Aquarium Problem Extension
The aquarium in the video was given as 15 feet long, 4 feet deep, and 6 feet high.
The video mentioned that 4000 gallons of water weighs 32,000 pounds.
It was stated that the height of the aquarium determines the thickness of the glass needed
to ensure that the glass is strong enough to support the weight of the water. Thicker glass
costs more, therefore, the taller the aquarium, the more expensive it will be to produce.
How many gallons of water will the aquarium in the video hold?
How much will the water weigh?
In order to answer this, there are at least two missing facts that are necessary for us to
find out. What are they? In the space below and using more paper if necessary,
 make a sketch of the aquarium
 write down what you already know and
 write down what you need to find out
 Solve the problem, making sure that your process is easily to follow, that you
include units, and that your answer is well marked.
4 ft
6 ft
15 ft
When you see this icon, it indicates a step towards the solution
When you see this icon, it indicates that we have to stop, think critically, and
make a decision that will get us closer to the solution.
When you see this icon, it indicates that we are taking a slight detour to
extend our answer, making it richer or making a connection that will
help us in future problems.
When you see this icon, it indicates that we are taking a
slight detour to explain a mathematical concept that will
help you with this problem and in future problems.
Given:
 We know that the volume of the aquarium is (4ft)(6ft)(15ft) = 360 ft3
 We know that 4000 gallons of water weighs 32000 pounds.
What do we want to know?
 We want to find out how many pounds the water in the aquarium weighs.
Below are the steps that I went through to solve this problem. This is not the only way to
solve the problem. There may be other ways that are more efficient or more elegant. In
true problem solving, you don’t always have the information that you need. You have to
figure out what you do have, what you still need, figure out a way to find what you need,
and then solve the problem. It’s a journey! Communication is an important part of the
journey, so you can share it with others.
We first need to find out how many gallons are in a cubic feet.
Then we can answer the question of how many pounds the water in the
aquarium weights.
Our volume is in cubic feet, not gallons.
Our weight is based on number of gallons, not cubic feet.
Where could we find that out?
The most efficient way is to look it up on the internet, but once you get there, you still
have to think!
I Googled “Conversion between cubic feet and gallons”.
Many sites were listed. Which is the best one to go to?
I looked at the first site listed and the text said “How many cubic feet in 1
gallon? The answer is 0.13368055556…” Although the answer did not have
a unit (not good!), it must be cubic feet. I don’t want cubic feet, I want
gallons. I could play with this fact and figure this out*(see note after the solution)….OR
I could look for another site that would help me. The second site was entitled “Cubic
Feet to Gallons Conversion.” This is the site that wanted:
http://www.metric-conversions.org/volume/cubic-feet-to-gallons.htm
When I got to the site, I was surprised to find out that there are 3 types of gallons! (U.S
liquid gallons, U.S. dry gallons, and UK gallons) Which do I choose?
I chose the U.S. liquid gallon. In this site, I was asked to enter the number
of cubic feet. The conversion formula was automatically applied to my
input of 360 cubic feet. The output was 2692.987 gallons.
If all I wanted to do was answer this problem, I’ve got the answer I need from the site.
For future reference, since I’m studying a lot about aquariums, it
would be nice to have a conversion factor at my fingertips so that I
would not have to look online each time I have a new aquarium.
It would be nice to know how many gallons are in 1 cubic foot. I enter 1 cubic foot as the
input and the output is 7.48051 gallons.
Do I want to use so many decimal points in my conversion?
It depends upon my goal for the answer…how my answer will be used.
Once I start rounding (using fewer decimal points), my final answer will
lose accuracy. Sometimes less accuracy is okay and sometimes accuracy is
very important.
360 ft3 7.48501gal

= 2694.6036 gal
1
1 ft3
360 ft3 7.49 gal

1
1 ft3
= 2696.4 gal
360 ft3 7.5 gal

1
1 ft3
= 2700 gal
(If you are confused by the “canceling”, see the Detour “Multiplying by One
Strategically”)
Which number do we use? For our purposes, we will use the 2700 gallons, since we
want to err on the side of caution…use the higher number so we don’t underestimate the
weight.
Now that we know that we have 2700 gallons in our aquarium,
we can find the final answer to our question:
How many pounds does the water in our aquarium weigh?
Going back to our givens, we know that 4000 gallons of water weighs 32000 pounds.
Above we converted between cubic feet and gallons. Now we need to convert between
gallons and pounds. We can use the same process of canceling out units by using our
conversion
Do we want “gallons per pound” or “pounds per gallon”?
The best unit rate to find here would be “pounds per gallon”, since we will are
given the number of gallons and we want to find the weight.
Let’s start by setting up the rate that will give us “pounds per gallon”, which means that
pounds is in the numerator and gallon is in the denominator.
Use the relationship given above (4000 gallons = 32000 pounds) to form “1”, then turn it
into a unit rate. (See Detour “Unit Rates” for more explanation.)
32000 pounds
32 pounds
=
4000 gallons
4 gallons
8 pounds
pounds
= 8
1gallon
gallon
So, we are almost there!
2700 gallons ∙ 8
pounds
= 21600 pounds
gallon
So the water in the tank weighs 21600 pounds!
*
Unit Rates
Earlier in the problem, a website told us that there are 0.13368055556 cubic feet in one
gallon. I mentioned that we could find out how many gallons there are in 1 cubic foot
from this fact. How do we do that?
One way to think about this is as a rate.
When saying a rate, we often use the word “per”.
When we see the word “per”, we can translate that mathematically into “divide”
Familiar rates are miles per hour, miles per gallon, feet per second, etc.
Rates can be expressed as fractions, which is a form of division. The first unit in the rate
goes in the numerator and the second unit in the rate goes in the denominator
35
miles
hour
29
miles
gallon
2
feet
sec ond
In each case mentioned above:
 the number (35, 29, 2) describes the unit in the numerator (miles, miles, feet)
 the number describing the unit in the denominator (hours, gallons, seconds) is
understood to be 1.
35 miles
1 hour
29 miles
1 gallon
2 feet
1 sec ond
These rates are called “unit rates”.
We write them without the “1”  35
miles
feet
miles
or 29
or 2
hour
sec ond
gallon
Vocabulary Alert!! The word “unit” used in the term “unit rate” is referring to the
number “one”.
This can be confusing. Above, we used the word “unit” in two different contexts.
We said that the units in the numerator were “miles, miles, and feet” and the units
in the denominator were “hours, gallons, and seconds”. Here we were using the
word “unit” to refer to “labels” or “units of measure”.
Another definition of “unit” is “one”. Think of the base ten number system. In
the number 432.5, the number 2 can be said to the in the “ones” place or in the
“units” place.
Unit rates always have an “understood 1” that describes the unit in the denominator.
These unit rates are so common that abbreviations are used (mph, mpg, fps)
So 35 mph means 35 miles in 1 hour
29 mpg means 29 miles for 1 gallon
Now, back to the fact that there are 0.13368055556 cubic feet in one gallon.
As a unit rate, this can be written
0.13368055556 ft 3
1 gallon
or 0.1336805556
Now, let’s go back to the unit rate
ft 3
gallon
2 feet
1 sec ond
If we travel 2 feet in 1 second, how many seconds does it take to go 1 foot?
It takes ½ seconds or .5 seconds to go 1 foot. This is an easy one to figure out in your
head, but what we did mathematically was to turn the rate above upside down.
1sec ond
2 feet
Now, take the numbers out front and keep the units as a fraction
or .5
1 sec onds
2
foot
sec onds
foot
When we switched the units in the numerator and denominator, their associated numbers
had to be switch as well. In effect, we found the reciprocal.
Back to
0.13368055556 ft 3
1 gallon
To find how many gallons per ft3, interpret “per” to mean “divide” and rewrite the above
as
1gallon
0.13368055556 ft 3
1
gallon
0.13368055556 ft 3
gallon
simplify by performing the division 7.48051948
ft 3
This matches the conversion we were given from the website.
take your numbers out front 
“Let your Units do the work for you!”
Cancelling Units as a Way to
Convert between Measurements
Above, we used the word “unit” in two ways:
 as a label
 as the value “one”
When we use units to convert between measurements, we use both of these definitions
for “unit”.
 We place the units strategically so that they will “cancel” (i.e. form “1”)
 We multiply by “1” in disguised forms.
Units are words, but they can be “cancelled” just like numbers can be “cancelled”. We
can use this fact to determine what form of “1” that we want to multiply by in order to
convert from one unit to another.
By “cancel”, we mean that, in a fraction, if we have a number as a factor in the numerator
and the same number as a factor in the denominator that we can use the commutative and
associative properties of multiplication to move them around and express them as “1”.
Example with numbers:
3 4
3 4
4
3 4
4
can be written as
, which is  or 1 =
53
35
5
3 5
5
Rather than go through these steps, we can cancel them out early…
3 4
4
=
53
5
We can also cancel units and doing this helps us to solve many real world problems. If
we cancel units strategically, the units will tell us whether we need to multiply or divide!
We don’t have to think as much!
Convert Between Measurements by Multiplying by One Strategically
(i.e. set up your units so that you can cancel them)
Notice how we are converting from ft3 to gallons below.
We are “multiplying by one”…making sure that our units “cancel”.
By “multiplying by one”, I mean that the numerator and the denominator in the
3
fraction are equivalent. We know that a fraction in the form of  1
3
In the same way, even if we are looking at things that seem different, such as
gallons and ft3, we know that 7.48501 gallons = 1 ft3
Therefore, because the numerator and denominator are equal,
7.48501gallons
=1
1 ft 3
So we can multiply by this form of 1 any way that will help us to get the answer we
are looking for. We used the above form of “1” when we solved our problem. Here
is another example that doesn’t use rates, but only uses units.
the fraction
Let’s say we want to convert 48 inches to yards.
We start with inches and strategically multiply by 1 in order to end up with yards.
48 inches 
1 foot
1yard

12inches
3 feet
If a number is on top, we multiply. If a number is on the bottom, we divide. We can
ignore the 1’s since multiplying by 1 does not change the value.
48 ÷12 = 4
4 ÷ 3 = 1.333
So 48 inches = 1.333 yards
(We could have also used the fact that 1 yard = 36 inches and used only one
conversion fraction.)
When we use rates, which have a unit on top and on bottom, the process is the
same. Change 60 miles per hour to feet per second.
60miles
1hour 1 min 5280 feet



hour
60 min 60 sec
1mile
Once we do all of the multiplication and division and cancel units, we have our
answer. We didn’t have to think, “Do I multiply? Do I divide?” The placement of
the units to form your “ones” so that you can cancel units automatically have you
do the correct operation.
60  5280
feet
feet
= 88
60  60 sec ond
sec ond