Chapter 7 - Decimals7.1 Decimals

Verbal Description

• Decimals are used to represent fractions in our usual base ten place-value notation

• The decimal point is placed between the ones column and the tenths column

• The number is read “three thousand four hundred fifty-seven and nine hundred sixty-eight thousandths? .

Hundreds square

• Used to visualize decimals .

Number line for decimals

• A number line can also be used to picture decimals .

Fractions with Terminating Decimal Representations

• Let be a fraction in simplest form.

Then has a terminating decimal

representation if and only if b contains only

2s and/or 5s in its prime factorization .

ab

ab

Ordering Decimals

• Terminating decimals can be compared using:

1. A hundreds square2. A number line3. Comparing them in their fraction form4. Comparing place values one at a time

from left to right .

Ordering Decimals

• Determine the larger of 0.7 and .231. Using a hundreds

square2. Using a number line3. In fraction form

• and

1. Compare place values– .7 >.2 .

10023

10070

Fraction equivalents

• Common decimals have fraction representations, the fraction equivalents can often be used to simplify decimal calculations

• Find the products using fraction equivalents:

• 68 X 0.5– = 68 x ½ = 34

• 0.25 x 48 = ?– ¼ x 48 = 24

• 56 x 0.125 = ?– 56 x 1/8 = 7 .

Multiplying/Dividing decimals by powers of 10

• Let n be any decimal number and mrepresent any nonzero whole number. Multiplying a number n by 10m is equivalent to forming a new number by moving the decimal point of n to the right m places.

• Dividing a number n by 10m is equivalent to forming a new number by moving the decimal point of n to the left m places .

• Use the following methods to estimate the answer

• $1.57 + $4.36+$8.78 =• Range

– Low $1+$4+$8 = $13– High $2+$5+$9 = $16

• Front-end with adjustment– Low estimate from above = $13 and– Sum of 0.57+0.36+0.78 = $1.50– $13 = $1.50 = $14.50

• Rounding to nearest whole or half– $1.50+$4.50+$9.00= $15.00 .

• Decimals can be rounded to any specified place as done with whole numbers

• Round 56.94352 to nearest:– Tenth– 56.9– Hundredth– 56.94– Thousandth– 56.944– Ten thousandth– 56.9435 .

Section 7.2Operations with Decimals

Addition of Decimals

1. Using Fractions• 3.56 + 7.95 = ?

• . 51.111001151

100795

100356

==+

Addition of Decimals

2. Decimal Approach –arrange the digits in columns according to their place values and add the numbers as if they were whole numbers, and insert a decimal point immediately beneath the decimal points in the numbers added

• 3.56+7.95 = ?• 0.0094+80.183 = ? .

Subtraction of Decimals

• 14.793-8.95

• 14793- 89505843

• 14.793-8.955.843

14.793-8.95 = ?

• Align Decimal Points

• Subtract as If Whole Numbers

(do mentally)

3. Insert Decimal Point in Answer .

Multiplication of Decimals

• Multiply 437.09 x 3.8– Fraction multiplication would be:

=××

=×=×10100

38709,431038

100437098.309.437

942.16601000

942,660,1==

Multiplication of Decimals

• Multiply 437.09 x 3.8– Algorithm 1. Multiply the numbers without the decimal

points– 43709 x 38 = 1,660,9422. Insert a decimal point – the number of

digits to the right of the decimal point in the answer is the sum of the number of digits to the right of the decimal points in the numbers being multiplied

– 2 digits in 437.09 and 1 digit in 3.8 = 3– 1,660.942 .

Division of Decimals

• Divide 154.63 4.7• Divide using fractions:

÷

9.32470

15463470100

100463,15

==×

=÷=÷=÷100470

100463,15

1047

100463,154763.154

Division of Decimals

• Divide

• Replace with an equivalent problem where the divisor is a whole number

• Now divide as if it is whole-number division.

• Replace the decimal point in the dividend and place a decimal point in the quotient directly above the decimal point in the dividend .

9.323.154647

63.1547.4

3.154647

3291546347

Repeating Decimals

• Theorem:– Every Fraction has a repeating decimal

representation, and every repeating decimal has a fraction representation .

Repeating Decimals

• Fractions whose denominators are of the form 2m*5n have terminating decimal representations. These fractions can be converted into decimals and long division (or a calculator).

• Express 7/40 in decimal form .

175.000.740

175.407 =÷

Repeating Decimals

• Theorem:– Fractions with Repeating, Nonterminating

Decimal Representations:

– Let a/b be a fraction written in simplest form. Then a/b has a repeating decimal representation that does not terminate if and only if b has a primer factor other than 2 or 5 .

Repeating Decimals

Express 1/3 in decimal form

• Note that the decimal quotient will never terminate

• Instead of writing dots a horizontal bar may be placed above the repetend .

...333.000.13

Repeating Decimals

• Decimals having a repetend are called repeating decimals

• The number of digits in the repetend is called the period of the decimal. Thus

• Has a period of 2 and

• Has a period of 6 .

Section 7.4Percent

Converting Percents

• Percents provide another common way to representing fractions

• Percents are alternative representations of fractions and decimals

Converting Percents

• Case 1 – Percents to Fractions– Use the definition of percent (per hundred)

to convert– 63% = 63/100

• Case 2 – Percents to Decimals– To convert a percent directly to a decimal,

“drop the % symbol and move the number’s decimal point two places to the left”

– 31% = 0.31– 213% = 2.13

Converting Percents

• Case 3 – Decimals to Percents– Reverse the shortcut in Case 2 – “move

the number’s decimal point two place to the right and add the % sign.

– 0.83 = 83%– 5.1 = 510%

• Case 4 Fractions to Percents– Some fractions that have terminating

decimals can be converted to percents by expressing the fraction with a denominator of 100

– 17/100 = 17%– 2/5 = 4/10 = 40/100 = 40%

Solving Percent Problems

• There are three approaches to solving percent problems1. Grid Approach2. Proportion Approach3. Equation Approach

Grid Approach

• Since percent means “per hundred” solving problems to find a missing percent can be visualized by using the 10-by-10 grids

• A car was purchased for $13,000 with a 20% down payment. How much was the down payment?

• 20 * $130 = $2600

Grid Approach

• One hundred sixty-two seniors, 90% of the senior class, are going on the class trip. How many seniors are there?

• 90 squares (90%) represent 162 students

• Then 1 square – 1.8 students

• 100*1.8 = 180 students

Grid Approach

• Susan scored 48 points on a 60-point test. What percent did she get correct?

• If the 100 squares represent 60 points then

• 1 square = 0.6 points• 10 squares = 6 points• 80 squares = 48

points (6 x 8)• 80%

Proportion Approach

• Percents can be written as a ratio, solving percent problems may be done using proportions. Think of a fuel gauge that varies from empty(0%) to full (100%)

Proportion Approach

• A car was purchased for $13,000 with a 20% down payment. How much was the down payment?

2600$5000,13,

10020

000,13=== xx

• One hundred sixty-two seniors, 90% of the senior class, are going on the class trip. How many seniors are there?

Proportion Approach

1809

10162,10090162

=⎟⎠⎞

⎜⎝⎛== x

x

Proportion Approach

• Susan scored 48 points on a 60-point test. What percent did she get correct?

8054*100,

1006048

=== xx

Equation Approach

• An equation can be used to represent the problem

• A car was purchased for $13,000 with a 20% down payment. How much was the down payment?– 20%*13,000 = x– 0.20(13,000)=$2600

Problems

• Rose bought a dress whose original price was $125 but was discounted 10%. What was the discounted price?– The discount is (10%)(125)=$12.50– New price is $125 - $12.50 = $112.50

• A television set is put on sale at 285 off the regular price. The sale price is $379. What was the regular price?– The sale price is 72% of the regular price

then:

379,72900,37,990,3772,100*379*72 ==== PPPP

P379

10072

=

Problems

A television set is put on sale at 285 off the regular price. The sale price is $379. What was the regular price?– The sale price is 72%

of the regular price then:

37972900,37

990,3772

=

=

=

P

P

P

P379

10072

=