DECIMALS ONE: (WB: p. 63-74; MF: 41-48, 51-52, 73-74, 121, 139-140)

PLACE VALUE & ROUNDING

10 = 10/1 (tens), 1 = 1/1, 0.1 = 1/10 (one decimal place to the right = tenths),
0.01 = 1/100 (two decimal places to the right = hundredths) (also = 1/102), etc……

Name place values: 44.143589

Round to nearest tenth, hundredth, 10,000th, etc.

Same basic rules as with integers.

--Round 6.759 to the nearest tenth.
--Round 816.3904 to the nearest thousandth.
--Round 86.607 to the nearest whole number.

ORDER RELATIONS

This section aims to help you recognize what decimals are larger or smaller.
Note that adding zeros after the last digit adds nothing; when extra zeros are shown, it’s usually for clearer comparisons: E.g., 0.3 = 3.300 = 0.300000… (note that 300/1000 has the same value as 3/10)

--Which is larger? 2.000, 0.2, 0.02, or 0.002 ; 0.19 or 0.9, 0.809 or 0.88 or 0.8, 14.3 or 14.3000…

ADDITION & SUBTRACTION

Be sure to line up digits according to place value.

--Find the sum of 17.89, 3.005, & 10.7.
--Subtract 6.274 from 7.05.

Practice problems on p. 63; note hints on p. 64

MULTIPLICATION

Multiply numbers as usual but remember that the number of decimal places in the answer (product) will be the sum of the decimal places in all the factors (i.e., add all decimal places in factors; if factors have a total of 3 decimal places, answer will have 3, etc).

E.g., 4 • 0.2 = 0.8, 4 • 0.002 = 0.008, 0.4 • 0.2 = 0.08, 12.301 • 0.001 = 0.012301.
--Find the product of 3.66 and 104:

--Multiply: (0.02)(0.02)

--Estimate the product of 0.842 and 5.7:

Practice problems p. 65

FRACTIONS & DECIMALS

To convert a fraction to a standard-form number (whole integer or decimal), just divide.

½ = 1 ÷ 2 = 0.5 (terminating decimal)

1/3 = 0.3333… = 0. 33 (repeating decimal)

To convert a decimal back into a fraction, remove the decimal and place over a denominator of equal place value (e.g., put a tenth over 10, a hundredth over 100, etc).
Then simplify (reduce) to the simplest fraction. E.g., 0.5 = 5/10 = ½, 0.07 = 7/100.

--Write as decimals & state in words: 67/100, 145/1000, ¼ (= 25/100) Just divide to find equivalent decimal.

--Write as a fraction; 0.017, 0.000006

--Convert to a decimal: ¾, 6 ¾ ,

--Which is larger? 1/6 or 0.167

--Convert to a fraction: 0.90, 0.003, 4.96

Practice problems p. 67

DIVISION: Make the divisor a whole number by moving the decimal point to the right, then move the decimal point of the dividend the same number of places to the right. Place the decimal in the quotient directly above the dividend’s decimal point and divide as with whole numbers.

--Find the quotients of: 6 and 0.2 (i.e., 6 ÷ 0.2), 6 and 0.002, 0.044 and 2.2.

Note that moving the decimal points in both divisor and dividend is no different than multiplying both denominator and numerator of a fraction by the same number to obtain equivalent fractions. Moving a decimal point one space (one number) to the right is the same as multiplying the number by 10 (e.g., 5 x 10 = 50; 0.44 x 10 = 4.4). Moving it one space to the left is the same as dividing by 10 (e.g., 5 ÷ 10 = 0.5; 1000 ÷ 10 = 100). Moving a decimal three spaces right or left is the same as multiplying or dividing by 1000, and so on. (See Powers of Ten below; exercises on p. 77.)

--Divide: 63 by 0.03, 24/200, 83.08 ÷ 6.2.

APPLICATIONS: The most common application of decimals involves money.

“Cent” means hundred; a cent is 1/100 of a dollar.
$0.25 = 25/100 = ¼ = “a quarter”

It is usually easier to solve problems using decimals than mixed numbers or fractions.

Try Practice problems, WB. p. 71 and review Quiz 5 on p. 73!

DECIMALS TWO: (WB: P. 75-82)

POWERS OF TEN (WB: p. 77; MF: P. 125-126, 169-170)

0.01, 0.1, 1, 10, 100, 1000, etc: each increase is a 10-fold increase (10X; i.e., you mulitplied a value by 10). It’s a 10-fold decrease if you read these numbers from right to left). A 10-fold change in anything (1 power of 10) is an order of magnitude. To increase something by an order of magnitude is to increase it 10-fold. xamples: 10 • 10 = 100 = 102 10 • 10 • 10 = 1000 = 103 10,000 = 104

We just add a “0” to the whole number when multiplying by 10 or 101.
Add 2 zeros if multiplying by 100 or 102.
Note that the number of 0’s equals the exponent value. Hence 1,000,000 = 106.
We move the decimal one place to right for each increased power of 10; or to the left if we are decreasing the value.

To multiply by powers of 10, just move the decimal to the right: 10 x 10 = 100;
100 x 10 = 1000 (we can also write this as 102 x 101 = 103);
2 x 10,000 = 20,000; 0.03 x 1000 = 30.
To divide by powers of 10: Just move the decimal to the left the same number of places as there are zeros: 2/1000 = 0.002; 999 ÷ 10 = 99.9.

Practice: Fill in the missing numbers: 33 ÷ 0.1 = _____ ; 33 x 0.1 = _____;

33 ÷ 10 = ______; 33 ÷ 100 = ______; 33 x ______ = 33,000;

33 ÷ 10,000 = _________ (Note that 33 ÷ 10,000 is the fraction, 33/10,000, or 33/104.)

33.456 x 1000 = __________; 33.456 ÷ 1000 = __________

Do Exercises: p. 77

(P. 79-80B are optional exercises and lab problems.)

Quiz 6 (p. 81) will also focus on decimals, with an emphasis on COMBINED OPERATIONS WITH DECIMALS (p. 75). You will also need to be able to combine decimals with other forms of numbers such as fractions, mixed numbers. As long as you remember how to convert a number into other types of numbers, this shouldn’t be a big problem. When you encounter mixed numbers, fractions, decimals, etc all mixed together, do them the most accurate and/or easiest way you can (decimals are often easier than fractions, but for repeating decimals, fractions are more accurate).
Remember this general rule: Use the same numerical format in your answer as used in the question. (Give a fraction if original numbers are all fractions, decimals if originally all decimals, etc.)

Try these: 2 ¼ – 4.4 x ½ ÷ 11 =

(2 ½) (0.22) =

(2/5)(0.5)/100 =

(5 – 4)2 =

103 x 102 x 10 =

3 ½ x 20 ÷ 0.01 =

(2/3 + 1/6) ÷ 1/66 – 5.05 =