Write each number as a fraction.
1.
SOLUTION:
To write a mixed number as a fraction, multiply the whole number by the denominator and then add the numerator.
Write the result over the denominator. So, = .
2.–9
SOLUTION:
To write a whole number as a fraction, write the whole number over 1. So, –9 = .
3.
SOLUTION:
To write a mixed number as a fraction, multiply the whole number by the denominator and then add the numerator.
Write the result over the denominator. So, = .
Write each decimal as a fraction or mixed number in simplest form.
4.0.07
SOLUTION:
The decimal 0.07 is 7 hundredths. So, 0.07 = .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 0 7 0 0
5.
SOLUTION:
Let N represent the number.
So, = .
6.
SOLUTION:
Let N represent the number.
So, = .
7.MEASUREMENT There are approximately 2.54 centimeters in 1 inch. Express 2.54 as a mixed number.
SOLUTION:
The decimal 2.54 is 2 and 54 hundredths. So, 2.54 = or .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 2 • 5 4 0 0
Identify all sets to which each number belongs.
8.–632
SOLUTION:
The number –632 is an integer because it is the opposite of a whole number. It is also rational, because it can be
written as the fraction .
9.
SOLUTION:
The number isrational,becauseitcanbewrittenasthefraction .
10.21
SOLUTION:
The number 21 is a natural number because it is > 0. It is a whole number and an integer, because all whole
numbers are integers. It is also rational because it can be written as the fraction .
Write each number as a fraction.
11.
SOLUTION:
To write a mixed number as a fraction, multiply the whole number by the denominator and then add the numerator.
Write the result over the denominator. So, = .
12.–12
SOLUTION:
To write a whole number as a fraction, write the whole number over 1. So, –12 = .
13.
SOLUTION:
To write a mixed number as a fraction, multiply the whole number by the denominator and then add the numerator.
Write the result over the denominator. So, = .
14.49
SOLUTION:
To write a whole number as a fraction, write the whole number over 1. So, 49 = .
Write each decimal as a fraction or mixed number in simplest form.
15.3.625
SOLUTION:
The number 3.625 is 3 and 625 thousandths. So, 3.625 = or .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 3 • 6 2 5 0
16.0.55
SOLUTION:
The number 0.55 is 55 hundredths. So, 0.55 = or .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 5 5 0 0
17.–5.36
SOLUTION:
The number –5.36 is –5 and 36 hundredths. So, –5.36 = or .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 5 • 3 6 0 0
18.–0.265
SOLUTION:
The number –0.265 is –265 thousandths. So, –0.265 = or .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 2 6 5 0
19.–1.3
SOLUTION:
The number –1.3 is –1 and 3 tenths. So, –1.3 = .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 1 • 3 0 0 0
20.0.9
SOLUTION:
The number 0.9 is 9 tenths. So, 0.9 = .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 9 0 0 0
21.FINANCIAL LITERACY Recently, one U.S. dollar was equal to 0.506 British pounds. Express 0.506 as a
fraction.
SOLUTION:
The number 0.506 is 506 thousandths. So, 0.506 = or .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 5 0 6 0
22.POPULATION The estimated portions for various age groups of the population for 2010 are shown in the table.
a. Find the fraction of the population that is 19 years of age or younger.
b. Find the fraction of the population that is 20 to 64 years of age.
SOLUTION:
a. The portion of the population that is 19 years of age or younger is 0.27.
The number 0.27 is 27 hundredths and 0.27 = . So, the fraction of the population that is 19 years of age or
younger is .
b. The portion of the population that is 20 to 64 years of age is 0.60.
The number 0.60 is 60 hundredths and 0.60 = or . So, the fraction of the population that is 20 to 64 years of
age is .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 2 7 0 0
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 6 0 0 0
Write each decimal as a fraction or mixed number in simplest form.
23.
SOLUTION:
Let N represent the number.
So, = .
24.
SOLUTION:
Let N represent the number.
So, = .
25.0.161616...
SOLUTION:
Let N represent the number.
So, 0.161616... = .
26.
SOLUTION:
Let N represent the number.
So, = .
27.
SOLUTION:
Let N represent the number.
So, = .
28.
SOLUTION:
Let N represent the number.
So, = .
Identify all sets to which each number belongs.
29.–8
SOLUTION:
The number –8 is an integer because it is the opposite of a whole number. It is also rational, because it can be
written as the fraction .
30.14
SOLUTION:
The number 14 is a natural number because it is > 0. It is a whole number and an integer, because all whole
numbers are integers. It is also rational because it can be written as the fraction .
31.9.23
SOLUTION:
The number 9.23 is rational because it can be written as the fraction .
32.
SOLUTION:
The number is rational because it can be written as the fraction .
33.0.323322333...
SOLUTION:
The number 0.323322333... is irrational, because it is a decimal that neither terminates nor repeats.
34.3.141516...
SOLUTION:
The number 3.141516... is irrational, because it is a decimal that neither terminates nor repeats.
35.JEWELRY Maria has a bead that is 0.6 inch long. She wants to use the bead to fill a space that is inchlong.
Will the bead fit? Explain.
SOLUTION:
To determine whether or not the bead will fit, write the fraction asadecimal.
=0.625
Because 0.625 inch is greater than 0.6 inch, a bead that is 0.6 inch long will fit into a space that is inch long.
36.FOOD All of the Calories in one cup of milk come from fat, protein, and carbohydrates. Use the table to find the
fraction of Calories that comes from protein. Write the fraction in simplest form.
SOLUTION:
To find the fraction of Calories that comes from protein, first find the decimal part. To do this, add the decimal part
of Calories that comes from fat and carbohydrates and then subtract the total from 1.
1 – (0.03 + 0.53) = 0.44
So, 0.44 of the Calories come from protein. Write 0.44 as a fraction.
The number 0.44 is 44 hundredths. So, 0.44 = or . The fraction of Calories that comes from protein is .
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 4 4 0 0
Replace each _ with <, >, or = to make a true sentence.
37.–0.23 _ –0.3
SOLUTION:
To compare the decimals, look at the values in the tenths place. The number –0.23 lies to the right of –0.3 on the
number line, so –0.23 > –0.3.
38. _ 0.888…
SOLUTION:
Write as a decimal and then compare the decimals.
Since both values are equal to 0.888…, .
39.0.714 _
SOLUTION:
Write as a decimal and then compare the decimals.
Since 0.714 is to the left of on a number line, 0.714 < .
40. _ –0.9
SOLUTION:
The mixed number islessthan–1, and –0.9 is greater than –1. So, < –0.9.
41. _
SOLUTION:
Write as a decimal and then compare the decimals.
Since istotherightof4.625onanumberline, > .
42. _ 5.333…
SOLUTION:
Because isnegativeand5.333...ispositive, < 5.333....
Write each decimal as a fraction or mixed number in simplest form.
43.
SOLUTION:
Let N represent the number.
So, = .
44.
SOLUTION:
Let N represent the number.
So, = .
45.
SOLUTION:
Let N represent the number.
So, = .
46.
SOLUTION:
Let N represent the number.
So, = .
47.
SOLUTION:
Let N represent the number.
So, = .
48.
SOLUTION:
Let N represent the number.
So, = .
49.MULTIPLE REPRESENTATIONS Pi (π) is a nonrepeating, nonterminating decimal. Two common estimates
for pi are 3.14 and .
a. Graphical Use a calculator to find the value of πto seven decimal places. Then graph the three values on a
number line.
b. Symbolic Write an inequality comparing the values.
c. Verbal To find the circumference of a circle, you multiply pi by the diameter d of the circle. Explain when you
might use 3.14 to find the circumference and when you might use tofindthecircumference.
SOLUTION:
a. A calculator gives 3.1415927 as the value of πto seven decimal places.
b. Read the values from left to right on the number line to order them from least to greatest. So, 3.14 < π< .
c. If the diameter of a circle is a multiple of 7, you would use as the approximation for π, because multiplying
byamultipleof7eliminatesthefraction.Otherwise,youshoulduse3.14.
50.HISTORY The mathematician Archimedes believed that πwas between and .
a. Express each mixed number as a decimal rounded to the nearest thousandth. Was Archimedes’theory correct?
Explain.
b. The Rhind Papyrus records that the Egyptians used forπ. Express the fraction as a decimal rounded to the
nearest thousandth.
Whichvalueisclosertotheactualvalueofπ, Arcihimedes’or the Egyptians’value?
SOLUTION:
a. To find the decimal portion of each mixed number, divide the numerator by the denominator. So, ≈3.143and
≈3.141.
Archimedes’theory was correct, because π≈3.1415927whichisbetween3.141and3.143.
b. Divide the numerator by the denominator. So, ≈3.160.Archimedes’value of πis closer than the Egyptian’s
value.
Order each set of rational numbers from least to greatest.
51.–3.4, , , 3.38
SOLUTION:
Write the mixed number asadecimalandthencomparethedecimals.
=3.36...
< –3.4 < 3.36... < 3.38
So, the numbers in order from least to greatest are , –3.4, , 3.38.
52. , , , 0.32
SOLUTION:
Write the fractions as decimals and then compare the decimals.
=0.33...
=0.384615...
<0.32<0.33...<0.384615....
So, the numbers in order from least to greatest are , 0.32, , .
53. , –1.9, , –1.95
SOLUTION:
Write the mixed numbers as decimals and then compare the decimals.
≈–1.929
= –1.81...
–1.95 < –1.929 < –1.9 < –1.81.
So, the numbers in order from least to greatest are –1.95, , –1.9, .
54. , , , 9.82
SOLUTION:
Write the mixed numbers as decimals and then compare the decimals.
=9.8
≈9.846
<9.8<9.82<9.846
So, the numbers in order from least to greatest are , , 9.82, .
55.ANIMALS A lion’s speed is thespeedofacheetah.Findtheleastrationalnumberwithadenominatorof9that
is greater than . Find the greatest rational number with a denominator of 8 that is less than . Write an inequality
comparing the three numbers.
SOLUTION:
The least rational number with a denominator of 9 that is greater than is . The greatest rational number with a
denominator of 8 that is less that is . To compare these fractions, first write them as decimals.
=0.714285...
=0.7...
=0.625
So, .
56.OPEN ENDED Choose a repeating decimal in which three digits repeat. Write the number as a fraction or mixed
number in simplest form.
SOLUTION:
Sample answer: Use the decimal 0.231....To write it as a fraction, let N represent the number.
57.WRITING IN MATH Explain why isgreaterthan0.76.
SOLUTION:
The number meansthatthe7and6repeatindefinitely.Thenumber0.76terminates.Thatis,ithasa0inall
decimal places to the right of the hundredths place. So, is greater than 0.76.
58.CHALLENGE Antonio stated that =1.Showthatheiscorrect.
SOLUTION:
To show that Antonio is correct, write asafraction.
Let N = 0.999... so, 10N = 9.999…
So, =1
59.REASONING Determine whether the following statements are true or false. If true, explain your reasoning. If
false, give a counterexample.
a. All integers are rational numbers.
b. All whole numbers are integers.
c. A rational number is always an integer.
d. All natural numbers are rational.
SOLUTION:
a. This statement is true. Integers include all whole numbers and their opposites. Further, all integers can be
expressed as fractions. Therefore, they belong to the set of rational numbers.
b. This statement is true. All whole numbers and their opposites belong to the set of integers.
c. This statement is false. Rational numbers are numbers that can be expressed as fractions. They are not
necessarily whole numbers or their opposites. For example, isarationalnumberbutitisnotaninteger
d. This statement is true. All natural numbers are rational because they can be expressed as fractions.
60.WRITING IN MATH How do you compare and order fractions and decimals? Give an example to explain your
reasoning.
SOLUTION:
Sample answer: Convert all the numbers to fractions or decimals. Then use a number line to compare.
61.Which fraction is between 0.12 and 0.15?
A
B
C
D
SOLUTION:
To find the fraction that is between 0.12 and 0.15, write the fractions as decimals.
=0.12 =0.125 = 0.15 =0.2
The fraction is between 0.12 and 0.15. So, Choice B is the correct answer.
62.Which of the following is not a rational number?
F
G –4.27
H
J –3.131131113…
SOLUTION:
The number –3.131131113... is a nonrepeating nonterminating decimal. So, it is not a rational number. Choice J is the
correct answer.
63.Last football season, Jason made 0.85 of his field goal attempts. Write this decimal as a fraction in simplest form.
A
B
C
D
SOLUTION:
The number 0.85 is 85 hundredths. So, 0.85 = or in simplest form. Choice C is the correct answer.
thousands hundreds tens ones tenths hundredths thousandths ten-
thousandths
0 0 0 0 • 8 5 0 0
64.EXTENDED RESPONSE The table shows the results of a survey about how students get to school.
a. Write each decimal in the table as a fraction.
b. List the methods of transportation in order from least to greatest.
c. Which method of transportation do most students use to get to school?
SOLUTION:
a. Each of the decimals is to the hundredths place. To write them as fractions, write the number of hundredths over
100 and then simplify.
b. To list the methods of transportation in order from least to greatest, compare the portion of students written as
decimals. The methods in order from least to greatest are other, bicycle, walk, car, bus.
c. To determine which method of transportation most students use to get to school, look at the decimals in the table.
Most (0.40) students use the bus.
Method of
Transportation Fraction of Students
bus
walk
car
bicycle
other
Write each fraction as a decimal. Use a bar to show a repeating decimal.
65.
SOLUTION:
So, =–0.625.
66.
SOLUTION:
So, = .
67.
SOLUTION:
So, =–0.2.
68.
SOLUTION:
So, = .
Graph the figure below and its image after the transformation indicated.
69.translation 3 units down and 2 units left
SOLUTION:
In the original figure, Point A is at (1, 1), Point B is at (4, 2) and Point C is at (0, 4). If the figure is translated 3 units
down and 2 units left, subtract 2 from the x-coordinates and 3 from the y-coordinates to get the coordinates of the
translated figure. So, Point A’ is at (–1, –2), Point B’ is at (2, –1) and Point C’ is at (–2, 1).
70.translation 4 units up and 1 unit right
SOLUTION:
In the original figure, Point A is at (1, 1), Point B is at (4, 2) and Point C is at (0, 4). If the figure is translated 4 units
up and 1 unit right, add 1 to the x-coordinates and 4 to the y-coordinates to get the coordinates of the translated
figure. So, Point A’ is at (2, 5), Point B’ is at (5, 6) and Point C’ is at (1, 8).
71.reflection across the x-axis
SOLUTION:
In the original figure, Point A is at (1, 1), Point B is at (4, 2) and Point C is at (0, 4). If the figure is reflected across
the x-axis, leave the x-coordinates the same and make the y-coordinates the opposite of those in the original figure to
get the coordinates of the reflected figure. So, Point A’ is at (1, –1), Point B’ is at (4, –2) and Point C’ is at (0, –4).
72.reflection across the y-axis
SOLUTION:
In the original figure, Point A is at (1, 1), Point B is at (4, 2) and Point C is at (0, 4). If the figure is reflected across
the y-axis, make the x-coordinates the opposite of those in the original figure and leave the y-coordinates to get the
coordinates of the reflected figure. So, Point A’ is at (–1, 1), Point B’ is at (–4, 2) and Point C’ is at (0, 4).
State the domain and range for each relation.
73.{(0, 0), (3, 2), (4, 6), (8, 12)}
SOLUTION:
The domain is the set of the x-coordinates and the range is the set of the y-coordinates. So, the domain is {0, 3, 4, 8}
and the range is {0, 2, 6, 12}.
74.{(1, 2), (3, 4), (5, 6), (7, 8)}
SOLUTION:
The domain is the set of the x-coordinates and the range is the set of the y-coordinates. So, the domain is {1, 3, 5, 7}
and the range is {2, 4, 6, 8}.
75.Mount Kilimanjaro’s altitude is 5895 meters. Lake Assal’s altitude is –155 meters. Find the difference between
these altitudes.
SOLUTION:
To find the difference between the altitudes, subtract –155 from 5895.
5895 – (–155) = 6050
So, the difference between the altitudes is 6050 meters.
Find each product.
76.–6(–12)
SOLUTION:
The product of two integers with the same sign is positive. So, –6(–12) = 72.
77.15(–3)(–4)(0)
SOLUTION:
The product of any number and zero is zero. So, 15(–3)(–4)(0) = 0.
78.–3(5)(–9)
SOLUTION:
79.14(–20)
SOLUTION:
The product of two integers with different signs is negative. So, 14(–20) = –280.
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3-2 Rational Numbers
