Decimal Notation
Chapter 4

OBJECTIVES

4.1

Given decimal notation, write a word name.

Convert between fraction notation and decimal notation.

Given a pair of numbers in decimal notation, tell which is larger.

Round decimal notation to the nearest thousandth, hundredth, tenth, ones, ten, hundred, or thousand.

4.2

Add using decimal notation.

Subtract using decimal notation.

Solve equations of the type x + a = b and a + x = b, where a and b may be in decimal notation.

4.3

Multiply using decimal notation.

Convert from notation like 45.7 million to standard notation, and between dollars and cents.

4.4

Divide using decimal notation.

Solve equations of the type , where a and b may be in decimal notation.

Simplify expressions using the rules for order of operations.

4.5

Convert from fraction notation to decimal notation.

Round numbers named by repeating decimals in problem solving.

Calculate using fraction and decimal notation together.

4.6

Estimate sums, differences, products, and quotients.

4.7

Solve applied problems involving decimals.

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4.1 DECIMAL NOTATION, ORDER, AND ROUNDING

In Chapter 4, we will study the use of decimal notation. The word decimal comes from the Latin word decima, meaning a tenth part. Using decimal notation, we can write 0.875 for 7/8 or 48.97 for 48 97/100.

Decimal Notation and Word Names

 

PLACE-VALUE CHART

Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

Ten-

Thousandths

Hundred-

Thousandths

100

10

1

 

Let's look at the meaning of the decimal 26.3385 min, the men's 10,000-meter run record held by Kenenisa Bekele from Ethiopia.

The decimal notation 26.3385 means

 

We read both of the meanings as: "Twenty-six and three thousand three hundred eighty-five ten-thousandths."

 

To write a word name from decimal notation: 397.085

1. Write a word name for the

whole number (the number

named to the left of the decimal

point).

397.085

Three hundred ninety-seven

2. Write the word "and" for the

decimal point.

397.085

Three hundred ninety-seven and

3. Write a word name for the

number named to the right of

the decimal point, followed by

the place value of the last digit.

397.085

Three hundred ninety-seven and

eighty-five thousandths

 

  

 

  

 

  

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CONVERTING BETWEEN DECIMAL NOTATION AND FRACTION NOTATION

Given decimal notation, we can convert to fraction notation as follows:

 

Decimal Notation

Fraction Notation

9.075

3 decimal places

3 zeros

 

To convert from decimal to fraction notation

Four and eight hundredths

4.08

 

1. Count the number of decimal places

4.08

2 places

2. Move the decimal point that many places

to the right to make a whole number.

408.

Moved 2 places

3. Write the answer over a denominator

with a 1 followed by that number of zeros.

2 zeros

 

  

 

To convert from fraction notation to decimal notation when the denominator is 10, 100, 1000, 10,000 and so on.

 

Eighty and seventy-nine thousandths

1. Count the number of zeros in the denominator

3 zeros

2. Move the decimal point that number of places

to the left. Leave off the denominator.

80.079

Move 3 places

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ORDER AND ROUNDING

ORDER

COMPARING NUMBERS IN DECIMAL NOTATION

To compare two numbers in decimal notation, start at the left and compare corresponding digits

moving from left to right. If two digits differ, the number with the larger digit is the larger of the two numbers. To ease the comparison, extra zeros can be written to the right of the last decimal place.

 

Example 1

Which of the two numbers is larger: 2.109 or 2.1

Number

Think

2.109

2.109

2.1

2.100

There is a tie in the ones place; a tie in the tenths; a tie in the hundreths; however 9 > 0. Thus, 2.109 > 2.1

 

Example 2

Which of the two numbers is larger: 2.04 or 2.039

Number

Think

2.04

2.040

2.039

2.039

There is a tie in the ones place; a tie in the tenths; however, 4 > 3. Thus, 2.04 > 2.039

 

  

 

  

 

Now we will look at  

ROUNDING

ROUNDING DECIMAL NOTATION

To round to a certain place:

a) Locate the digit in that place.

b) Consider the next digit to the right.

c) If the digit to the right is 5 or higher, round up; if the digit to the right is 4 or lower, keep the number the same.

 

PLACE VALUE CHART

Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

Ten-

Thousandths

Hundred-

Thousandths

 

Example 1

Round 14.8973 to the nearest hundredth.

a) Locate the digit in the hundredths place.

14.8973

b) Consider the next digit to the right.

14.8973

c) Since the digit, 7, is 5 or higher, round up. When we

add 1 to the 9 to round up, we carry 1 to the tenths

place since 9 + 1 = 10.

d) Note that the 0 in 14.90 indicates that the answer is

correct to the nearest hundredth.

14.90

 

Example 2

Round 3872.2459 to the nearest:

Round to:

Number

Answer

Tenths

3872.2459

3872.2

Hundreds

3872.2459

3900

Thousandths

3872.2459

3872.246

Ones

3872.2459

3872

Hundredths

3872.2459

3872.25

Thousand

3872.2459

4000

 

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4.2 ADDITION AND SUBTRACTION

Adding and subtracting with decimal notation is similar to adding and subtracting whole numbers. First we line up the decimal points so that we can add or subtract corresponding place-value digits. Then we add or subtract the digits from the right.

 

ADDITION

SUBTRACTION

Add: 789 + 23.67

 

Subtract: 37.5 - 26.74

 

ADDITION:

When you see a whole number like 789, think of money and write it as $789.00 (no dimes or pennies). If more zeros are needed, then add the zeros. Zeros are place holders showing that you have no tenths, hundredths, thousandths, and so on.

 

SUBTRACTION:

When you subtract you must add your zeros for place holders in order to borrow.

 

DO NOT ADD OR SUBTRACT THIS WAY

ADDITION NO-NOS

SUBTRACTION NO-NOS

Add: 789 + 23.67

 

Subtract: 37.5 - 26.74

 

 

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SOLVING EQUATIONS

Now let's solve equations x + a = b and a + x = b, where a and b may be in decimal notation.

 

Example 1

Solve: x + 17.78 = 56.314

Subtract 17.78 on both sides.

Check your answer in the original

equation.

 

Example 2

Solve: 8.906 + t = 23.07

Subtract 9.906 on both sides.

Check your answer in the orginal

equation.

 

Example 3

Solve: 241 + y = 2374.5

Subtract 241 on both sides.

Check your answer in the original

equation.

 

Try These

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4.3 MULTIPLICATION

Let's find the product of 2.3 x 1.12.

To understand how we find such a product, we first convert each factor to fraction notation. Next, we multiply the whole numbers 23 and 112, and then we divide by 1000.

2.3 x 1.12 = ?

 

2 decimal places in 1.12 (100ths)

1 decimal place in 2.3 (10ths)

 

 

 

3 decimal places all together. (1000ths)

 

  

 

 

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MULTIPLYING BY 0.1, 0.01, 0.001, AND SO ON

0.1 x 38

0.01 x 38

0.001 x 38

0.0001 x 38

 

  

 

MULTIPLYING BY 10, 100, 1000, AND SO ON

10 X 97.34

100 X 97.34

1000 X 97.34

10,000 X 97.34

973.4

9734

97,340

973,400

 

  

 

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4.3 Application Using Multiplication with Decimal Notation

NAMING LARGE NUMBERS

We often see notation like the following in newspapers, magazines and on television.

bulletdots.gif The largest building in the world is the Pentagon, which has 3.7 million square feet of floor space.

bulletdots.gif In 2003, consumers spent $51.7 billion on online retail products.

To understand such notation, consider the information in the following table.

Pay attention to the number of zeros in each number.

1 hundred

100

10 x 10

102

1 thousand

1000

10 x 10 x 10

103

1 million

1,000,000

10 x 10 x 10 x 10 x 10 x 10

106

1 billion

1,000,000,000

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

109

1 trillion

1,000,000,000,000

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

1012

EXAMPLE: Convert the number in this sentence to standard notation:

In 1999, the U.S. Mint produced 11.6 billion pennies.

11.6 billion

11.6 x 1 billion

 

11.6 x 1,000,000,000 (9 zeros)

 

11,600,000,000 (move the decimal point 9 places to the right.)

 

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MONEY CONVERSION

Converting from dollars to cents is multiplying by 100. To see why, consider $1.00. If you went to the bank to exchange $1.00 for pennies, how many pennies would you receive for your $1.00? Answer.

dollarbill.jpg

 equal sign.jpg

pennies100.jpg

 

Convert $19.43 to cents.

$19.43

19.43 x 1 dollar

 

19.43 x 100 cents

 

1943 cents

 

Dollars to Cents

To convert from dollars to cents, move the decimal point two places to the right and change the $ sign in the front to a cent sign at the end.

 

  

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Converting from cents to dollars is multiplying by 0.01. To see why, consider 65 cents.

65 cents

65 x 1 cent

 

65 x $0.01

 

$0.65

 

Cents to Dollars

To convert from cents to dollars, move the decimal point two places to the left and change the cent sign at the end to a $ sign in the front.

 

  

 

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LET'S REVIEW (click here for PowerPoint)

  

 

4.4 DIVISION

First let's go over some terms that we will be using with division.

 

 

defcolordiv1.png

defcolordiv.png

 

 

  

 

  

WHOLE NUMBER DIVISORS

We use the following method when we divide a decimal quantity by a whole number.

 

To divide by a whole number,

a) place the decimal point directly above

the decimal point in the dividend, and

 

b) divide as though dividing whole numbers.

 

Try These:

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DIVISORS THAT ARE NOT WHOLE NUMBERS

Consider the division:

Write your division problem as a fraction. Then multiply by 1 to change to a whole number divisor:

 

To divide when the divisor is not a whole number,

a) move the decimal point (multiply by 10, 100, and

so on) to make the divisor a whole number;

 

b) move the decimal point (multiply the same way) in

the dividend the same number of places; and

 

c) place the decimal point directly above the new

decimal point in the dividend and divide as though

dividing whole numbers.

 

 

Try these:

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DIVIDING BY 10, 100, 1000 and SO ON

If is often helpful to be able to divide quickly by a ten, hundred, or thousand, or by a tenth, hundredth, or thousandths. Consider the following example:

 

 

Mental Math Method

To divide by 10, 100, 1000, and so on,

a) count the number of zeros in the divisor, and

 

b) write the equation by moving the decimal point

in the dividend that number of places to the left.

 

We are dividing by a number greater than 1 so the

result is smaller than 713.49.

 

7.1349

 

7.1349 < 713.49

 

Try These:

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DIVIDING BY 0.1, 0.01, 0.001, AND SO ON

Now consider the following example:

 

 

Mental Math Method

To divide by 0.1, 0.01, 0.001 and so on,

a) count the number of decimal places in the divisor.

 

b) write the quotient by moving the decimal point

in the dividend that number of places to the right.

 

We are dividing by a number less than 1 so the

result is larger than 23.789.

 

2378.9

 

2378.9 > 23.789

 

Try These:

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4.4 ORDER OF OPERATIONS: DECIMAL NOTATION

The same rules for order of operations used with whole numbers and fraction notation apply when simplifying expressions with decimal notation.

RULES FOR ORDER OF OPERATIONS

1. Do all calculations within grouping symbols before operations outside.

2. Evaluate all exponential expressions

3. Do all multiplications and divisions in order from left to right.

4. Do all additions and subtractions in order from left to right.

  

 

Try These:

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4.5 CONVERTING FROM FRACTION NOTATION TO DECIMAL

When a denominator has no prime factors other than 2's and 5's, we can find decimal notation by multiplying by 1. We multiply to get a denominator that is a power of ten, like 10, 100, or 1000.

 

Example 1: Find decimal notation for 3/5.

 

 

Example 2: Find decimal notation for 7/20.

 

 

Example 3: Find decimal notation for 87/25.

 

 

Example 4: Find decimal notation for 9/40.

 

 

We can always divide to find decimal notation.

 

Example 5: Find decimal notation for 3/5.

 

 

Example 6: Find decimal notation for 7/8.

 

 

In examples 5 and 6, the division terminated, meaning that eventually we got a remainder of 0. A terminating decimal occurs when the denominator has only 2's or 5's, or both, as factors, as in 17/25, 5/8, or 83/100. This assumes that the fraction notation has been simplified.

Let's consider a different situation:

Since 6 has a 3 as a factor, the division will not terminate. Although we can still use division to get decimal notation, the answer will be a repeating decimal, as follows.

Example 7: Find decimal notation for 1/6.

 

 

Try These:

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4.6 ESTIMATING

 

jellybeans1.jpg grads.jpg brochure.jpg

 

How many jelly beans are in the jar? How large was your graduating class? How much is this vacation going to cost? In all of these situations you use estimation to make a guess of what the actual number might be. We use estimation to get a quick answer instead of doing the long calculation, or you use estimation after your long calculation to make sure your answer makes sense in your addition, subtraction, multiplication and division problems.

 


refrigerator.jpg computer.jpg television.jpg

 

Estimate by rounding to the nearest ten the total cost of one refrigerator priced at $474, and one computer priced at $595. Which of the following is an appropriate estimate?

a) $1000

b) $900

c) $1070

d) $800

 

Estimate the total cost of 4 Televisions at 351. Which of the following is an appropriate estimate?

a) $1200

b) $1400

c) $1600

d) $2000

 

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4.7 APPLICATIONS AND PROBLEM SOLVING

 

FIVE STEPS FOR PROBLEM SOLVING

1. Familiarize yourself with the situation.

2. Translate the problem to an equation using the letter or variable.

3. Solve the equation.

4. Check the answer in the original wording of the problem. (estimate to make sure your answer makes sense)

5. State the answer to the problem clearly with the appropriate units.

 

Example 1: At a printing company, the cost of copying is 11 cents per page. How much, in dollars, would it cost to make 466 copies?

1. Familiarize yourself with the situation.

divisionart3.png

2. Translate the problem to an equation using a letter of variable.

3. Solve the equation.

4. Check the answer in the original wording of the problem. (estimate to make sure your answer makes sense)

5. State the answer to the problem clearly with the appropriate units.

 

Example 2: A loan of $4425 is to be paid off in 12 monthly payments. How much is each payment?

1. Familiarize yourself with the situation.

calendar.png

2. Translate the problem to an equation using a letter of variable.

3. Solve the equation.

 

4. Check the answer in the original wording of the problem. (estimate to make sure your answer makes sense)

 

5. State the answer to the problem clearly with the appropriate units.

 

Try These

 

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