Objective a - Writing Decimals in Standard Form and in Words

3.1INTRODUCTION TO DECIMALS

Objective A - Writing Decimals in Standard Form and in Words

Decimals are really just another way to write fractional numbers when the denominator is equal to,ora multiple of, the number10.

Ex.Ex.Ex.

In Chapter #1 it’s shown the position of each digit in a whole number has aPlace Value.

Recall:

_____

,

_____

Billions

Hundred-Millions

Ten-Millions

Millions

Hundred- Thousands

Ten- Thousands

Thousands

Hundreds

Tens

Ones

Similarly, with decimals, there is a Place Valuefor each digit.
Below are the place values for the first six positions:

_____

Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

Ten-Thousandths

Hundred-Thousandths

Millionths

Notice that there are no commas used to the right of the decimal point.

Ex. Write as a decimal. The final-digitmust be in the hundredths position, since the fraction is, “56 hundredths.”

Ex. Write as a decimal.

Note: Whole number part will be just like a regular number to the left of the decimal point.

Ex. Write as a decimal. Ex. Write as a decimal.

Now let’s write decimals in words:

Ex. Write 0.823 in words:

First, write the name of the decimal part as if there was no decimal point:

Eight hundred twenty-three

Next, look at the place value column of the last digit.

For example, it’s the ___ column; so the name of that column is placed after the number.

Answer:Eight hundred twenty-three thousandths

Ex. Write 56.77 in words:

Note: The decimal point is read as the word, “and.” If you look back at Chapter #1, section #1 (on first page of notes) there was not supposed to be an “and” in any of those numbers.

Ex. Write 103.025 in words:

Ex. Write six hundred eleven thousandths as a decimal:

Ex. Write twelveandthirteen hundredths as a decimal:

Ex. Write twelveandthirteenthousandths as a decimal:

Objective B - Rounding a Decimal to a Given Place Value

Rounding decimals is similar to what you’ve already done in Chapter #1

You need to pay attention to which place value you being asked to round.

1.You will locate the digit in that place value.

2.Look at the digit to its right.

a.If digit to the right is 0, 1, 2, 3, or 4→ round down. (Leave located digit alone)

b.If digit to the right is 5, 6, 7, 8, or 9→ round up. (Increase located digit by one)

3.DO NOT write any digits to the right of the digit rounded.

Ex.Round 7.305 to the nearest tenth.Ex.Round 534.2981 to the nearest thousandth.

Ex. Round 534.2981 to the nearest hundredth.Ex. Round 534.2981 to the nearest tenth.

3.2ADDITION OF DECIMALS

Objective A - Adding Decimals

Add decimals just like whole numbers – requires the samedecimal points lined up with each other. To do this, write the addition problem vertically (Chapter#1) and make sure the decimal points are lined up!

Ex. Add: 59.823 + 164.96 + 3,426.1034

Ex. Add: 63.25 + 0.344 + 1.0005

3.3SUBTRACTION OF DECIMALS

Objective A - Subtracting Decimals

Subtraction is done the sameas with whole numbers.
“Borrowing”may be necessary.
Identical to “Addition of Decimals” (from section 3.2), you need the same decimal points lined up with each other.
Do this by making sure theyare lined up!

Ex. Subtract: 672.083 – 245.171 (Remember how to check your subtraction!)

Ex. Subtract: 5,743.9 – 632.16 (Hint: If one number does not have asmany

decimal places as the other,writeadditional0’s so bothnumberswill have the samenumber of decimalplaces.)

3.4MULTIPLICATION OF DECIMALS

Objective A - Multiplying Decimals

Procedure:

You will need to set up the multiplication exactly as you did when you multiplied whole numbers.
Count how many decimal places there are in each factor, and add to find the total number of decimal places.
The product must have the number of decimal places that you found when you added in Step 2. To determine where to position the decimal point in the answer (i.e. the product), count over that many places starting with the digit on the right.

(Hint: Set it up vertically)

Ex. Ex. Ex.

“Shortcut” - If you have to multiply a number times a multiple of 10 (i.e. 10, 100, 10000, 10000, etc.) move the decimal point to the right (to make the number larger). How do you know how many places should you move the decimal point?

Ex.Ex.Ex.

Note: Above problems could have been written with exponents of 10 as they are below.

Ex.Ex.Ex.

3.5DIVISION OF DECIMALS

Objective A - Dividing Decimals

Procedure:

Move the decimal point of the divisor to the right so that the number looks like a whole number (i.e. there are no decimal places left). Count how many places you moved it.
Move the decimal point of the dividend that same number of places to the right.
Place decimal point in the quotient directly over the decimal point in the dividend.
Divide as you did with whole numbers.

Ex.Ex.

If the division does not end up with a zero remainder, DONOT write the remainder as a fraction, which is what was done in Chapter #2. Instead you will round the quotient to a specified place value.

Ex.Round to the nearest tenth:Hint: Since you need to round to the nearest tenth, carry the division out to the nearest hundredth and

use that digit to determine whether to round “up” or “down.” May need to insert zeros in the dividend to accomplish it.

Ex.Round to the nearest hundredth:

Ex.Round to nearest hundredth:Ex.Round to nearest thousandth:

Ex.Round to nearest whole number:Ex.Round to nearest whole number:

Recall that there was a “shortcut” when you multiply by a multiple of 10.
Similarly, there is a “shortcut” when you divide by a multiple of 10.

“Shortcut”

oIf you divide a number by a multiple of 10 (i.e. 10, 100, 10000, 10000, etc.), move the decimal point to the left (to make the number smaller).

oHow do you know how many places should you move the decimal point?

Ex.Ex.Ex.

Note: Above problems could have been written with exponents of 10 as they are below.

Ex.Ex.Ex.

3.6 COMPARING AND CONVERTING FRACTIONS AND DECIMALS

Objective A - Converting Fractions to Decimals

Because fractions are really division statements - convert a fraction to a decimal by dividing the numerator by the denominator.
That means that if you wanted to change into a decimal, you would need to make 1 = dividend and 8 = divisor. →

The last example divided so that there was a remainder of zero.
If you were asked to round that to the nearest thousandth, just leave it as it is because it ended with the thousandths place.

Ex.Convert to a decimal and round it to the nearest thousandth.

This decimal ended with a digit in the tenth’ place. How do you think you could “round that to the nearest thousandth?

Ex. Convert to a decimal and round it to the nearest thousandth.

Point: When converting a fraction to decimal form, sometimes the decimal will terminate (which means it ends); other times it repeats (which means the digit(s) continue a pattern).

Objective B - Converting a Decimal to a Fraction

Procedure
To convert a decimal to a fraction you will:

Write the digits to the right of the decimal point in the numerator.
Look at the place value of the right-most digit – that will be the denominator.
Reduce the fraction if possible.

Ex. Convert 0.63 to a fraction.

Ex. Convert 2.151 to a fraction.(Hint: when there are digits to the left of the decimal point, they become the whole number part of a mixed number.)

Ex. Convert 0.75 to a fraction.

Ex. Convert 19.250 to a fraction.

Objective C - Comparing Decimals and Fractions

You have already compared two fractions (section 2.8) and place the correct inequality symbol (< or >) between them.

In this section you will compare two decimals or compare a decimal and a fraction and then determine the correct inequality symbol that belongs between them.

Remember: The smaller number is always to the ______of the larger number on the number line.

Procedure:

Write both numbers as decimals with the same number of decimal places. Meaning you have to:
Convert a fraction to decimal representation.
“Pad” the left of any decimal number that has fewer digits with zeros.
Look at these decimal numbers and know which is the larger and the smaller.

Place the correct symbol ( or ) between the numbers in the following examples.

Ex.0.56____0.81Ex.6.87____6.32

Ex.0.1____0.099Ex.0.0231____0.17

Ex.____0.55Ex.0.161____