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DEFINITION. A decimal is a number that represents a fraction whose denominator is a power of ten. That is, the denominator is 10 or 100 or 1000 or 10,000, etc.

Being a "power of ten" simply means that the denominator is 10 multiplied by itself a certain number of times. The "power" shows how many times 10 is multiplied by itself to obtain the number. The number 1000, for example, is 10 x 10 x 10. This shows that 1000 is 10 multiplied by itself three times. 1000 is 10 to the third power (usually written as 10³).

a. What is the denominator of a fraction if the denominator is equal to 10 to the sixth power?

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READING AND WRITING DECIMALS. Each digit in a decimal has a place value. A decimal point (period or dot) is used to separate the whole number from the decimal numerals (fraction). Like the place values shown in Frame 1-2, each place value has a name. Like whole numbers, the value decreases by one-tenth (1/10) each time you move to the right. (Likewise, the place value increases by 10 if you go to the left.) The names of some of the place values are shown below.

Note: If the entire number has a value that is less than one (there are no whole numbers to the left of the decimal), a zero is usually placed in the ones place to make reading easier (it emphasizes the decimal point).

NOTE: Commas are not used to the right of the decimal.

Solution to Frame 3-1.

n

1,000,000

n

10

(the "n" represents the numerator.)

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As indicated in Frames 1-2 and 3-2, place values have names based upon the powers of ten. Sometimes, they are written as 10^X with the "x" being the power of ten (the number of times ten is multiplied by itself). For example, ten to the third power is one thousand (10³ = 10 x 10 x 10 = 1000).

This works for whole numbers, but how about decimals? Think about it as relating to fractions. If the denominator is 10³, for example, then the fraction would be one-tenth (1/10) multiplied by itself ten times (1/10 x 1/10 x 1/10 = 1/1000).

If the power of ten refers to whole numbers (numerators, if you will), then the power number is expressed as a positive number. If the power of ten refers to a decimal (denominator), then the power number is expressed as a negative number. Negative numbers are denoted by a minus sign; numbers with no negative symbol are assumed to be positive.

10³ = 10 x 10 x 10 = 1000 (third power; three zeros)

10^-3 = 1 _X 1 _X 1 ₌ 1 ₌ 0.001 (negative three; three places

10 10 10 1000 to the right of the decimal)

Solution to Frame 3-2.

hundred-millionths

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If you combine the information in Frames 1-2, 3-2, and 3-3, you might come up with something like this:

6 5 4 3 2 1 0 · 1 2 3 4 5 6

| | | | | | | | | | | | |

10⁶ 10⁵ 10⁴ 10³ 10² 10¹ 10^? 10^-1 10^-2 10^-3 10^-4 10^-5 10^-6

Everything falls into place, except for the "ones" value place, which is also referred to as the "units" place.

What do you think the "?" (unknown power of 10) might be?

Solution to Frame 3-3.

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10⁰, or any number to the zero power, is defined as that number divided by itself. Any number divided by itself is equal to 1 (10⁰ = 10/10 = 1).

As you probably noticed in Frame 3-2, the place to the right of the decimal point always ends in "ths." The decimal is read as though it were a fraction with the numerator followed by the denominator. (The denominator is the place value of the last digit.) For example:

0.46 ₌ 46

100 ,which is read, "forty-six hundredths."

NOTE: Don't forget the "ths." It is this sound which notifies you that you are dealing with a decimal instead of a whole number.

a. Write the number meaning "one hundred twenty-seven thousand."

b. Write the number meaning "one hundred twenty-seven thousandths."

Solution to Frame 3-4.

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Remember, the "ths" of the right-most digit is read. For example, 0.032 is read as "thirty-two thousandths," not as "three hundredths and two thousandths."

a. How is 0.3736 read? .

b. How is 0.000002 read? .

a. 127,000

b. 0.127 or 127

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The decimal point is read as "AND." For example, 35.362 is read as "thirty-five AND three hundred sixty-two thousandths."

a. How is 404.404 read?

b. Write fourteen and five tenths.

a. three thousand seven hundred thirty-six ten-thousandths

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CHANGING FRACTIONS TO DECIMALS. Fractions can be changed to a decimal by dividing the numerator by the denominator.

The steps for changing a fraction into a decimal form are:

(1) Write a division problem in which the numerator is divided by

the denominator.

(2) Place a decimal point to the right of the numerator.

(3) Add zeros to the right of the decimal point, as needed.

(4) Place a decimal point in the quotient DIRECTLY OVER the

decimal point in the division bracket.

(5) Divide as normal (see Frames 1-20 through 1-24).

(6) Continue dividing until your remainder is zero or until you

have reached the needed level of accuracy. [Some

problems, such as 2/3, never have a remainder of zero. You

have already run across one such problem in Frame 1-24.]

Change 7/8 to a decimal.

Solution to Frame 3-7.

a. four hundred four and four hundred four thousandths

Note: "And" can mean "decimal" or "plus." If you said "four hundred and four and four hundred and four thousandths," you would really be saying

400 + 4 + 400.004.

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Let's try a couple more problems. Carry out your division though four places to the right of the decimal (no further than the ten-thousandths place).

a. Change 1/12 to a decimal.

b. Change 8/900 to a decimal.

NOTE: If a digit in the quotient is zero and it is TO THE RIGHT OF THE DECIMAL POINT, the zero must be written in the quotient.

Solution to Frame 3-8.

0. 875

8 / 7.000

0

7 0

6 4

60

56

40

40

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If you have a mixed number and wish to convert the fraction to a decimal, then the whole number goes to the left of the decimal and the fraction goes to the right of the decimal. For example "five and one-half" (or "five and five-tenths) is written as "5.5."

a. Write 3 1/12 as a decimal (carry out to the fourth decimal place).

Solution to Frame 3-9.

a. 0.0833

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To change an improper fraction to a decimal, divide the numerator by the denominator. Remember to keep the decimal in the quotient above the decimal point in the dividend. The improper fraction 3/2 is shown below being changed to its decimal form.

1.5

2 / 3.0

2

1 0

1 0

0

Solution to Frame 3-10.

a. 3.0833

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CHANGING DECIMALS TO FRACTIONS. Frames 3-1 through 3-7 have given you the basic information you need to change a decimal to a fraction. Just put the numerator over the appropriate denominator (a power of 10).

For example: 0.045 = 45 thousandths = 45/1000

If you want the fraction reduced, divide the numerator and denominator by their common factors (whole numbers which divide into both the numerator and denominator without leaving a remainder). For example

45 ₌ 45 ) 5 ₌ 9

1000 1000 ) 5 200

a. Change 0.004 to a fraction and reduce.

Solution to Frame 3-11.

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ADDING DECIMALS. Adding decimals is much the same as the addition of whole numbers. The difference is that there is a decimal point to keep in mind. The decimals are put in a straight column; that is, DECIMAL POINTS ARE UNDER DECIMAL POINTS (see example). The decimal point is brought down to the sum, and the addition is carried on just as it is in whole number addition. Rules for carrying still apply.

₁

Example: 6.9

0.01

22.2201

29.1301

₁

NOTE: Zeros may be added after 6 . 9 0 0 0

the last number of a decimal to 0 . 0 1 0 0

help keep the digits in the proper 2 2 . 2 2 0 1

alignment (or columns) as shown 2 9 . 1 3 0 1

Add these decimals: 33.79 + 0.0097 + 2.4 + 6

Solution to Frame 3-12.

a. 4/1000 = 1/250

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Solve these problems:

a. 729.75309 - 0.0077

b. 3 - 0.003

Solution to Frame 3-13.

_{1 1}

33.79

0.0097

2.4

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Sample problem: 1.11 x 0.15

Multiply just as you do in whole numbers. 1.11 111

(Ignore the decimal for now.) x 0.15 x 15

NOTE: Normally, no space is left under 555

the decimal when working a 111

multiplication problem. 1665

Now, you need to place the decimal point in the answer. To do this, you:

(1) Count the number of digits to the right of the decimal point in

the top factor (multiplicand),

(2) Count the number of digits to the right of the decimal point in

the bottom factor (multiplier),

(3) Add the results together,

(4) Count off that many places from the RIGHT in the

PRODUCT (number at far right is one, number

immediately to its left is two, etc.) , and

(5) Place a decimal point to the left of that location.

In the example, the multiplicand (1.11) has 2 places to the right of the decimal and the multiplier (0.15) has 2 places to the right of the decimal. Adding the results (2 places + 2 places = 4 places) tells how many decimal places (places to the right of the decimal) you have in the produce. The answer then is .

Solution to Frame 3-14.

_{4 12 10}

a. 729.75309

- 0.00770

729.74539

_{2 9 9 10}

b. 3.000

0.003

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Place the DECIMAL POINT in 3.217

the product of this problem: x 4.71

3217

22519

12868

1515207

Solution to Frame 3-15.

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Work the following problems. Be sure to place the decimal point correctly in the product.

NOTE: If there are not enough digits in the product, put zeros to the left of the product until you have enough digits to place the decimal. Normally, you will also put another zero to the left of the decimal to indicate that there are no whole numbers.

a. 0.0035 b. 22.222 c. 0.001

x 3.28 x 0.11 x 0.1

Solution to Frame 3-16.

15.15207 (the multiplicand has 3 decimal places and the multiplier has 2 decimal places.

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DIVIDING DECIMALS. A very important rule in dividing by a decimal is that the divisor must be "changed" into a whole number before division is started. This is done by moving the decimal point in the divisor all the way to the right (that is, make the divisor a whole number).

For example: 0.25 / becomes 25./ by moving the decimal two places to the right. In reality, you have just multiplied the divisor by 100. In order to offset this change, you must multiply the dividend by 100 also. To do this, move the decimal in the dividend the SAME NUMBER of places to the right. Add zeros to the dividend as needed.

For example: 0.25 / 1.25 becomes 25./ 125.

If you prefer, think of the problem as a fraction in which the denominator must be changed to a whole number.

For example: 1.25 _X 100 ₌ 125

0.25 100 25

Rewrite the following problems to remove the decimal in the divisor.

3.3 / 0.066 0.0033 / 66 0.000033 / 0.0066

a. 0.0035 (4 places)

x 3.28 (2 places)

0.011480 (6 places)

(Note: When an answer has a zero in the last decimal place, the zero is often dropped -- 0.01148)

b. 22.222 (3 places)

x 0.11 (2 places)

2.44442 (5 places)

c. 0.001 (3 places)

x 0.1 (1 place)

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Complete the problems by dividing. Don't forget to put your decimal point in the quotient directly above the decimal point in the dividend. Remember: Zeros between the decimal point and the non-zero numbers to the right of the decimal must be written.

a. 3.3 / 0.066

b. 0.0033 / 66

c. 0.000033 / 0.0066

33 / 0.66

(moved decimals one place)

33 / 660000

(moved decimals four places)

33 / 6600

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For additional practice, divide "1" by:

a. 0.1

b. 0.001

c. 0.000001

a. 0.02

b. 20,000

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"ROUNDING" DECIMALS. In many cases, a large, cumbersome, accurate decimal value is not necessary. In cases when less accuracy (fewer digits to the right of the decimal point) will do, you may round (or round off) the decimal.

To make a long decimal number shorter and easier to use without losing too much accuracy, you can the number.

Solution to Frame 3-20.

a. 1/0.1 = 10

b. 1/0.001 = 1000

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Suppose jellybeans cost $1.99 per pound and you bought exactly a quarter-pound (0.25 pounds). How much money should you give the clerk? [Assume that there is no tax on the purchase.]

1.99 (2 decimal places)

0.25 (2 decimal places)

9 95

49 75 (4 decimal places)

The answer is $0.4975 . If you had changed $1.99 into pennies when you started, then you would owe 49.75 (49 3/4) pennies. You could take a penny, divide it into four equal parts, and give the clerk three of them (along with the 49¢), but I don't think the clerk will be very happy. Instead, we usually round the cost to the nearest penny. That means $0.4975 will be rounded to the nearest of a dollar (penny or cent).

Solution to Frame 3-21.

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In the previous example, you will pay either $0.49 or $0.50 (the amount just below the calculated true price and the amount just above the calculated true price). Look at the number line drawn below.

$0.49 $0.4975 $0.50

Is 0.4975 (¨ ) closer to 0.49 or 0.50?

Solution to Frame 3-22.

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Therefore, 0.4975 rounded to the nearest hundredth is 0.50. You owe the clerk $0.50 for the candy.

The previous example showed a problem that required rounding to the nearest hundredth. Other problems may involve rounding to the nearest tenth, to the nearest thousandth, to the nearest millionth, to the nearest whole number, to the nearest thousand, to the nearest billion, etc.

As long as you are dealing with whole numbers and/or decimals, you can use some basic steps to determine how to round a given number. These steps are givenin the next frame.

Remember, you will be rounding to the nearest number. The theory is that if several numbers are rounded, some will go to the higher number (round up) while others will go to the lower number (round down). If all of the original (unrounded) numbers were added together and rounded, the results should be about the same as the sum of the rounded numbers.

Solution to Frame 3-23.

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Rounding to the nearest number or value involves these steps:

(1) Determine the PLACE you want to round to (tenths,

hundredths, etc. -- call it the "place value to be retained)."

(2) Locate the digit in that place value (call this the "digit to be

rounded."

(3) Either leave that digit unchanged (round down) or increase

that digit by 1 (round up) using these rules.

(a) Locate the digit directly to the right of the digit to be

rounded.

(b) If that digit less than 5 (that is, a 0, 1, 2, 3, or 4), leave

the digit to be rounded unchanged.

(c) If that digit is 5 or more (that is, a 5, 6, 7, 8, or 9),

increase the digit to be rounded by one (1).

(4) Once you have rounded up or down, drop all of the digits to

the right of the place value to be retained (the rounded digit).

For example, round 28.034697 to the nearest thousandth.

a. What is the digit in the place value to be retained?

b. What is the digit to the immediate right of that digit?

c. Based upon the information in "b," should the digit in the place value to be retained be left unchanged or be increased by 1?

d. What is 26.034697 rounded to the nearest thousandth?

Solution to Frame 3-24.

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Round 28.034697 to the nearest:

a. hundredth

b. ten-thousandth

c. hundred-thousandth

d. whole number

e. ten

f. hundred

Solution to Frame 3-25.

a. 4 (in the thousandths position)

b. 6 (the ten-thousandths position)

c. Increased (6 is 5 or more)

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Did you have any problems? The information given below may help if you did.

a. Round 28.034697 to the nearest hundredth.

The digit to be rounded is 3. The digit to the right of the hundredths is 4, so you leave the 3 unchanged.

NOTE: Even though this digit rounded to 5 when you rounded to the thousandths position in the previous problem, you must use the actual (unrounded) digit when working this problem.

b. Round 28.034697 to the nearest ten-thousandth.

The digit to be rounded is 6. The digit to the right is 9. Round up to 7.

c. Round 28.034697 to the nearest hundred-thousandth.

The digit to be rounded is 9. The digit to the right is 7. Round up. When you add 1 to 9, you get 10. Write down the zero and carry the one. 26.03469 + 0.00001 = 26.03470. When writing the answer, you can include the zero at the end (26.03470) or drop the zero (26.0347).

d. Round 28.034697 to the nearest whole number.

The digit to be rounded is in the units (ones) position, which is 8. The digit to the right is 0 (tenths position). Round down.

e. Round 28.034697 to the nearest ten.

The digit to be rounded is in the tens position, which is 2. The digit to the right is 8 (units position). Round up. The 2 becomes 3, but you cannot just drop the digits as you do when the number to be rounded is to the right of the decimal. Although the digits are dropped, the place values to the left of the decimal must be shown. They are filled with zeroes. Digits to the right of the decimal are dropped without putting zeroes in their place.

f. Round 28.034697 to the nearest hundred

The digit to be rounded is in the hundreds position, which has no number now. Change 28.034697 to 028.034697 (adding a zero to the front does not change the value of the number). The number to be rounded is now 0. The digit to the right is 2 (tens position). Round down. The 0 remains zero. Like exercise "e" above, you put zeroes in place of the digits to the left of the decimal that are being dropped. The result is "000," which is usually written as just "0."

Solution to Frame 3-26.

a. 28.03

b. 28.0347

c. 28.03470

d. 28

e. 30

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SPECIAL ROUNDING PROCEDURES. In the preceding frames, you used rules to round to the nearest place (nearest hundredth, nearest whole number, etc.). The basic theory is that sometimes you round up and sometimes you round down, but in the end it balances out.

Some organizations, however, may use different rules. For example, suppose you are rounding off the weights of individual products to the nearest pound, then adding the weights together to determine the total weight of the shipment. One organization may not care if the estimated weight is more than the actual total weight, but will be very upset if the estimate is below the actual weight. In such a case, you may be told to round up at all times to prevent an underestimate. Likewise, you may be told to always round down by an organization that must make sure that the actual weight is not under your estimate.

Some other organizations may use modified rules of rounding. Refer back to Frame 3-23. Suppose that the amount you wished to round to the nearest cent ($0.01) had been exactly in the middle (0.$4950).

$0.49 $0.4925 $0.4950 $0.4975 $0.50

According to our rules of rounding, you would always round half-cents up to the next penny. But suppose you knew that you would have a lot of halfs (say a lot of half pounds in the above example). You might want a system to round the halfs up sometimes and round them down sometimes. One such rule is the "engineer's rule of rounding." When using this modified rule of rounding, if the digit(s) to the right of the digit to be rounded is "5" or "50", you round down if the digit to be rounded is even (0, 2, 4, 6, or 8) and round up if the digit is odd (1, 3, 5, 7, or 9).

For example, 2 1/2 (2.5) pounds rounded to the nearest pound would round to 2 pounds (the 2 is even). 3 1/2 (3.5) pounds, however, would round to 4 pounds (the 3 is odd).

If you used the engineer's rule of rounding given above to round $0.4950 to the nearest penny, the results would be (chose one -- $0.49 $0.50).

NOTE: The information presented in this frame was for your information. In this subcourse, you will only be tested on the rounding rules given in Frame 3-25 and not on any system of rounding presented in this frame.

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PERCENTS. A percent (%) is a special type of decimal form. Percent means "per one hundred." It tells how many hundredths. (Think of cents. One cent is 1/100 [1%] of a dollar.)

For example, 24 percent means 24/100, which is 0.24.

To change from a percent to a decimal, simply move the decimal point two pieces to the left. If no decimal point is shown, put one after the last digit. Add zeroes to the left of the percentage number if needed. For example, 2% = 0.02.

Change these percent forms to their decimal forms.

a. 20% =

b. 5.5% =

c. 1/8% =

d. 0.2% =

e. 350% =

Solution to Frame 3-28.

$0.50

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In math problems, the word "of" frequently indicates that you are to multiply. Solve these problems by changing the percent to a decimal and multiplying. Round any answer involving money to the nearest cent.

a. 20% of $300

b. 8.5 % of $255

c. 150% of 10

d. 1/2 % of 1000

a. 0.20 (or 0.2)

b. 0.055

c. 0.00125

(Change the fraction to a decimal form, then move the decimal point 2 places to the left.)

d. 0.002

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To change from a decimal form to a percent form, move the decimal two places to the right and add the percent symbol (%). For example, the percent form of 0.25 is 25%.

Change from the decimal form to the percent form. Add zeroes as needed.

a. 0.5 =

b. 0.153 =

c. 1.25 =

d. 0.0003 =

Solution to Frame 3-30.

a. $60 (or $60.00)

b. $21.68

c. 15

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Sometimes you are asked what percent one number is of another. For example, what percent of 20 is 5? (or 5 is what percent of 20?)

To solve, change the information to a fraction, then to a decimal, then to a percent. For example:

5/20 = 0.25 = 25%

Solve these problems. Round to the nearest hundredth of a percent, if needed.

a. What percent of 100 is 3?

b. 120 is what percent of 60?

c. 1 is what percent of 3?

Solution to Frame 3-31.

a. 50%

b. 15.3%

c. 125%

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A variation of the above problems is to tell you that a number is a certain percent of the original number, then ask you to find the original number.

For example, 25% of what number is 5? (or 5 is 25% of what number?) Let "N" stand for the original number. The question can then be restated as 25% of N is 5. The mathematical form of this statement is 25% x N = 5. You can either state the problem in decimal form or as a fraction:

0.25N = 5 or 1/4 x N = 5

NOTE: 0.25N is another way of writing (0.25)(N). 1/4 is the reduced form of 25/100.

Solution to Frame 3-32.

a. 3% (3/100)

b. 200% (120/60)

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After you have stated the problem as an equation (that is, the mathematical statements on both sides of the "=" symbol are equal), solve for N. Multiply or divide both sides of the equation by the same number or fraction in order to change one side of the equation to N (or 1N). The example can be worked as shown.

0.25N = 5 or 1/4 x N = 5

0.25N ₌ 5 N _X 4 ₌ 5 _X 4

0.25 0.25 4 1 1 1

1N = 20 4N ₌ 5 x 4

4 1

N = 20

0.20N = 30

or

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To check your answer, simply substitute your answer for N in the equation.

For example: 25% of what number is 5?

0.25 x N = 5 or 1/4 x N = 5

0.25 x 20 = 5 1/4 x 20 = 5

5 = 5 20/4 = 5

5 = 5

Check your answer to Frame 3-34.

0.20N = 30

N = 30/0.20 = 150

or

1/5 x N = 30

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