Perimeter
inches, feet, miles;
centimeters, meters, kilometers
Geometric
Formulas
Area
in2, ft2, mi2;
cm2, m2, km2
Volume
in3, ft3, mi3;
cm3, m3, km3
Rectangle
Box
Free Pre-Algebra
(
)
A = LW
P = 2 L + W = 2L + 2W
Lesson 39
Units in Triangle
Ratios and Rates
Lesson 39 ! page 1
V = LWH
1
bh
A = bh =
For the last several lessons, we’ve been
exploring
ratios,
rates,
and
proportions
P = a +b +c
2
2as ways of comparing data using division.
The many examples show how thinking proportionally is a very important part of how we understand data in everyday life.
We’ve seen that there are two types of comparisons with division, ratios and rates. In this lesson we compare and contrast
those concepts so that
you can use them more confidently in your own life and work.
Circle
( )
Sphere
Rates
C = !d = 2!r
4 3
A = !r 2
V = seen
!r in many
We’ve been using rates in this text(The
since
the
distance-rate-time
formula
was introduced in Lesson 6. As you’ve
perimeter of a circle
3
examples and homework problems,
the
formula
really
just
states
the
relationship
between
the
units
in
the
problem.
In the
is called the circumference.)
formula summary in Lesson 34, we saw:
Rates
Rate (Speed) is Distance over Time
r=
d
t
The related multiplication is
Distance = Rate • Time
d = rt
Examples
with
units:
miles =
miles
• hours
hour
feet =
Other
rates:
items =
items
• boxes
box
gas mileage =
feet
• seconds
second
miles
gallon
actions =
actions
• minutes
minute
servings per container =
amount per container
amount per serving
Since units
in the to
numerator
in the
denominator,
version
Fahrenheit
Celsius cancel with unitsProfit
Total
Cost =d = rt could be translated
P =multiplicative
R !C
= Revenue
– Cost the
C=
(
5 F ! 32
Cost per item • Number of Items
Net Pay numerator
= Gross Pay – units
Deductions
) = (Fnumerator
• denominator units
+ Fixed Costs
! 32 ) / 1.8 units =
denominator
P = G ! Dunits
9
T = CN + F
which seems almost too obvious to count as a formula. When you read a rate in words, you say the word per in place of the
fraction Celsius
bar. Sotowe
have “miles per hour,” “feet
perofsecond,”
per minute,” “items per box,” “miles per gallon.” The
Fahrenheit
Height
an object“actions
in feet t seconds
last example in the formula list, “servings perafter
container”
is made
falling (until
it hits):up of two other rates, “amount per container” and “amount
9
per serving.”
how
the C
units
work in this especially
complicated example:
F = Watch
C + 32
= 1.8
+ 32
h = !16t 2 + initial
t + initial
5
velocity
height
amount
amount
amount
amount
serving
distance 50
miles servings
servings per container = container =
÷
=
•
=
time
2
hours container
amount
container serving container amount
rate r miles per hour
serving
(
) (
)
distance
miles
distance
miles
© 2010
If we have
any Cheryl
two ofWilcox
the numbers in a rate,
we can50
find the
third by writing a proportion
and 50
cross-multiplying.
time
2
hours
Example: Find the missing quantity in each
problem.
rate
r miles per hour
distance 50 miles
time 2 hours
rate r miles per hour
distance 50 miles
time t hours
rate 25 miles per hour
distance
miles
50
miles 50x miles
time =t hours
2 hours
1 hour
rate 25 miles per hour
50 = 2x
distance x d
= 25 miles
miles per hour
time 2 hours
rate 25 miles per hour
50distance
miles d25 miles
time = 2 hours
t hours
1 hour
rate 25 miles per hour
50 = 25t
t = 2 hours
© 2010 Cheryl Wilcox
time t hours
rate 25 miles per hour
distance d
miles
time 2 hours
rate 25 miles per hour
d miles
2 hours
=
25 miles
1 hour
d = 25 • 2
d = 50 miles
rate 25 miles per hour
distance 50 miles
time t hours
rate 25 miles per hour
Free Pre-Algebra
distance d
miles
time 2 hours
rate 25 miles per hour
distance d
miles
time
2
hours example to the density
weight
20 below:
g
Compare the distance-rate-time
example
rate 25 miles per hour
volume 5 cm3
Example: Find the missing quantity in each problem.
density d g / cm3
weight 20 g
volume 5 cm3
density d g / cm3
20 weight
g
d20g g
= v 3cm3
volume
3
5 cm
1 cm
density 4 g / cm3
20 = 5d
d = 4 g/cm3
weight 20 g
volume v cm3
density 4 g / cm3
20 gweight 4wg
=
volume
5 3
3
v cm
density1 cm
4
20 = 4v
g
cm3
g / cm3
v = 5 cm3
rate 25 miles per hour
weight 20 g
volume 5 Lesson
cm3 39 ! page 2
density d g / cm3
weight 20 g
volume v cm3
density 4 g / cm3
weight w
volume 5
density 4
w g
3
=
g
cm3
g / cm3
4g
5 cm
1 cm3
w = 20 g
weight w g
volume 5 cm3
3
g / cm
Becausedensity
the units 4are part
of the
rate, the number that we get for the rate is dependent on the units used.
If you change the units of speed from miles per hour to feet per second, the speed of the car is the same but the
number given in the rate is different because the units are different.
If you measure density in pounds per cubic inch instead of grams per cubic centimeter the numbers you get will
be different although the material is the same.
If you calculate the unit price in dollars per ounce you will get a different number than euros per gram.
The Body Mass Index in Lesson 38 is a rate of weight in kilograms to height in meters. The number for the BMI
will change if we use weight in pounds and height in inches, which is why we used the multiplier 703 in the
formula with pounds and inches.
A change in unit in a rate can be used to bring large numbers “down to size” so we can understand them better. For
example, if you know that there were 4.3 million babies born in the U.S. in 2007, you could write this fact as a rate of
4,300,000 babies per year. But you could help people understand it better if you used a smaller unit of time in the
denominator. Since 1 year = 365 days, you could write
4,300,000 babies 4,300,000 babies
=
! 11,781 babies per day
1 year
365 days
This is still to large a number for easy comprehension. Using 24 hours = 1 day, and then 60 minutes = 1 hour, we get
11,781 babies 11,781 babies
=
! 491 babies per hour
1 day
24 hours
491 babies 491 babies
=
! 8 babies per minute
1 hour
60 minutes
Babies are not really born at this exact rate per minute, as if they were cars on an assembly line. Probably minutes pass in
which no babies are born, and then 20 or so are born all at once. But the rate gives us a clearer feeling for what the birthrate
of 4.3 million per year means. A rate like this is called an “average rate” because the occurrences are not really perfectly
regular.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 39 ! page 3
Ratios
A ratio is different from a rate because it compares quantities of the same kind, so that it is possible to cancel units. Instead
to the word “per,” we usually use “to” as below:
L = 3 inches
W= 1 inch
The ratio of length to width is
3 inches
=
1 inch
3
=3
1
The ratio of length to width is 3 to 1. (or 3:1)
Because the units cancel in the ratio, it doesn’t matter what units we use to measure the rectangle. If the same rectangle
were measured in centimeters, the ratio of length to width would still be 3 to 1.
L = 7.62 cm
W= 2.54 cm
The ratio of length to width is
7.62 cm
2.54 cm
= 7.62 ÷ 2.54 = 3
In this case we supply a denominator of 1 to write the two parts of
the ratio. It’s still a ratio of 3 to 1.
Because the units cancel, the ratio directly compares the length and width. The division in the ratio has a related
multiplication that is helpful.
length
= ratio
length = ratio • width
width
The related multiplication gives us an important interpretation of the ratio: “The length is 3 times the width.”
Because the units are the same for the numerator and denominator they don’t give us a clue about which numbers go
where, so it’s especially important to be careful in arranging ratios so that corresponding quantities are aligned. Putting the
numbers in a table is helpful.
Remember Thales in Lesson 2? Thales measured the height of the pyramid by measuring the length of the pyramid’s
shadow. In that lesson we had to wait for the moment that shadows were the same length as the objects that cast them.
Now that we know ratios, we can use shadows at any time of day. At any particular time the angle of the sun makes the ratio
of the height of an object to the length of the shadow the same for all the objects in the vicinity.
Example: The length of the shadow is proportional to the height of the animal casting the shadow. Duck is 3 feet
tall, and right now her shadow measures 2 feet. At the same time, Raging Dino’s shadow measures 2 meters long.
Find Raging Dino’s height.
Note that this is a ratio since we are comparing the length of the shadow and the height of the animal, measured in the same
units. It doesn’t matter that Dino and Duck are using different units, so long as the units within the ratio are the same.
Here written words really help to organize the information. The shadow
measurement can go on either the top or bottom of the fraction, as long as it goes
in the same place for both animals.
height
shadow
3 feet x meters
=
2 feet 2 meters
The units will cancel from the ratios.
x has to be 3 to make the ratios equal.
Raging Dino is 3 meters tall.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 39 ! page 4
If we put the information about Dino and Duck into a table, it would look like
Duck Dino Remember that the row ratios should also be proportional?
height
3 ft 3 m
3 feet
2 feet
shadow 2 ft 2 m That gives us 3 meters = 2 meters , which is true, but are these ratios?
Recall the map and model scales that worked more practically with a change of unit? The dollhouse scale is a true ratio,
1:12, and the dollhouse dimensions are 1/12 of a real house dimensions. But it’s far more practical to just change the name
of the unit involved, because 12 inches is 1 foot. The ratio is 1:12, but practically, we just change 5 inches to 5 feet to make
a conversion.
So technically, the division
3 feet
3 meters
is a ratio. To write the true ratio without units, we’d have to convert either feet to
meters or meters to feet. As long as the conversion is possible (both units measure length), we have a (potential) ratio. If the
conversion is not possible (miles are not the same kind of unit as hours, since one measures length and the other time), we
have a rate.
Example: Write the true ratio of 3 feet to 3 meters. (1 foot = 0.3048 meters)
First we convert 3 feet to meters.
3 feet 0.3048 meters
•
= 0.9144 meters
1
1 foot
Then we re-write the ratio in terms of meters:
3 feet
0.9144 meters
=
= 0.3048
3 meters
3 meters
You can see that the ratio is just the conversion factor originally given. Conversion factors for changing one unit to another
have a true ratio of 1.
0.3048 meter
=1
1 foot
60 minutes
=1
1 hour
5280 feet
=1
1 mile
The conversion factors can be built into a ratio formula so that you can put units in conveniently. For example, Widmark’s
formula for blood alcohol concentration yields a true ratio of ounces of alcohol in the blood to ounces of blood. But the
multiplier (7 for men, 7.8 for women) allows us to enter weight in pounds instead of ounces of blood into the formula,
because our weight in pounds is easier to find and use.
Ratios of Part to Part or Part to Whole
Suppose you make lemonade from a can of concentrate. The instructions tell you to add 4 cans of water to the concentrate.
In the end you will have a whole pitcher of lemonade, which contains 5 total cans of liquid. You can write several different
ratios for this situation.
concentrate
1 can
concentrate
1 can
or
water
4 cans
lemonade
5 cans
The first gives a ratio of part to part, the second a ratio of part to whole. We would say “The concentrate is 1/5 of the
lemonade,” or “There is 4 times as much water as concentrate,” depending on what was important to convey in the situation.
© 2010 Cheryl Wilcox
&'()*+%!,-.!
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Free Pre-Algebra
Lesson 39 ! page 5
Because the two parts of a ratio must be distinguished by some criteria, it’s sometimes difficult to tell if the “units” are the
4,23#!556!6786!9#,.!:0(-*!&-1033;#-*!)-!<)$*,-"#!&.(",*)0-!! same or different. For example, if we are comparing students taking
distance education courses to all students enrolled at
9#,.!:0(-*!&-1033;#-*!)-!<)$*,-"#!&.(",*)0-!:0(1$#$
the college with a quotient – is that a ratio or a rate? It is a ratio,
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college. The unit is “students enrolled at the college.” When the
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first quantity
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the/41+,)C!;;;;3!9#0#!D#+0!#$2!;;;;9!9#0#0)"!
whole quantity “young people in Surf City.”
Ratios of part to whole are often expressed as percents. That’s the subject of the next lesson.
=)>(1#!556!6?@6!9#,.!:0(-*!&-1033;#-*!)-!<)$*,-"#!&.(",*)0-!
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%$,+)#*)2! *%&$%L%,#$0"M! 4N)+! 0-)! O#*0! )%&-0! M)#+*'! P%0-! )N)+M! 0)+5! *)00%$&! #! +),4+2! 0-#0!
*1+O#**)2! 0-#0! 4L! 0-)! O+)N%41*! 4$)Q! R$! 7AAH"AI'! 4$"%$)! ,41+*)*! &)$)+#0)2! 6'FHF! <ST/'!
#,,41$0%$&!L4+!BQEU!4L!0-)!040#"!<ST/!L4+!0-)!M)#+Q!V%0-!0-)!,4$N)$%)$,)!4L!4$"%$)!,41+*)*!#$2!
0-)! %$,+)#*%$&! 0),-$4"4&%,#"! *4O-%*0%,#0%4$! 4L! *012)$0*'! 0-%*! 542)! 4L! ,41+*)! 2)"%N)+M! P%""!
,4$0%$1)!04!)WO#$2!%$!L101+)!M)#+*Q!!
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© 2010 Cheryl Wilcox
678!
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Free Pre-Algebra
Lesson 39 ! page 6
Lesson 39: Units in Ratios and Rates
Worksheet
Name___________________________________________
This side of the page has RATES. Solve with proportions by cross-multiplying.
1. A gardener weeded 3 rows per hour. If she worked 6
hours, how many rows were weeded?
2. Another gardener finished 20 rows in 5 hours. What was
her rate?
3. A pipe fills a tank at the rate of 19 gallons per minute. The
tank holds 250 gallons. How long will it take to fill the tank?
4. The population density of a region is measured in people
per unit area. Find the population density and write it with
units for the city of Manila, in the Phillipines, with the data
given. Population 1,660,714; Area 14.88 square miles
5. “In 2003, each person [in the U.S.] consumed about 142
pounds of sugar per year.” U.S. News and World Report
6. If you look at the information in problem 5 a different way,
there is a rate of 142 pounds of sugar per person. If the
U.S. population in 2003 was approximately 294,043,000
people, what was the total amount of sugar consumed
nationwide that year?
What is the amount of sugar the average person consumed
per day?
Convert the answer to ounces (16 oz = 1 lb) per day.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 39 ! page 7
This side of the page has RATIOS. Solve with proportions by cross-multiplying.
4. The shadow of a tree is 40 feet at the same time and
place that the shadow of a 5 foot tall woman is 6 feet long.
What is the height of the tree?
5. The ratio of students taking distance ed to students
enrolled is 0.24. If 21,673 students are enrolled, how many
are taking distance ed?
6. Fill in the blank to complete the sentence.
7. Fill in the blank to complete the sentence.
a. The ratio of length to width of a rectangle is 1.5 to 1.
a. The length of the rectangle is 2 times the width.
The length of the rectangle is ______ times the
width.
The ratio of length to width of the
rectangle is ______ to 1.
b. The ratio of students who smoke to the number of
students is 0.34.
b. There are 4 times as many students enrolled as are taking
distance education courses.
The number of students who smoke is _______
times the number of students.
The ratio of students enrolled to
students taking distance education courses
is _____ to 1.
8. The length of a rectangle is 1.5 times the width of the
rectangle. If the length is 4.8 meters, what is the width?
9. Find the true ratio of length to width if the length is 8.8
meters and the width is 4.4 cm. (100 cm = 1 m).
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 39 ! page 8
Lesson 39: Units in Ratios and Rates
Homework 39A
1. Use the nutrition label to make the
comparisons.
Name_______________________________________
a. About how many crackers are in a container?
b. How many grams per container?
c. If you eat 12 crackers, how many servings have you
eaten?
d. How many calories in 12 crackers?
Figure2. 2.(a)(a)Data
Datapoints
pointsfrom
fromU.S.
U.S.Navy
Navycruises
cruisesused
usedbybyRPW08,
RPW08,and
andthethedata
datarelease
releasearea
area(irregular
(irregularpolygon).
polygon).
Figure
(b)
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
(b)Figure
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
2. (a) Data points from U.S. Navy cruises used by RPW08, and the data release area (irregular polygon).
error
bars
show
residuals
in
the
regression
and
quality
of
ICESat
data.
(c,
d)
Spatial
patterns
of
ice
thickness
in
winter
error
bars
show
residuals
in
the
regression
and
quality
of
ICESat
data.
(c,
d)
Spatial
patterns
of
ice
thickness
in
winter
(b) Interannual changes in winter and summer ice thickness from RPW08 and K09 centered on the ICESat campaigns. Blue
(Feb
–bars
Mar)
and
fall
(Oct–in
Dec)
1988.
Mean
concentration
at
summer
minimum
(1978
Spatial
(Feb
–Mar)
and
fallresiduals
(Oct–Dec)
of of
1988.
(e)(e)
Mean
seasea
iceice
concentration
summer
minimum
(1978
(f, (f,
g) g)
Spatial
error
show
the
regression
and
quality
of
ICESat data.at (c,
d) Spatial
patterns
of –2000).
ice–2000).
thickness
in
winter
patterns
of
mean
winter
(Feb–
Mar)
and
fall
(Oct
–
Dec)
ice
thickness
from
ICESat
(2003–
2008).
(h)
Mean
sea
(Feb – Mar)
and fall
(Oct(Feb–Mar)
– Dec) of 1988.
sea ice ice
concentration
at summer
minimum
(1978 –2000).
(f, g)
patterns
of mean
winter
and (e)
fallMean
(Oct –Dec)
thickness from
ICESat
(2003–2008).
(h) Mean
seaSpatial
iceice
concentration
at
summer
minimum
(2003
–
2008).
Quantities
in
Figures
2c,
2d,
2f,
and
2g
are
mean
and
standard
deviation
patterns of mean
winter
(Feb – Mar)
fall (Oct
– Dec) iniceFigures
thickness
from
(2003
– 2008).
(h) Mean
sea ice
concentration
at summer
minimum
(2003and
–2008).
Quantities
2c, 2d,
2f, ICESat
and 2g are
mean
and standard
deviation
of
thickness
within
DRA. (2003 – 2008). Quantities in Figures 2c, 2d, 2f, and 2g are mean and standard deviation
atwithin
summer
minimum
ofconcentration
iceice
thickness
thethe
DRA.
2.ofThe
is from
the article
Decline in Arctic Sea Ice a. Compare the mean ice thickness in Nansen Basin in Period 3 to
ice table
thickness
within
the DRA.
Table 1. Mean
Ice
Thickness
at the End
Melt Season in thethe
Six thickness
Regions ofinthethe
Arctic
Ocean
FrominSubmarine
in 1958
Thickness.
Use
the
table
to
answer
theofquestions.
same
location
Period 1 Cruises
in one or
more– 1976,
Table 1. Mean Ice Thickness at the End of Melt Seasonain the Six Regions of the Arctic Ocean From Submarine Cruises in 1958 – 1976,
1993 –1.1997,
and
ICESat
Acquisitions
inof2003
– 2007
a
sentences.
Compare
directly
and
using
the
difference.
Table
Mean
Ice
Thickness
at
the
End
Melt
Season
in
the
Six
Regions
of
the
Arctic
Ocean
From
Submarine
Cruises
in
1958
– 1976,
1993 – 1997, and ICESat Acquisitions in 2003 – 2007
1993 – 1997, and ICESat Acquisitions
in 2003 – 2007a
Period
Change
Period
Change
Period
Change
(2)
–
(1)
(3)
– (1)
Period 1,
Period 2,
Period 3,
(2) – (1)
(3) – (1)
Period
1,76
Period
2,97
Period
3,07
(2)
–
(1)
(3)
–
(1) Percent
Region
58
–
93
–
03
–
Thickness
Percent
Thickness
Period
Period
Period
Region
58
– 76 1,
93
– 97 2,
03
– 07 3,
Thickness
Percent
Thickness
Percent
RegionCap
581.95
– 76
930.98
– 97
030.70
– 07
Thickness
Percent
Thickness
Percent
Chukchi
!0.97
!50
!1.25
!64
Chukchi
Cap
1.95
0.98
0.70
!0.97
!50
!1.25
!64
BeaufortCap
Sea
1.95
0.98
0.97
!0.97
!50
!0.97
!50
Chukchi
1.95
0.98
0.70
!0.97
!50
!1.25
!64
Beaufort
1.95
0.98
0.97
!0.97
!50
!0.97
!50
CanadaSea
Basin
3.45
2.05
1.70
!1.40
!40
!1.75
!51
Beaufort
Sea
1.95
0.98
0.97
!0.97
!50
!0.97
!50
Canada
3.45
2.05
1.70
!1.40
!40
!1.75
!51
NorthBasin
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
Canada
Basin
3.45
2.05
1.70
!1.40
!40
!1.75
!51
North
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
North
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
All Regions
3.02
1.62
1.43
!1.40
!46
!1.59
!53
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
aRegions
AllAll
Regions
3.02
1.62
1.43
!1.40
!1.59
!53
3.02
1.62 and changes
1.43
!1.40
!46 and percent.
!1.59
!53
Mean ice thickness
is shown in meters,
in thickness
are shown in !46
meters
a a
Mean
iceice
thickness
is is
shown
in in
meters,
and
changes
in in
thickness
areare
shown
in in
meters
and
percent.
Mean
thickness
shown
meters,
and
changes
thickness
shown
meters
and
percent.
c. What is the ratio of mean ice thickness in “All
Regions” during Period 3 to that during Period 2?
Round to two decimal places.
(3) – (2)
(3) – (2)
(3) – (2) Percent
Thickness
Thickness
Percent
Thickness
Percent
!0.28
!29
!0.28
!29 0
0.00
!0.28
!29
0.00
0 !17
!0.35
0.00
0
!0.35
!17
!0.38
!17
!0.35
!17
!0.38
!17
0.06
3
!0.38
!17
0.06
3 !5
!0.06
0.06
3
!0.06
!5!5
!0.19
!12
!0.06
!0.19
!12
!0.19
!12
b. What is the ratio of ice thickness at the North Pole in Period 1 to
of 5
ice3thickness
at the North Pole in Period 3? Round to two decimal
3ofof5 5
3places.
Fill in the blank:
The ice at the North Pole was __1.82___ times as thick during
Period 1 as it was during Period 3.
© 2010 Cheryl Wilcox
Free Pre-Algebra
3. The legend of a map shows 1 inch = 1 mile. What is the
real scale (the true ratio)?
Lesson 39 ! page 9
4. The triangles are similar. Find side c of the larger triangle.
(1 mile = 5280 feet, 1 foot = 12 inches)
5. The shadow of a bell tower is 80 feet long at the same
time a person 5.5 feet tall has a shadow of 4.8 feet. How tall
is the bell tower?
6. Find the population density (rate of people per square
mile) in San Francisco, California. Write the units with the
rate.
Population 776,733; Area 46.69 square miles
7. An earring made of silver with density 10.5 g/cm3 weighs
7 grams. What is the volume of the earring?
8. A man 6 feet 1 inch tall is aiming for a BMI of 24. What is
his desired weight?
BMI = 703
© 2010 Cheryl Wilcox
W (lb)
H (inches)
2
Free Pre-Algebra
Lesson 39 ! page 10
Lesson 39: Units in Ratios and Rates
Homework 39A Answers
1. Use the nutrition label to make the
comparisons.
a. About how many crackers are in a container?
28 servings
1 container
•
5 crackers
1 serving
= 140 crackers per container
b. How many grams per container?
28 servings
1 container
c. If you eat 12 crackers, how many servings have you
eaten?
5 crackers 12 crackers
=
1 serving
x servings
•
16 grams
1 serving
= 448 grams per container
d. How many calories in 12 crackers?
80 calories
x calories
=
5 crackers 12 crackers
5xthedata
= data
80release
•release
12 =area
960
Figure2. 2.(a)(a)Data
Datapoints
pointsfrom
fromU.S.
U.S.Navy
Navycruises
cruisesused
usedbybyRPW08,
RPW08,and
andthe
area(irregular
(irregularpolygon).
polygon).
Figure
5
x
=
12
5
x
/
5
=
12
/
5
(b)
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
(b)Figure
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
2. (a) Data points from U.S. Navy cruises used by RPW08,
the data 5
release
area
(irregular
5xand
= 960
x /5=
960
/ 5 polygon).
error
bars
show
residuals
in
the
regression
and
quality
of
ICESat
data.
(c,
d)
Spatial
patterns
of
ice
thickness
in
winter
error
bars
show
residuals
in
the
regression
and
quality
of
ICESat
data.
(c,
d)
Spatial
patterns
of
ice
thickness
in
winter
(b)
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
xand
= 2.4
(Feb
–bars
Mar)
fall
(Oct–in
Dec)
1988.
Mean
concentration
at
minimum
(1978
Spatial
=summer
(Feb
–Mar)
and
fallresiduals
(Oct–Dec)
of of
1988.
(e)(e)
Mean
seasea
iceice
concentration
summer
minimum
(1978
(f, (f,
g) g)
Spatial
error
show
the
regression
and
quality
of
ICESat data.atx(c,
d)192
Spatial
patterns
of –2000).
ice–2000).
thickness
in
winter
patterns
of
mean
winter
(Feb–
Mar)
and
fall
(Oct
–
Dec)
ice
thickness
from
ICESat
(2003–
2008).
(h)
Mean
sea
(Feb – Mar)
and fall
(Oct(Feb–Mar)
– Dec)
of 1988.
sea ice ice
concentration
at summer
minimum
(1978 –2000).
(f, g)
patterns
ofYou’ve
mean
winter
and (e)
fallMean
(Oct –Dec)
thickness from
ICESat
(2003–2008).
(h) Mean
seaSpatial
iceice
eaten
2.4 servings.
There
are
192
calories
in
12
crackers.
concentration
at
summer
minimum
(2003
–
2008).
Quantities
in
Figures
2c,
2d,
2f,
and
2g
are
mean
and
standard
deviation
patterns of mean
winter
(Feb – Mar)
fall (Oct
– Dec) iniceFigures
thickness
from
(2003
– 2008).
(h) Mean
sea ice
concentration
at summer
minimum
(2003and
–2008).
Quantities
2c, 2d,
2f, ICESat
and 2g are
mean
and standard
deviation
of
thickness
within
DRA. (2003 – 2008). Quantities in Figures 2c, 2d, 2f, and 2g are mean and standard deviation
atwithin
summer
minimum
ofconcentration
iceice
thickness
thethe
DRA.
2.ofThe
is from
the article
Decline in Arctic Sea Ice a. Compare the mean ice thickness in Nansen Basin in Period 3 to
ice table
thickness
within
the DRA.
Table 1. Mean
Ice
Thickness
at the End
Melt Season in thethe
Six thickness
Regions ofinthethe
Arctic
Ocean
FrominSubmarine
in 1958
Thickness.
Use
the
table
to
answer
theofquestions.
same
location
Period 1 Cruises
in one or
more– 1976,
Table 1. Mean Ice Thickness at the End of Melt Seasonain the Six Regions of the Arctic Ocean From Submarine Cruises in 1958 – 1976,
1993 –1.1997,
and
ICESat
Acquisitions
inof2003
– 2007
a
sentences.
Compare
directly
and
using
the
difference.
Table
Mean
Ice
Thickness
at
the
End
Melt
Season
in
the
Six
Regions
of
the
Arctic
Ocean
From
Submarine
Cruises
in
1958
– 1976,
1993 – 1997, and ICESat Acquisitions in 2003 – 2007
1993 – 1997, and ICESat Acquisitions
in 2003 – 2007a
Period
Change
Period
Change in Nansen Basin during
The mean ice thickness
Period
Change
(2)
–
(1)
(3)
– (1)
(3) – (2)
Period 1,
Period 2,
Period 3,
Period
3 was less than
(2)
– (1)
(3) – (1) the mean ice thickness
(3) – (2)
Period
1,76
Period
2,97
Period
3,07
(2)
–
(1)
(3)
–
(1)
(3)
– (2) Percent
Region
58
–
93
–
03
–
Thickness
Percent
Thickness
Percent
Thickness
Period
Period
Period
duringPercent
Period Thickness
1.Thickness
The thickness
declined
1.77
Region
58
– 76 1,
93
– 97 2,
03
– 07 3,
Thickness
Percent
Thickness byPercent
RegionCap
581.95
– 76
930.98
– 97
030.70
– 07
Thickness
Percent
Percent
Thickness
Percent
Chukchi
!0.97
!50
!1.25
!64
!0.28
!29
meters,
from
3.88
meters
to
2.11
meters.
Chukchi
Cap
1.95
0.98
0.70
!0.97
!50
!1.25
!64
!0.28
!29 0
BeaufortCap
Sea
1.95
0.98
0.97
!0.97
!50
!0.97
!50
0.00
Chukchi
1.95
0.98
0.70
!0.97
!50
!1.25
!64
!0.28
!29
Beaufort
1.95
0.98
0.97
!0.97
!50
!0.97
!50
0.00
0 !17
CanadaSea
Basin
3.45
2.05
1.70
!1.40
!40
!1.75
!51
!0.35
Beaufort
Sea
1.95
0.98
0.97
!0.97
!50
!0.97
!50
0.00
0
Canada
3.45
2.05
1.70
!1.40
!40
!1.75
!51
!0.35
!17
NorthBasin
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
!0.38
!17
Canada
Basin
3.45
2.05
1.70
!1.40
!40
!1.75
!51
!0.35
!17
North
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
!0.38
!17
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
0.06
3
North
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
!0.38
!17
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
0.06
3 !5
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
!0.06
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
0.06
3
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
!0.06
!5!5
All Regions
3.02
1.62
1.43
!1.40
!46
!1.59
!53
!0.19
!12
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
!0.06
b. What
the
ratio
ice!1.59
thickness!53
at!53
the North!0.19
Pole
1 to
aRegions
AllAll
Regions
3.02
1.62
1.43
!1.40
!1.59
!12
3.02
1.62 and changes
1.43
!1.40
!46
!0.19in Period
!12
Mean ice thickness
is shown in meters,
in thickness
are
shown is
in !46
meters
andof
percent.
a a
ice
thickness
at
the
North
Pole
in
Period
3?
Round
to
two
decimal
Mean
ice
thickness
is
shown
in
meters,
and
changes
in
thickness
are
shown
in
meters
and
percent.
Mean ice thickness is shown in meters, and changes in thickness are shown in meters and percent.
places.
3 of 5
c. What is the ratio of mean ice thickness in “All
Regions” during Period 3 to that during Period 2?
Round to two decimal places.
Period 3
Period 2
© 2010 Cheryl Wilcox
1.43 m
! 0.88
1.62 m
3 3ofof5 5
Period 1
Period 3
3.77 m
! 1.82
1.89 m
Fill in the blank:
The ice at the North Pole was __1.82___ times as thick during
Period 1 as it was during Period 3.
Free Pre-Algebra
Lesson 39 ! page 11
3. The legend of a map shows 1 inch = 1 mile. What is the real
scale (the true ratio)?
4. The triangles are similar. Find side c of the larger
triangle.
(1 mile = 5280 feet, 1 foot = 12 inches)
1 mile
1
•
5280 feet
1 mile
1 inch
1 mile
=
•
12 inches
1 foot
= 63,360 inches
1 inch
1
=
63,360 inches 63,360
a
c
( )( )
2.1 1.5
=
c
2.1
1.5c = 4.41
c = 2.94
1.5c = 2.1 2.1 = 4.41
1.5c / 1.5 = 4.41/ 1.5
Side c is 2.94 cm.
5. The shadow of a bell tower is 80 feet long at the same time
a person 5.5 feet tall has a shadow of 4.8 feet. How tall is the
bell tower?
5.5 h
=
4.8 80
height (ft)
shadow (ft)
6. Find the population density (rate of people per square
mile) in San Francisco, California. Write the units with the
rate.
Population 776,733; Area 46.69 square miles
776,733 people
2
46.69 mi
( )( )
4.8h = 5.5 80 = 440
4.8h = 440
4.8h / 4.8 = 440 / 4.8
x people
1 mi2
46.69x = 776,733
46.69x / 46.69 = 776,733 / 46.69
x = 16,635.96059
h = 91.6
The tower is about 91.7 feet tall.
7. An earring made of silver with density 10.5 g/cm3 weighs
7 grams. What is the volume of the earring?
10.5 g
3
1 cm
10.5x = 7
=
=
7g
3
x cm
10.5x / 10.5 = 7 / 10.5
x = 0.6
The volume is about 0.7 cm3
There are about 16,636 people per square
mile in San Francisco.
8. A man 6 feet 1 inch tall is aiming for a BMI of 24. What
is his desired weight?
BMI = 703
W (lb)
H (inches)
2
6 feet 1 inch = 72 inches + 1 inch = 73
inches
24 = 703
W
732
703W
5329 703W
5329
= 24
•
= 24 •
5329
703 5329
703
W = 181.9288762
His desired weight is about 182 pounds.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 39 ! page 12
Lesson 39: Units in Ratios and Rates
Homework 39B
1. Use the nutrition label to make the
comparisons.
Name_________________________________________
a. How many brownies are in a container?
b. How many grams per container?
c. If you eat 2 brownies, how many servings have you
eaten?
d. How many calories in 2 brownies?
Figure2. 2.(a)(a)Data
Datapoints
pointsfrom
fromU.S.
U.S.Navy
Navycruises
cruisesused
usedbybyRPW08,
RPW08,and
andthethedata
datarelease
releasearea
area(irregular
(irregularpolygon).
polygon).
Figure
(b)
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
(b)Figure
Interannual
changes
in
winter
and
summer
ice
thickness
from
RPW08
and
K09
centered
on
the
ICESat
campaigns.
Blue
2. (a) Data points from U.S. Navy cruises used by RPW08, and the data release area (irregular polygon).
error
bars
show
residuals
in
the
regression
and
quality
of
ICESat
data.
(c,
d)
Spatial
patterns
of
ice
thickness
in
winter
error
bars
show
residuals
in
the
regression
and
quality
of
ICESat
data.
(c,
d)
Spatial
patterns
of
ice
thickness
in
winter
(b) Interannual changes in winter and summer ice thickness from RPW08 and K09 centered on the ICESat campaigns. Blue
(Feb
–bars
Mar)
and
fall
(Oct–in
Dec)
1988.
Mean
concentration
at
summer
minimum
(1978
Spatial
(Feb
–Mar)
and
fallresiduals
(Oct–Dec)
of of
1988.
(e)(e)
Mean
seasea
iceice
concentration
summer
minimum
(1978
(f, (f,
g) g)
Spatial
error
show
the
regression
and
quality
of
ICESat data.at (c,
d) Spatial
patterns
of –2000).
ice–2000).
thickness
in
winter
patterns
mean
winter
(Feb–ofMar)
(Oct
icethickness
thicknessfrom
ICESat
(2003–
2008).
Mean
seaiceice
(Feb
– Mar)
and
fall
(Oct(Feb–Mar)
– Dec)
1988.
(e)
Mean
sea– Dec)
ice ice
concentration
at from
summer
minimum
(1978
–2000).
(f, g)
patterns
of ofmean
winter
andand
fallfall
(Oct
–Dec)
ICESat
(2003–2008).
(h)(h)Mean
seaSpatial
concentration
summer
minimum
(2003
– fall
2008).
Quantities
in
Figures
2d,
are
mean
standard
deviation
patterns
of mean
winter
(Feb – Mar)
and
(Oct
– Dec) inice
thickness
from
ICESat
(2003
– 2008).
(h)
Mean
sea ice
concentration
at at
summer
minimum
(2003
–2008).
Quantities
Figures
2c,2c,
2d,
2f,2f,
andand
2g2g
are
mean
andand
standard
deviation
of
ice
thickness
within
the
DRA.
concentration
at
summer
minimum
(2003
–
2008).
Quantities
in
Figures
2c,
2d,
2f,
and
2g
are
mean
and
standard
deviation
of ice thickness within the DRA.
2.ofThe
is from
the article
Decline in Arctic Sea Ice a. Compare the mean ice thickness in the Chukchi Cap in Period 3
ice table
thickness
within
the DRA.
Table 1. Mean
Ice
Thickness
at the End
Melt Season in theto
Sixthe
Regions
of theinArctic
Oceanlocation
From Submarine
in or
1958
– 1976,
Thickness.
Use
the
table
to
answer
theofquestions.
thickness
the same
in PeriodCruises
1 in one
more
Table 1. Mean Ice Thickness at the End of Melt Seasonain the Six Regions of the Arctic Ocean From Submarine Cruises in 1958 – 1976,
1993 –1.1997,
and
ICESat
Acquisitions
inof2003
– 2007
a
sentences.
Compare
directly
and
using
the
difference.
Table
Mean
Ice
Thickness
at
the
End
Melt
Season
in
the
Six
Regions
of
the
Arctic
Ocean
From
Submarine
Cruises
in
1958
–
1976,
1993 – 1997, and ICESat Acquisitions in 2003 – 2007
1993 – 1997, and ICESat Acquisitions
in 2003 – 2007a
Period
Change
Period
Change
Period
Change
(2)
–
(1)
(3)
– (1)
(3) – (2)
Period 1,
Period 2,
Period 3,
(2) – (1)
(3) – (1)
(3) – (2)
Period
1,76
Period
2,97
Period
3,07
(2)
–
(1)
(3)
–
(1)
(3)
– (2) Percent
Region
58
–
93
–
03
–
Thickness
Percent
Thickness
Percent
Thickness
Period
Period
Period
Region
58
– 76 1,
93
– 97 2,
03
– 07 3,
Thickness
Percent
Thickness
Percent
Thickness
Percent
Region
58
–
76
93
–
97
03
–
07
Thickness
Percent
Thickness
Percent
Thickness
Percent
Chukchi Cap
1.95
0.98
0.70
!0.97
!50
!1.25
!64
!0.28
!29
Chukchi
Cap
1.95
0.98
0.70
!0.97
!50
!1.25
!64
!0.28
!29 0
Beaufort
Sea
1.95
0.98
0.97
!0.97
!50
!0.97
!50
0.00
Chukchi Cap
1.95
0.98
0.70
!0.97
!50
!1.25
!64
!0.28
!29
Beaufort
1.95
0.98
0.97
!0.97
!50
!0.97
!50
0.00
0 !17
CanadaSea
Basin
3.45
2.05
1.70
!1.40
!40
!1.75
!51
!0.35
Beaufort
Sea
1.95
0.98
0.97
!0.97
!50
!0.97
!50
0.00
0
Canada
3.45
2.05
1.70
!1.40
!40
!1.75
!51
!0.35
!17
NorthBasin
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
!0.38
!17
Canada
Basin
3.45
2.05
1.70
!1.40
!40
!1.75
!51
!0.35
!17
North
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
!0.38
!17
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
0.06
3
North
Pole
3.77
2.27
1.89
!1.51
!40
!1.89
!50
!0.38
!17
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
0.06
3 !5
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
!0.06
Nansen
Basin
3.88
2.05
2.11
!1.83
!47
!1.77
!46
0.06
3
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
!0.06
!5!5
All Regions
3.02
1.62
1.43
!1.40
!46
!1.59
!53
!0.19
!12
Eastern
Arctic
3.24
1.30
1.24
!1.94
!60
!2.00
!62
!0.06
b. What is the
ratio
of !1.59
ice!1.59
thickness!53
at!53
the North!0.19
Pole
3 to
aRegions
AllAll
Regions
3.02
1.62
1.43
!1.40
!12
3.02
1.62 and changes
1.43
!1.40
!46
!0.19in Period
!12
Mean ice thickness
is shown in meters,
in thickness
are shown in !46
meters
and percent.
a a
ice
thickness
at
the
North
Pole
in
Period
1?
Round
to
two
decimal
Mean
iceice
thickness
is is
shown
in in
meters,
and
changes
in in
thickness
areare
shown
in in
meters
and
percent.
Mean
thickness
shown
meters,
and
changes
thickness
shown
meters
and
percent.
places.
3 of 5
c. What is the ratio of mean ice thickness in “All
Regions” during Period 2 to that during Period 1?
Round to two decimal places.
3 3ofof5 5
Fill in the blank:
The ice at the North Pole was _____ times as thick during Period
3 as it was during Period 1.
© 2010 Cheryl Wilcox
Free Pre-Algebra
3. The legend of a map shows 1 centimeter = 0.5 kilometers.
What is the real scale (the true ratio)?
Lesson 39 ! page 13
4. The triangles are similar. Find side a of the smaller
triangle.
(1 km = 1000 m, 1 m = 100 cm)
5. The shadow of a bell tower is 90 feet long at the same
time a woman 5 feet tall has a shadow of 5.5 feet. How tall is
the bell tower?
6. Find the population density (rate of people per square
mile) in Pleasant Hill, California. Write the units with the rate.
Population 34,199; Area 7.294 square miles
7. A bracelet made of silver with density 10.5 g/cm3 weighs
80 grams. What is the volume of the bracelet?
8. A man 5 feet 4 inches tall is aiming for a BMI of 22. What
is his desired weight?
BMI = 703
© 2010 Cheryl Wilcox
W (lb)
H (inches)
2