Download Chapter 7 - Decimals

Chapter 7 - Decimals
7.1 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 a be a fraction in simplest form.
b
a
Then
has a terminating decimal
b
representation if and only if b contains only
2s and/or 5s in its prime factorization .
Ordering Decimals
•
Terminating decimals can be compared
using:
1.
2.
3.
4.
A hundreds square
A number line
Comparing them in their fraction form
Comparing place values one at a time
from left to right .
Ordering Decimals
•
Determine the larger
of 0.7 and .23
1. Using a hundreds
square
2. Using a number line
3. In fraction form
23
70
•
and
100
100
1. Compare place
values
– .7 >.2 .
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 m
represent 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 7.2
Operations with Decimals
Addition of Decimals
1. Using Fractions
•
3.56 + 7.95 = ?
•
356 795 1151
. 100 + 100 = 100 = 11.51
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 = ?
•
•
3.
Align Decimal
Points
Subtract as If
Whole Numbers
(do mentally)
Insert Decimal Point
in Answer .
• 14.793
-8.95
• 14793
- 8950
5843
• 14.793
-8.95
5.843
Multiplication of Decimals
• Multiply 437.09 x 3.8
– Fraction multiplication would be:
43709 38 43,709 × 38
437.09 × 3.8 =
× =
=
100 10
100 × 10
1,660,942
=
= 1660.942
1000
Multiplication of Decimals
•
Multiply 437.09 x 3.8
– Algorithm
1. Multiply the numbers without the decimal
points
– 43709 x 38 = 1,660,942
2. 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:
15,463 47 15,463 470
154.63 ÷ 47 =
÷
=
÷
=
100
10
100
100
15,463 100 15463
×
=
= 32.9
100
470
470
Division of Decimals
• Divide
47 1546.3
4.7 154.63
• Replace with an equivalent problem
where the divisor is a whole number
329 • Now divide as if it is whole-number
division.
47 15463
32.9
47 1546.3
• Replace the decimal point in the dividend
and place a decimal point in the quotient
directly above the decimal point in the
dividend .
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 .
7 ÷ 40 = .175
.175
40 7.000
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
.333...
3 1.000
• Note that the decimal quotient will never
terminate
• Instead of writing dots a horizontal bar
may be placed above the repetend .
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 7.4
Percent
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 problems
1. Grid Approach
2. Proportion Approach
3. 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 sixtytwo 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?
x
20
13,000
=
,x =
= $2600
13,000 100
5
Proportion Approach
• One hundred sixtytwo seniors, 90% of
the senior class, are
going on the class
trip. How many
seniors are there?
162 90
⎛ 10 ⎞
=
, x = 162⎜ ⎟ = 180
x
100
⎝9⎠
Proportion Approach
• Susan scored 48
points on a 60-point
test. What percent
did she get correct?
48
x
4
=
, x = 100 * = 80
60 100
5
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:
72 379
100
=
P
72 * P = 379 *100,72 P = 37,990, P =
37,900
, P = 379
72
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:
72 379
=
100
P
72 P = 37,990
37,900
P=
72
P = 379