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Mathematics: Applications and Concepts, Course 3 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 2-1 Fractions and Decimals
Lesson 2-2 Comparing and Ordering Rational Numbers
Lesson 2-3 Multiplying Rational Numbers
Lesson 2-4 Dividing Rational Numbers
Lesson 2-5 Adding and Subtracting Like Fractions
Lesson 2-6 Adding and Subtracting Unlike Fractions
Lesson 2-7 Solving Equations with Rational Numbers
Lesson 2-8 Powers and Exponents
Lesson 2-9 Scientific Notation
Example 1 Write a Fraction as a Decimal
Example 2 Write a Mixed Number as a Decimal
Example 3 Round a Repeating Decimal
Example 4 Write a Terminating Decimal as a Fraction
Example 5 Write a Repeating Decimal as a Fraction
Write
as a decimal.
0.1875
Add a decimal point and zeros to the
dividend:
Division ends when the
remainder is 0.
You can also use a calculator.
3
16
ENTER
0.1875
Answer: The fraction
can be written as 0.1875.
Write
as a decimal.
Answer: 0.0625
Write
as a decimal.
means
To change
to a decimal,
divide 2 by 11.
0.1818 …
The three dots means the one and eight
keep repeating.
The remainder after each step
is 2 or 9.
You can also use a calculator.
2
11
ENTER
0.1818181818
Answer: The mixed number
can be written as
3 + 0.1818… or 3.1818… .
Write
as a decimal.
Answer: 5.1111…
AGRICULTURE A Florida farmer lost the fruit on 8
of 15 orange trees because of unexpected freezing
temperatures. Find the fraction of the orange trees
that did not produce fruit. Express your answer as
a decimal to the nearest thousandth.
To find his fraction of trees that did not produce fruit,
divide the number of lost trees, 8, by the total number
of trees, 15.
8
15
ENTER
0.5333333333
Look at the digit to the right of the thousandths place.
Round down since 3 < 5.
Answer: The fraction of fruit trees that did not
produce fruit was 0.533.
SCHOOL In Mrs. Townley’s eighth grade science
class, 4 out of 22 students did not turn in their
homework. Find the fraction of the students who
did not turn in their homework. Express your
answer as a decimal to the nearest thousandth.
Answer: 0.182
Write 0.32 as a fraction.
0.32 is 32 hundredths.
Simplify. Divide by the greatest common
factor of 32 and 100, 4.
Answer: The decimal 0.32 can be written as
Write 0.16 as a fraction.
Answer:
ALGEBRA Write
Let
as a mixed number.
Then
Multiply N by 10 because 1 digit repeats.
Subtract
0.777… .
to eliminate the repeating part,
Divide each side by 9.
Simplify.
Answer: The decimal
can be written as
ALGEBRA Write
Answer:
as a mixed number.
Example 1 Compare Rational Numbers
Example 2 Compare Negative Rational Numbers
Example 3 Order Rational Numbers
Replace
with <, >, or
to make
a true sentence.
Method 1 Write as fractions with the same denominator.
For
the least common denominator is 91.
Method 2 Write as decimals.
Write
3
as decimals. Use a calculator.
7
Since
Answer:
ENTER
0.42857…
8
13
ENTER
0.61538…
Replace
Answer:
with <, >, or
to make
a true sentence.
Replace with <, >, or
sentence.
Write
to make
as a decimal.
.
Since
a true
Check Use a number line to check the answer.
The answer is correct.
Answer:
Replace with <, >, or
sentence.
Answer:
to make
a true
CHEMISTRY The values
for the approximate
densities of various
substances are shown in
the table. Order the
densities from least to
greatest.
Density
(g/cm3)
aluminum
2.7
beryllium
1.87
brick
crown glass
fused silica
marble
nylon
1.1
pyrex glass
2.32
rubber neoprene
Source: CRC Handbook of Chemistry and Physics
Substance
Write each fraction as a decimal.
Answer: From the least to the greatest, the densities are
So, nylon is the least dense, and aluminum is
the most dense.
AMUSEMENT PARKS The ride times for five
amusement park attractions are shown in the table
below. Order the lengths from least to greatest.
Ride Time
(min)
Big Dipper
Double Loop
1.5
Mind Eraser
1.8
Serial Thriller
X–Flight
Answer:
Source: www.coasterglobe.com
Coaster
Example 1 Multiply Fractions
Example 2 Multiply Negative Fractions
Example 3 Multiply Mixed Numbers
Example 4 Evaluate an Algebraic Expression
Example 5 Use Dimensional Analysis
Find
Write in simplest form.
1
Divide 3 and 9 by their GCF, 3.
3
Multiply the numerators.
Multiply the denominators.
Simplify.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
–1
Divide –3 and 12 by their GCF, 3.
4
Multiply the numerators.
Multiply the denominators.
The factors have different signs,
so the product is negative.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
4
Divide 16 and 4 by their GCF, 4.
1
Multiply the numerators.
Multiply the denominators.
Simplify.
Check
is about 3, and
is about
reasonable.
Answer:
is about 2. Therefore,
or 6. The answer is
Find
Answer:
Write in simplest form.
ALGEBRA Evaluate prq if
Replace p with
, r with
and q with
1
1
Divide out common factors.
1
Simplify.
Answer:
,
ALGEBRA Evaluate prq if
Answer:
WATER USE Low–flow showerheads use
gallons
of water per minute. If family members shower a
total of
hours per week, how much water does
the family use for showers each week?
Words
Variables
Water used equals the time multiplied
by the water flow rate.
w =
t

r
Equation
w =
hours 
gallons per minute
Write the equation.
70
Divide by common factors and
units.
1
Check The problem asks for the amount of water
used a week. When you divide the common
units, the answer is expressed in gallons.
So, the answer is reasonable.
Answer: If the family showers
rate of
hours per week at a
gallons per minute, they will use
about 350 gallons of water.
WATER USE Some showerheads use
gallons
of water per minute. If family members shower a
total of
hours per week, how much water does
the family use for showers each week?
Answer: about 897 gallons
Example 1 Find a Multiplicative Inverse
Example 2 Divide Fractions
Example 3 Divide by a Whole Number
Example 4 Divide by Negative Fractions
Example 5 Divide Mixed Numbers
Example 6 Use Dimensional Analysis
Write the multiplicative inverse of
Write
Answer: Since
of
as an improper fraction.
the multiplicative inverse
Write the multiplicative inverse of
Answer:
Find
Write in simplest form.
Multiply by the multiplicative inverse
of
which is
1
Divide 5 and 10 by their GCF, 5.
2
Simplify.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
Write 12 as
Multiply by the multiplicative
inverse of 12, which is
1
Divide 6 and 12 by their GCF, 6.
2
Simplify.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
Multiply by the multiplicative
inverse of
which is
.
1
Divide 2 and 8 by their GCF, 2.
4
The fractions have different signs,
so the quotient is negative.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
The multiplicative inverse of
is
.
2
Divide 4 and 8 by their GCF, 4.
1
Simplify.
Check Since
is about 3 and
is about –2, you
can estimate the answer to be about
which is
reasonable because
Answer:
. The answer seems
Find
Answer:
Write in simplest form.
MOVIES One summer day, a cinema shows a hit
action movie repeatedly for
hours. It takes
about 150 minutes to run the movie and prepare for
the next showing. How many times does the
cinema show the movie that day?
Since 150 minutes equals
hours, divide
by
Multiply by the multiplicative
inverse of
5
which is
1
Divide by common factors.
1
1
Simplify.
Check Use dimensional analysis to examine the units.
Convert hours to
minutes.
Divide out the
units.
Simplify.
The result is expressed as showings. This agrees with
your answer of 5 showings that day.
Answer: The cinema shows the movie 5 times that day.
ZOOS A zoo has a reptile show that runs repeatedly
for
hours. It takes about 105 minutes to conduct
the show and prepare for the next one. How many
times does the zoo run the reptile show each day?
Answer: 6 times
Example 1 Add Like Fractions
Example 2 Subtract Like Fractions
Example 3 Add Mixed Numbers
Example 4 Subtract Mixed Numbers
Find
Write in simplest form.
Add the numerators.
The denominators are
the same.
Simplify.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
Subtract the numerators.
The denominators are
the same.
Rename
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
Add the whole numbers
and fractions separately.
Add the numerators.
Simplify.
Answer:
Find
Answer:
Write in simplest form.
HEIGHTS In the United States, the average height of
a 9–year–old girl is
of a 16–year–old girl is
inches. The average height
inches. How much does
an average girl grow from age 9 to age 16?
Estimate 64 – 54 = 10
Write the mixed numbers
as improper fractions.
Subtract the numerators.
The denominators are the
same.
Rename
Answer: The average girl grows
from age 9 to age 16.
inches
HEIGHTS Ainsley was
inches tall when she was
4 years old. When she was 10 years old, she was
inches tall. How much did she grow between the ages
of 4 and 10?
Answer:
inches
Example 1 Subtract Unlike Fractions
Example 2 Add Mixed Numbers
Example 3 Estimate the Sum of Mixed Numbers
Example 4 Evaluate Expressions
Find
Write in simplest form.
The LCD is 7  5 or 35.
Rename each fraction
using the LCD.
Subtract
inverse,
by adding its
Add the numerators.
Simplify.
Answer:
Find
Answer:
Write in simplest form.
Find
Write in simplest form.
Write the mixed numbers
as fractions.
The LCD is
or 24.
Rename each fraction
using the LCD.
Add the numerators.
Simplify.
Answer:
Find
Answer:
Write in simplest form.
MULTIPLE- CHOICE TEST ITEM During January, Nikki
watched television
the second week,
hours the first week,
hours
hours the third week, and
hours the fourth week. How much time did she watch
television during the four-week period?
A
hours
B
hours
C
hours
D
hours
Read the Test Item
You need to find the sum of four mixed numbers.
Solve the Test Item
It would take some time to change each of the fractions to
ones with a common denominator. However, notice that
all four of the numbers are about 2. Since 2 x 4 = 8, the
answer will be about 8. Notice that only one of the
choices is close to 8.
Answer: B
MULTIPLE- CHOICE TEST ITEM During July, Carsten
watched television
hours the first week,
hours the second week, and
hours the third
week. How much time did she watch television
during the 3-week period?
A
hours B
Answer: C
hours C
hours D
hours
ALGEBRA Find the value of p – q if
Replace p with
and
and q with
Rename each fraction using the
LCD, 45.
Subtract the numerators.
Simplify.
Answer:
ALGEBRA Find the value of
Answer:
Example 1 Solve by Using Addition or Subtraction
Example 2 Solve by Using Addition or Subtraction
Example 3 Solve by Using Multiplication or Division
Example 4 Solve by Using Multiplication or Division
Example 5 Write an Equation to Solve a Problem
Solve
. Check your solution.
Write the equation.
Subtract 2.84 from each side.
Simplify.
Check
Write the equation.
Replace g with 0.78.
Simplify.
Answer: 0.78
Solve
Answer: 3.08
. Check your solution.
Solve
Write the equation.
Add
to each side.
Simplify.
Rename each fraction using
the LCD, 15.
Simplify.
Answer:
Solve
Answer:
Solve
Check your solution.
Write the equation.
Multiply each side by
Simplify.
Check
Write the equation.
Replace c with –33.
Simplify.
Answer: –33
Solve
Answer: –45
Check your solution.
Solve
Write the equation.
Divide each side by 9.7.
Simplify. Check the solution.
Answer: –7
Solve 3.4t = –27.2.
Answer: –8
PHYSICS You can determine the rate an object is
traveling by dividing the distance it travels by the
time it takes to cover the distance
If an object
travels at a rate of 14.3 meters per second for
17 seconds, how far does it travel?
Words
Rate equals distance divided by time.
Variables
r
=
Equation
14.3
=
Write the equation.
Multiply each side by 17.
Simplify.
Answer: The object travels 243.1 meters.
PHYSICS You can determine the rate an object is
traveling by dividing the distance it travels by the
time it takes to cover the distance
If an object
travels at a rate of 73 miles per hour for 5.2 hours,
how far does it travel?
Answer: 379.6 miles
Example 1 Write an Expression Using Powers
Example 2 Evaluate Powers
Example 3 Evaluate Powers
Example 4 Evaluate Powers
Write
using exponents.
Commutative
Property
Associative
Property
Definition of
exponents
Answer: p4q3
Write
Answer: x3y4
using exponents.
Evaluate
.
Definition of exponents
Simplify.
Check using a calculator.
9
5
ENTER
59049
Answer: 59,049
Evaluate
.
Answer: 7,776
Evaluate
Definition of exponents
Simplify.
Answer:
Evaluate
Answer:
ALGEBRA Evaluate
if
and
Replace x with 4
and y with 2.
Definition of
exponents
Simplify.
Simplify.
Answer: 2,048
ALGEBRA Evaluate
Answer: 2,304
if
and
Example 1 Express Numbers in Standard Form
Example 2 Express Numbers in Standard Form
Example 3 Write Numbers in Scientific Notation
Example 4 Write Numbers in Scientific Notation
Example 5 Compare Numbers in Scientific Notation
Write
in standard form.
100,000
or 100,000
Notice that the decimal place
moves 5 places to the right.
Answer: 962,000
Write
in standard form.
Answer: 53,200
Write
in standard form.
Notice that the decimal point
moves 6 places to the left.
Answer: 0.00000285
Write
in standard form.
Answer: 0.000381
Write 931,500,000 in scientific notation.
The decimal point
moves 8 places.
The exponent is
positive.
Answer:
Write 35,600,000 in scientific notation.
Answer:
Write 0.00443 in scientific notation.
The decimal point moves 3
places.
The exponent is negative.
Answer:
Write 0.000653 in scientific notation.
Answer:
PLANETS The following table lists the average
radius at the equator for each of the planets in our
solar system. Order the planets according to radius
from largest to smallest.
Radius (km)
Earth
6.38 x 103
Jupiter
7.14 x 104
Mars
3.40 x 103
Mercury
2.44 x 103
Neptune
2.43 x 104
Pluto
1.5 x 103
Saturn
6.0 x 104
Uranus
2.54 x 104
Venus
6.05 x 103
Source: CRC Handbook of Chemistry and Physics
Planet
First order the numbers according to their exponents.
Then order the numbers with the same exponents by
comparing the factors.
Jupiter, Neptune,
Saturn, Uranus
Earth, Mars,
Mercury, Pluto,
Venus
Step 1
7.14 x 104
2.43 x 104
6.0 x 104
2.54 x 104
>
6.38 x 103
3.40 x 103
2.44 x 103
1.5 x 103
6.05 x 103
Step 2
7.14 x 104 > 6.0 x 104 > 2.54 x 104 > 2.43 x 104
Jupiter
Saturn
Uranus
Neptune
6.38 x 103 > 6.05 x 103 > 3.40 x 103 > 2.44 x 103 > 1.5 x 103
Earth
Venus
Mars
Mercury
Pluto
Answer: The order from largest to smallest is Jupiter,
Saturn, Uranus, Neptune, Earth, Venus, Mars,
Mercury, and Pluto.
PLANETS The following table lists the mass for
each of the planets in our solar system. Order the
planets according to mass from largest to smallest.
Mass (in tons)
Mercury
3.64 x 1020
Venus
5.37 x 1021
Earth
6.58 x 1021
Mars
7.08 x 1020
Jupiter
2.09 x 1024
Saturn
6.25 x 1023
Uranus
9.57 x 1022
Neptune
1.13 x 1023
Pluto
1.38 x 1019
Source: nssdc.gsfc.nasa.gov
Planet
Answer: Jupiter, Saturn, Neptune, Uranus, Earth, Venus,
Mars, Mercury, and Pluto
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