Transcript Document

Rational Numbers
3-1
Warm Up
Divide.
1. 36  3
3. 68  17
12
4
5. 1024  64 16
2. 144  6
4. 345  115
24
3
Learn to write rational numbers in
equivalent forms.
Vocabulary
rational number
relatively prime
A rational number is any number that can
n
be written as a fraction
, where n and d
d
are integers and d  0.
Decimals that terminate or repeat are
rational numbers.
Numerator
n
d
Denominator
The goal of simplifying fractions is to make
the numerator and the denominator
relatively prime.
Relatively prime numbers have no
common factors other than 1.
You can often simplify fractions by dividing
both the numerator and denominator by
the same nonzero integer. You can simplify
12
4
the fraction
to
by
dividing
both the
15
5
numerator and denominator by 3.
12 of the 15 boxes
are shaded.
12
15
4 of the 5 boxes
are shaded.
=
4
5
The same total area is shaded.
Example: Simplifying Fractions
Simplify.
5
A.
10
5 =1•5
10 = 2 • 5
5÷5
5
= 10 ÷ 5
10
1
=
2
;5 is a common factor.
Divide the numerator
and denominator by 5.
Example: Simplifying Fractions
Simplify.
B. 16
80
16 = 1 • 16 ;16 is a common factor.
80 = 5 • 16
16 ÷ 16
16
= 80 ÷ 16
80
1
=
5
Divide the numerator
and denominator by 16.
Example: Simplifying Fractions
Simplify.
C. –18
29
18 = 2 • 9
29 = 1 • 29
–18
–18
=
29
29
;There are no common
factors.
–18 and 29 are relatively prime.
Try This
Simplify.
A.
6
30
6 = 1 • 6 ;6 is a common factor.
30 = 5 • 6
6÷6
6
=
30 ÷ 6
30
1
=
5
Divide the numerator
and denominator by 6.
Try This
Simplify.
18 = 3 • 3 • 2 ;9 is a common factor.
27 = 3 • 3 • 3
B. 18
27
18 = 18 ÷ 9
27 ÷ 9
27
=
2
3
Divide the numerator
and denominator by 9.
Try This
Simplify.
C.
17
–35
17 = 1 • 17 ;There are no common
factors.
35 = 5 • 7
17
17 17 and –35 are
=–
–35
35 relatively prime.
To write a finite decimal as a fraction, identify
the place value of the farthest digit to the right.
Then write all of the digits after the decimal
points as the numerator with the place value as
the denominator.
Example: Writing Decimals as Fractions
Write the decimal as a fraction in simplest form.
A. –0.8
–8 is in the tenths place.
–0.8 =
–8
10
=–
4
5
Simplify by dividing by
the common factor 2.
Example: Writing Decimals as Fractions
Write the decimal as a fraction in simplest form.
B. 5.37
37
5.37 = 5
100
7 is in the hundredths place.
Example: Writing Decimals as Fractions
Write the decimal as a fraction in simplest form.
C. 0.622
622
0.622 =
1000
311
=
500
2 is in the thousandths
place.
Simplify by dividing by the
common factor 2.
Try This
Write the decimal as a fraction in simplest form.
–4 is in the tenths place.
A. –0.4
–4
–0.4 =
10
=–
2
5
Simplify by dividing by
the common factor 2.
Try This
Write the decimal as a fraction in simplest form.
B. 8.75
5 is in the hundredths place.
75 Simplify by dividing by the
8.75 = 8
100 common factor 25.
3
= 8
4
Try This
Write each decimal as a fraction in simplest form.
C. 0.2625
5 is in the ten-thousandths
place.
2625 Simplify by dividing by
0.2625 =
10,000 the common factor 125.
21
=
80
To write a fraction as a decimal, divide the
numerator by the denominator. You can
use long division.
numerator
denominator
denominator numerator
When writing a long division problem from
a fraction, put the numerator inside the
“box,” or division symbol. It may help to
write the numerator first and then say
“divided by” to yourself as you write the
division symbol.
Example: Writing Fractions as Decimals
Write the fraction as a decimal.
A. 11
9
The fraction
1 .2
9 11 .0
–9
20
–1 8
2
The pattern repeats, so
draw a bar over the 2
to indicate that this is a
repeating decimal.
11
is equivalent to the decimal 1.2.
9
Example: Writing Fractions as Decimals
Write the fraction as a decimal.
7
B.
20
0.3 5 This is a terminating decimal.
20 7.0 0
–0
70
–6 0
1 00
–1 0 0
0 The remainder is 0.
The fraction
7
is equivalent to the decimal 0.35.
20
Try This
Write the fraction as a decimal.
A. 15
9
The fraction
1 .6
9 15 .0
–9
60
–5 4
6
The pattern repeats, so
draw a bar over the 6 to
indicate that this is a
repeating decimal.
15
is equivalent to the decimal 1.6.
9
Try This
Write the fraction as a decimal.
0.2 2 5
40 9.0 0 0 This is a terminating decimal.
–0
90
–8 0
1 00
– 80
200
– 2 00
0 The remainder is 0.
9
The fraction
is equivalent to the decimal 0.225.
40
B.
9
40
Lesson Quiz: Part 1
Simplify.
18
1.
42
3
7
15
2.
21
5
7
Write each decimal as a fraction in
simplest form.
5
27
–
3. 0.27
4.
–0.625
8
100
13
5. Write
as a decimal
6
2.16
Lesson Quiz: Part 2
6. Tommy had 13 hits in 40 at bats for
his baseball team. What is his batting
average? (Batting average is the
number of hits divided by the number
of at bats, expressed as a decimal.)
0.325