place value chart, or moving a decimal point in 0.2, two steps tothe right.

is multiplied by 100, so eachone of those moves two “slots” in the place value

3 9 1 . 5

chart. This is identical to thinking that the decimal point moves two steps to the right.

The similar shortcut for division works because division is the opposite operation of multiplication—it “undoes” multiplication. If we move the decimal point to the right when multiplying by 10, 100, 1000 and so on, then it is quite natural that the rule for division would work the “opposite” way.

Fractions vs. division. If we move the decimal point to solve 6 ÷ 100, we get:

0 0 6 . 0 ÷ 100 = 0.060 = 0.06

Let’s write 6 ÷ 100 using the fraction line: it is 6

100 or 6 hundredths, which is

written 0.06 as a decimal. Therefore, in this case you do not need the “shortcut,” but you canjust think of fractions and decimals. These kinds of “connections” make mathematics so neat!

6. Divide. Think of fractions to decimals, or use the shortcut. Compare the problems in each box!

a. 2

100 =

2.1

100 =

b. 49

1000 =

490

1000 =

c. 6

10 =

6.5

10 =

d. 5

10 =

5.04

10 =

e. 4.7

10 =

4.7

100 =

f. 72

100 =

72.9

100 =

7. A 10-lb sack of nuts costs $72. How much does one pound cost?

8. Find the price of 100 ping-pong balls if one ping pong ball costs $0.89.

Thinking more about fractions and decimals

If we divide any whole number by 1,000, the answer will have thousandths or three decimal digits. This makes it easy to divide whole numbers by 1,000: simply copy thedividend as your answer (without the commas), and then make it have three decimal digits:

Examples: 819,302

1000 = 819.302

41,300

1000 = 41.300 = 41.3

8,000

1000 = 8.000 = 8

Notice in the last two cases, we can simplify the results: 41.300 to 41.3 and 8.000 to 8.

9. Divide whole numbers by 1000. Simplify the final answer by dropping any ending decimal zeros.

a. 239

1000 = b.

35,403

1000 = c.

67

1000 =

d. 263,000

1000 = e.

3,890

1000 = f.

1,692,400

1000 =

g. 12,560,000

1000 = h.

9

1000 = i.

506,940

1000 =

Similarly:

If you divide any whole number by 10, copy the dividend and make it have one decimal digit.

If you divide any whole number by 100, copy the dividend and make it have two decimal digits.

Examples: 72

10 = 7.2

3,090

100 = 30.90 = 30.9

74,992

100

= 749.92

82,000

10

= 8200.0 = 8,200

10. Divide whole numbers by 10 and 100.

a. 239

100 = d.

89,803

100 = g.

69

10 =

b. 239

10 = e.

26,600

100 = h.

69

100 =

c. 23,133

100 = f.

3,402

100 = i.

9

10 =

11. Find one-tenth of... a. $8 b. $25.50 c. $126

12. Find one-hundredth of... a. $78 b. $4 c. $390

13. A pair of shoes that cost $29 was discounted by 3/10 of its price. What is the new price? (Hint: First find 1/10 of the price.)

14. Find the discounted price:

a. A bike that costs $126 is discounted by 2/10 of its price.

b. A $45 cell phone is discounted by 5/100 of its price. (Hint: First find 1/100 of the price.)

15. One-hundredth of a certain number is 0.03. What is the number?

16. Which vacuum cleaner ends up being cheaper? Model A, with the initial price $86.90, is discounted by 3/10 of its price. Model B costs $75 now, but you will get a discount of 1/4 of its price.

An important tip

In the problem ____ × 3.09 = 309, the number 3 becomes 300, so obviouslythe missing factor is 100. You do not even have to consider the decimal point!

The same works with division, too. In the problem 7,209 ÷ _____ = 7.209, the missing divisoris one thousand, because the value of the digit 7 was first 7000, and then it became 7.

Of course, in some problems it will be easier to think in terms of “moving the decimal point.”

17. It is time for some final practice. Find the missing numbers. Match the letter of each problem with the right answer in the boxes, and solve the riddle. There are two sets of boxes. The first boxes belong to the first set of exercises, and the latter boxes belong to the latter set.

Why didn’t 7 understand what 3.14 was talking about?

E ____ × 0.04 = 40

D ____ × 9.381 = 938.1

H 1,000 × 4.20 =

D ____ × 7.31 = 731

T ____ × 0.075 = 0.75

I 10 × 3.55 = ______

N 100 × ______ = 4.2

S 1,000 × ______ = 355

E ____ × 60.15 = 60,150

4,200 1000 100 35.5 100 0.042 10 0.355 1000 1000

’

T _____ ÷ 100 = 0.42

P _____ ÷ 10 = 2.3

N _____ ÷ 1000 = 4.2

H

100 = 2.3

I

10 = 0.42

S 0.31 ÷ _____ = 0.031

O 4,360 ÷ _____ = 4.36

I 304.5 ÷ _____ = 3.045

230 100 10 23 1000 4.2 4,200 42

Decimal Division By A Whole NumberDividing a decimal by a whole number can be done reliably (that is, with minimum chance of making mistakes) using long division. Of course, once you've had enough practice, you can use shorter and more advance techniques to get the answers faster.

You'll see how to do long division in the examples below. And yeah, be sure to line up the decimal points vertically.

ExampleDivide the following.(a) 73.8 ÷ 12(b) 143.55 ÷ 15

Solution

Notice that the decimal points are in line. No mis-alignments like shown below.

Decimal Division By A DecimalWhen dividing a decimal with a decimal, we can reuse the techniques we learned from dividing a decimal by a whole number. Simply change the divisor (at the right side of the ÷) into a whole number first by multiplying both divisor and dividend with the same number, then proceed with the long division.

ExampleDivide the following.(a) 24.032 ÷ 16(b) 0.304 8 ÷ 0.48Solution(a) First, convert the denominator (1.6) into a whole number by multiplying both the numerator (24.032) and denominator by 10.

Therefore, 24.032 ÷ 1.6 = 15.02

Therefore, 0.304 8 ÷ 0.48 = 0.635

Decimal Division By The Powers of 10Dividing decimals by powers of 10 (e.g. 0.001, 0.01, 0.1, 10, 100, 1000, and so on) is a special case. You can use long division to solve it, but it's not the fastest and most efficient way to do it. Here's a method that can only be used in this case.

When dividing a decimal with a power of 10, shift the decimal to the left according to the number of zeros in the power of 10. For instance,

When dividing a decimal with 0.1, 0.01 or 0.001, shift the decimal point to the right according to the number of decimal places in 0.1, 0.01 or 0.001. For instance,

ExampleDivide the following.(a) 2.033 ÷ 1000(b) 19.253 ÷ 0.001

Solution(a) 2.033 ÷ 1000 = 0.00233(b) 19.253 ÷ 0.001 = 19253

Here's a Tip...When multiplying or dividing a decimal by a power of 10 (10, 100, 1000, …) or a decimal like 0.1, 0.01 or 0.001, the decimal point can be shifted as follows:

Decimal Division By A FractionIt's not difficult to divide a decimal by a fraction. You'll first need to change “÷” into “×”, `then change the fraction into its reciprocal. It's a fancy way of saying that you need to flip the fraction upside down (numerator becomes the denominator, and vice versa). Solve it after that. See the example below.

ExampleDivide the following.

Solution

Dividing Decimals Word ProblemsExampleA rope is 32.16 m long. It is cut into identical pieces, each measuring 8.04 m long. How many pieces were cut from the rope?

Solution

Solving More Decimal

Word Problems

Example 1: School lunches cost $14.50 per week. About how much would 15.5 weeks of lunches cost?

Analysis: We need to estimate the product of $14.50 and 15.5. To do this, we will round one factor up and one factor down.

Estimate:

Answer: The cost of 15.5 weeks of school lunches would be about $200.

Example 2: A student earns $11.75 per hour for gardening. If she worked 21 hours this month, then how much did she earn?

Analysis: To solve this problem, we will multiply $11.75 by 21.

Multiply:

Answer: The student will earn $246.75 for gardening this month.

Example 3: Rick's car gets 29.7 miles per gallon on the highway. If his fuel tank holds 10.45 gallons, then how far can he travel on one full tank of gas?

Analysis: To solve this problem, we will multiply 29.7 by 10.45

Multiply:

Answer: Rick can travel 310.365 miles with one full tank of gas.

Example 4: A member of the school track team ran for a total of 179.3 miles in practice over 61.5 days. About how many miles did he average per day?

Analysis: We need to estimate the quotient of 179.3 and 61.5.

Estimate:

Answer: He averaged about 3 miles per day.

Example 5: A store owner has 7.11 lbs. of candy. If she puts the candy into 9 jars, how much candy will each jar contain?

Analysis: We will divide 7.11 lbs. by 9 to solve this problem.

Divide:

Answer: Each jar will contain 0.79 lbs. of candy.

Example 6: Paul will pay for his new car in 36 monthly payments. If his car loan is for $19,061, then how much will Paul pay each month? Round your answer to nearest cent.

Analysis: To solve this problem, we will divide $19,061.00 by 36, then round the quotient to the nearest cent (hundredth).

Divide:

Answer: Paul will make 36 monthly payments of $529.47 each.

Example 7: What is the average speed in miles per hour of a car that travels 956.4 miles in 15.9 hours? Round your answer to the nearest tenth.

Analysis: We will divide 956.4 by 15.9, then round the quotient to the nearest tenth.

Step 1:

Step 2:

Answer: Rounded to the nearest tenth, the average speed of the car is 60.2 miles per hour.

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dividing decimals.docx