Lesson 1-3
Lesson
1-3
Vocabulary
Rational Numbers and
Their Uses
rate
rate unit
ratio
rational number
BIG IDEA
Measures usually require rational numbers.
Rates and Rate Units
Mental Math
In 2001, Tim Waterson, the DrumCanMan, set the record for world’s
fastest foot drummer, pounding out 1,030 beats in 60 seconds—with
his feet. We can ask, “On average, how many beats does Tim pound
out in each second?”
a. Give the floor area of a
rectangular room 15 feet
by 10 feet.
1,030 beats
_________
≈ 17.17 beats per second
60 seconds
b. Give the floor area of a
rectangular store 150 feet
by 100 feet.
Although “1,030 beats” and “60 seconds” are counts, “17.17 beats per
second” is not a count. It is a rate.
A quantity is a rate when it has a unit that contains the word “per” or
“for each” or some other synonym. Here are two examples of rates.
Every rate has a rate unit. The rate units below are in red.
The maximum speed limit in some states is 65 miles per hour.
The amount of snow Marquette, Michigan, gets is about
191.8 inches per year.
See Quiz Yourself 1 at the right.
QUIZ YOURSELF 1
Measures
Rates are measures, not counts. You can find many examples of
whole numbers used as measures, but these measures have often
been rounded. For instance, people are exactly 13 years old for only
an instant. But they will often say they are 13 for an entire year.
The key difference between a measure and
a count is that a measure unit can be divided
into smaller units. For instance, your age
can be measured in months, days, hours,
seconds, or even nanoseconds. Money
may be in dollars and in cents; but in some
calculations, parts of pennies are used.
Find the rate and the
rate unit in the following
sentence. It was estimated
that there were 1.108
cell phones per person in
Taiwan in 2005.
The thick black image at the
left is a human hair. Wrapped
around the hair is a nanowire
that is about 0.000000050
meter, or 50 nanometers,
thick. A nanometer is one
billionth of a meter.
Rational Numbers and Their Uses
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Chapter 1
Here are some examples of numbers used as measures. The measure
units are in red.
Marta’s little sister is 16 months old.
Kaya caught a trout that was 1.5 feet long.
Ann earned $2,709.25 at her summer job. (The measure unit
is dollars.)
See Quiz Yourself 2 at the right.
QUIZ YOURSELF 2
Ratio Comparisons
Counts have counting units. Measures have measure units. But some
uses of numbers do not have units. One of these uses is a ratio. A
ratio is a type of comparison involving division with the same unit.
Ratios often are given as percents.
In the sentence below,
identify the count and its
unit and the measure and
its unit.
Helen has two braids,
each 8.5 inches long.
GUIDED
Example 1
Here are three instances of ratios. Determine the ratio, the numerator, the
denominator, and the equivalent fraction.
a. The ratio of girls to boys in the first period English class is 18:12.
b. According to the Census Bureau, in 2005 the United States population
was about 11.3 times the Iraqi population.
c. The local stationery store recently reduced the price of pens to 75% of
the original price.
Solution To find the values, it helps to organize the information in a table.
What things are being compared?
Numerator Denominator
in
comparison in comparison
Situation
Unit
Ratio
a. Ratio of girls
to boys
people
? : ?
18 people
?
_____
12
?
? : ?
?
?
_____
?
? : ?
?
?
_____
b. Ratio of U.S.
population
to Iraqi
population
c. Ratio of new
price to old
price of pens
Fraction
?
?
?
?
?
QUIZ YOURSELF 3
See Quiz Yourself 3 at the right.
18
Ratios are often written in
lowest terms. Write the
ratio 18 :12 in lowest
terms.
Reading and Writing Numbers
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Lesson 1-3
Rational Numbers
All of the numbers that have been mentioned in this lesson are
rational numbers. The word rational comes from ratio. A rational
number is a number that can be written as a fraction with integers in
its numerator and denominator. (Of course, the denominator cannot
be 0 because it is impossible to divide by 0.) It is easy to recognize
rational numbers when they are written as fractions.
3
__
4
125
___
32
7
__
1
Other numbers that do not look like rational numbers may still
represent rational numbers.
Example
Written form
Rewritten to show the number is rational
0
Decimal notation
0
__
1
–12
Negative decimal notation
–12
___
1
25%
Percent
25
___
100
5:7
Ratio
5
_
7
4.392
Terminating decimal
4,392
____
1,000
4 __12
Mixed number
9
_
2
In general, a rational number can be written in the form __ab , where a
and b are integers and b is not 0.
See Quiz Yourself 4 at the right.
QUIZ YOURSELF 4
Write 0.3 as a fraction of
a
the form _b to show that
0.3 is rational.
Activity
Refer back to the numbers you found in the newspaper activity of
Lesson 1-1. On your list of numbers, find two examples of rational numbers
used as measures and used as ratio comparisons. Look at another
newspaper if your list did not have any examples.
Names for Decimal Places
Leap years occur because the length of a year is not exactly 365 days.
More accurately, it takes 365.242198 days for Earth to make
a revolution around the Sun.
Rational Numbers and Their Uses
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Chapter 1
Using this number, we can name the places to the right of the
decimal point.
s
hs
th
d
an
t us
s ths and tho s
h
s
t d
s
- h
ed an ou ed nt
ed
dr ns es nths ndr ous n-th ndr llio
n
i
hu te on te hu th te hu m
3 6 5. 2 4 2 1 9
8
The names of the places to the right of the decimal point are similar
to the names of the places to the left. Think of the ones place and the
decimal point as the center. Then there is perfect balance of names to
the right and to the left.
READING MATH
Notice the place-value
names at the left.
Ten-thousandths and
hundred-thousandths
have a hyphen. Make sure
to include the hyphen
when writing decimals in
word form because nine
hundred-thousandths
is 0.00009, but nine
hundred thousandths
is 0.900.
We often require many decimal places in everyday use. Some
grinding tools are accurate to within two millionths of an inch.
(That’s much less than the thickness of this page.) Computers work
at speeds often measured in nanoseconds.
In 1585, Simon Stevin, a Flemish mathematician, first extended the
use of decimal places to the right of the ones place. Before then,
fractions were used. In 1614, John Napier, a Scottish mathematician,
used Stevin’s idea to create tables of numbers with seven decimal
places, called logarithms, which greatly simplified computation. The
use of Napier’s tables quickly spread decimals throughout Europe.
Decimals are now more common than fractions for measurements.
Questions
COVERING THE IDEAS
In 1–3, write a rate that can be calculated from the given information.
Include the rate unit.
1. There are 40 cars parked in the 6 rows of that parking lot.
2. Shanté read 3 pages in 5 minutes.
3. Latisha read 3 _12 pages in 5 minutes.
In 4 and 5, the sentence contains a measure. Indicate the measure
and the measure unit.
4. There are four windows in the room, each 36 inches wide.
5. Luisa received a gift of $100 from a relative when she graduated
from 8th grade.
20
Reading and Writing Numbers
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Lesson 1-3
In 6–8, the sentence contains a ratio. Name
the ratio and indicate what two quantities are
being compared.
6. Kansas has a 5.3% sales tax.
7. Those are 10X binoculars.
8. About _38 of people in the United States
have blood type O positive.
In 9–12, rewrite each number as a ratio of two
whole numbers to show that it is rational.
9. 0.453
11. 15 __27
10. 6%
12. 15 thousand
In 13–15, include an example with your
explanation.
The red blood cells above are
shown magnified 1,520X, or
1,520 times their actual size.
13. Explain the difference between a rate and a ratio.
14. Contrast a measure with a count.
15. What is a rational number?
APPLYING THE MATHEMATICS
16. The length of a year in the Mayan calendar is an average
of 365.242129 days long. In this measure, identify the
indicated digit.
a. thousandths
b. thousands
c. millionths
d. tens
e. tenths
f. ten-thousandths
For 17 and 18, use the examples you gathered from the newspaper in
the Activity (or use the newspaper statements given in the Activity).
17. Classify each type of number used for measuring (for example,
whole number, negative integer, terminating decimal, simple
fraction, or mixed number).
18. Identify each ratio or rate.
19. The following sentence, which appeared in a newspaper article,
contains three numbers. “Revenues for the top 12 movies came
in at $116.5 million, down 16 percent from the same weekend
last year.”
a. Which of the numbers are whole numbers? (Be careful!)
b. Which of the numbers are rational numbers but not whole
numbers? (Be careful!)
Rational Numbers and Their Uses
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Chapter 1
In 20–23, make up a situation that could lead to the given rate.
20. 16 pairs of jeans per load
22.
1
_
2
cup per batch
21. 30 minutes per day
23. $0.37 for each ounce
24. In the Gregorian calendar that we use in the United States, a
year has an average length of 365.242500 days. Which calendar
year is closer to the actual length, the Gregorian calendar or the
Mayan calendar mentioned in Question 16?
REVIEW
25. Graph the numbers –40, 30, and –20 on a number line. Then
write a double inequality relating them. (Lesson 1-2)
26. Write the inequality suggested by the following sentence. A
profit of $500 is better than a loss of $1,000. (Lesson 1-2)
27. Multiple Choice If p is an integer and p > 2, then p could be
which of the following? (There may be more than one correct
choice.) (Lesson 1-2)
A 0
B 2
C 2.5
D 3
E 1,000
28. Identify whether the number is used for a count, for
identification, or for ordering. (Lesson 1-1)
a. Derrick and his grandfather sat in Section 152 to watch the
hockey game.
b. Calvin Coolidge was the thirtieth president of the United States.
EXPLORATION
29. When traveling abroad, you usually need local currency.
Currency-exchange rates are used to determine the equivalent
value of currency between two countries.
a. Use the Internet or a newspaper to locate the currencyexchange rates between the Mexican peso and the
U.S. dollar.
b. Convert $50 U.S. to Mexican pesos.
QUIZ YOURSELF ANSWERS
1. The rate is 1.108 cell
phones per person; the
rate unit is cell phones
per person.
2. Count: two, counting unit:
braids; measure: 8.5,
measuring unit: inches
3. 3:2
Mexican pesos
22
3
4. __
10
Reading and Writing Numbers
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