Showing posts with label Math. Show all posts
Showing posts with label Math. Show all posts

How to Learn Algebra Fast and Easy, Online, on Your Own, and Free — Rules, Equations, Solutions — For Beginners

Latest update: October 23, 2020
I originally published this article elsewhere on 10/22/2010 and have been updating it ever since. Copyright and historical notes at end of article.

Here is your online, complete, free, beginner algebra and equations tutorial. Easily and quickly learn algebra on your own. It is recommended that one does not attempt to do the entire tutorial in a single session; bookmark and return as desired. If you already know arithmetic (including fractions and decimals), then you already know algebra. You just don't know you know yet. If you understand the answers to the following statements, proceed with this page; otherwise, it is probably not a good idea.
  • 4 + 5 = 9
  • 17 - 13 = 4
  • 5 times 7 = 35
  • 70 divided by 35 = 2
  • 80 divided by 25 = 3.2

The Basics


Example #1

Algebra is nothing more than merely substituting letters for numbers. As an example:

       4 + 2 = 6

So, if we say the letter A is temporarily equal to 4, i.e.:

      A = 4

And the letter B is temporarily equal to 2, i.e.:

      B = 2

Then A plus B must equal 6, i.e.:

      A + B = 6

Example #2


      A = 5

      B = 3

So,

      A + B = ?

Well, if we replace the letter A with 5, then the question becomes:

      5 + B = ?

And then when we replace the letter B with 3, we have:

      5 + 3 = ?

Problem solved.

A side note: Algebra likes to use the letter X in place of the question mark. So the correct way to have stated the above question would have been to say:

      A = 5

      B = 3

      X = A + B

What is X? The answer is:

      X = 8

You have just learned the basic concept of algebra.

Example #3: Subtraction


      A = 19

      B = 14

      X = A - B

What is X?

We plug in the numbers and we get:

      X = 19 - 14

      X = 5

Multiplication and Division

Of course multiplication and division in algebra are just the same as in arithmetic.

Multiplication Example

(The asterisk sign (“*”) is used to replace the word “multiply.”)

      A = 20

      B = 5

      X = A * B

What is X? We plug in the numbers and we get:

      X = 20 * 5

      X = 100

Division Example

(The “/” sign is used to replace the word “divide.”)

      A = 20

      B = 5

      X = A/B

What is X? We plug in the numbers and we get:

      X = 20/5

      X = 4

Let's Mix Things Up

You now know all the arithmetic functions of algebra. Algebra lets you mix and combine these functions.

For example:


      A=1

      B=2

      C=3

      D=4

      X=A+B+C+D

      X=10

Let’s include subtraction:


      X=A + B + C - D

      X=(1+2+3) - 4, or X = 6 - 4, which is 2, or

      X = 6 - 4 = 2

Yes, there can be more than one equal sign in an equation. Instead of saying,

      A=42
      B=42
      C=42
      D=42

You can say,

      A=42
      A=B=C=D

Or just say,

      A=B=C=D=42

Side note. You have been solving equations since the first paragraph.

Just Some Random Example NASA Formulas

In Algebra How Do You Solve for V? Basic / beginner algebra volume formulas.

About the NASA Formula Examples

Note the "d²" in the volume formula for the cylinder. Yes, the upper "2" means the variable "d" is squared or itself times itself or "d" to the second power.

Note the "a³" in the volume formula for the cube. Likewise, the upper "3" means the variable "a" is cubed or itself times itself times itself or "a" to the third power.

Notice how some of the variables in the formulas are directly adjacent to each other. This is the standard used to indicate the variables are multiplied.

Examples

  • The rectangular prism formula or equation, V = a b h, means volume is equal to "a" times "b" times "h".
  • The top half of the volume for the sphere formula or equation, "πd³", means pi times d after d has been cubed. If d was equal to 5, then d³ would equal 125, making the equation π times 125 or 125π.
  • Yes, the horizontal slash in the sphere and cylinder formulas means divide by the lower number, 6 and 4 respectively.
  • As mentioned, "π" is the well-known symbol for pi. The approximate value of pi is 3.14159; this approximation serves most everyday purposes just fine.

More Multiplication Practice


Example #1


      A=1
      B=2
      C=3
      D=4

      X=A+B*C-D

What is X?

Simplify and solve.

When you see an equation has multiplication and division mixed into it, the rule is to do the multiplication and division first, then do the +’s and -‘s.

So the equation above really means,

      X = A + (B*C) - D or

      X = 1 + (2*3) - 4 or

      X = 1 + (6) - 4

      X = 3

The “(“ and “)” are used to indicate what parts of the equation to do first.

It should be noted X=A and A=X are mathematically equivalent.

Just Like the Pros

What you have been and are doing is just simplifying, a.k.a breaking down, the equation one piece at a time; just like the mathematicians do it. The mathematicians are no more able to look at an equation and instantly come up with the answer any better than the rest of us can. In other words, they can’t grasp the whole equation either. They just solve and proceed from line to line, trusting they solved the previous line(s) correctly.

Example #2

Here is another one:

      A=1, B=2, C=3, D=4, E=5, F=6

      ((D * B) + (F - 7)) + A) * C = X.

What is X?

This time there is more than one set of parentheses. When that happens, the rule is to do the innermost ones first. So let’s start solving this equation by breaking it down.

The (D*B) and the (F-7) are the innermost parts of the equation.

Let’s start with the (D*B).

      D * B = 4 * 2 = 8,

so we simplify the equation to,

      (8 + (F-7) + A) * C = X

Next is the (F-7).

      F - 7 = 6 - 7

This results in a number one less than zero, so we say negative one or -1.

(Another example would be 15-20. This results in a number 5 less than zero, so we say negative 5 or -5.)

The equation now looks like,

      (8 + (-1) + A) * C = X

Let’s get the A and C taken care of; the equation is now,

      (8 + (-1) + 1) * 3 = X

Next we add up the numbers inside the parenthesis.

-1 plus 1 equals zero of course.

Or you could have said: -1 plus 8 equals 7. The 8 is called a positive number, just as the -1 is called a negative number. Adding a positive number to a negative number is really just subtracting the negative number from the positive number. In other words:

      8 + (-1) = 8 - 1 = 7 or 1 + (-1) = 1 -1 = 0

Either way, our equation now looks like,

      (8 - 1 + 1) * 3 = X, which is

      (8) * 3 = 24 = X, or

      8 * 3 = 24 = X, or

      X = 24

Simplified a step at a time and solved.

If you didn’t know negative numbers before, now you do. For the sake of completeness, the next section is about what else one should know about negative numbers.

More About Negative Numbers


Numbers plus negative numbers result in lesser numbers. Keep in mind -10 is a lesser number than -5, etc.

Numbers minus negative numbers result in larger numbers. For example, whereas 9-5 = 4, but 9-(-5) = 14. In other words, minus minus results in a positive increase a.k.a a lesser lesser or a larger larger. Minus a minus is exactly the same as plus a plus, e.g. -(-25)=25.

This is a good time to mention that in mathematics, two negatives equal a positive when applied to minus a minus subtraction, or any multiplication, or any division.

For multiplication:
  • Negative numbers times positive numbers equal negative numbers, e.g. -5 * 4 = -20.
  • Negative numbers times negative numbers equal positive numbers, e.g. -5 * -4 = 20.
  • You already knew positive numbers times positive numbers equal positive numbers.
For division, the same rules apply:
  • Negative numbers divided by positive numbers (or vice versa) equal negative numbers, e.g. -5/4 = -1.25 and 5/-4 = -1.25.
  • Negative numbers divided by negative numbers equal positive numbers, e.g. -5/-4 = 1.25.
You already knew positive numbers divided by positive numbers equal positive numbers.

More Example NASA Formulas

In Algebra How Do You Solve for V? Learning and doing volume formulas.


Using Spreadsheets

Spreadsheet software or applications will happily do the arithmetic and sort out the negatives versus the positives for you once you have replaced all the variables. It even knows to do the innermost before the outermost, etc. As an example, suppose you have simplified an equation to the following mess:

      X=((5-3)* 52)-21+((6+7)/(34-12))

If your spreadsheet software is MS Excel or you are using cloud Google Drive, you can exclude the X and just copy/paste the following into a single cell:

      =((5-3)* 52)-21+((6+7)/(34-12))

The spreadsheet will immediately solve the equation and give back the answer of 83.5bunchmoredigits. If you have the software or Google Drive access, go ahead and try it.

If you are experienced at spreadsheet calculations, you can, of course, do equations with the variables still in place; substituting the variables with cell locations or range names.

Another Division Example

Might as well keep it simple and use the previous variables.

      A = 5
      B = 34
      C = 21

      X=((A-3)* 52)-C+((6+7)/(B-12))

We replace the variables with the assigned numbers and we are right back where we started from:

      X=((5-3)* 52)-21+((6+7)/(34-12))

The arithmetic then gives us:

      X = 83.59090909...

Dividing by Zero

This is a good time to mention you cannot divide by zero.

For example:

      If A=1
      If A=2
      If A=3

      X = 5 + 10/(3-A)

Now if A=1, then

      X=5+10/(3-1)=5+10/2=5+5=10

Now if A=2, then

      X=5+10/(3-2)=5+10/1=5+10=15

If, however, we attempt to declare the variable A as A=3, the following occurs:

      X=5+10/(3-3)=5+10/0. (invalid)

At this point the equation becomes invalid. There is no answer to the question, “What is 10 divided by 0?”. An equation immediately becomes invalid when a divide-by-zero scenario occurs. Software applications are designed to recognize this when it happens. Plugging whatever-divided-by-zero into a spreadsheet used to give interesting results, before applications were modified to detect this.

What You've Learned

The basic concept of algebra is just plugging the numbers into the variables, and then doing the arithmetic. One merely keeps simplifying the equation until it is solved. You now have a full understanding of that concept. Yes, you have been using variables since the first paragraph.

Final Example

Here is the last example. It is presented in a different format. The question, however, remains the same. What is X? You already know everything needed to solve this equation.

      A=1, B=2, C=3, D=4, E=5
      T=-1, U=-2, V=-3

      (6X/8)+(2T+4)=((CD/2)-AD)+V

It should be noted 6X means the same as 6*X; and AD means the same as A*D. Other examples would be: 3A=3*A=A*3, 5Y=5*Y=Y*5, -2C=-2*C=C*-2, etc.

We plug the numbers into the variables, and the equation now is:

      (6X/8)+((2*-1)+4)=((3*4)/2)-(1*4)+-3

Some simplifying arithmetic gives us:

      (6X/8)+-2+4=(12/2)-4+-3

More arithmetic then gives us:

      6X/8 +2=6-4+-3

More arithmetic gives us:

      6X/8+2=-1

We can’t solve X as the equation is currently stated; so we will have to move things around and do more arithmetic.

Important Note

Whenever you change the actual value on one side of the equation, you must do the same on the other side of the equation. Example: 7=7. If you subtract 3 from the left side, then you must subtract 3 from the right side; thus 4=4. The same rule applies for addition, multiplication, and division.

Let’s subtract 2 from both sides of our equation.

      6X/8+2=-1

Then becomes:

      6X/8=-3

We have to get rid of the “divide by 8” part of the left side of the equation. So we multiply both sides of the equation by 8.

      6X/8=-3

Then becomes:

      6X=-24

We must make the X stand alone, so we divide both sides by 6.

      6X=-24

Then becomes:

      X = -4 (The Answer!)

How Do We Know If We Have the Right Answer?

To find out, we go back to the original equation and replace X with -4. We then simplify (reduce) the equation as before to its simplest form. If the simplest possible construct is valid; then, by definition, the statement “X=-4” is valid.

Here is the original equation.

      A=1, B=2, C=3, D=4, E=5
      T=-1, U=-2, V=-3

      (6X/8)+(2T+4)=((CD/2)-AD)+V

We don’t have to re-solve the parts that didn’t have the X in it to begin with, so we have:

      (6X/8)+2 = -1

We replace the X with -4, giving us:

      ((6*-4)/8)+2=-1

Simplifying gives:

      (-24/8)+2=-1

Which is:

      -3+2=-1

Which is:

      -1=-1

This construct is valid and simple enough to know X=-4 is valid.

To take it to the very end, you can multiply both sides by -1, giving us:

      1=1

What Would Have Happened, If Instead of Correctly Calculating X=-4, We Had Erroneously Calculated X=16?

The equation simplification/reduction would have proceeded smoothly to this point:

      (6X/8)+2 = -1 (as above)

When the 6X is replaced with 6*16, we get:

      (96/8)+2=-1 (false)

When further simplified says:

      12+2=-1 (false)

Which is

      14=-1 (false)

The resulting false statement by definition means the original calculation of "X=16" is a false statement.

The Adventure Continues...


There is a lot more (much, much more) to algebra, but it is really only an expansion of what you have already learned. Algebra is the basis of all other mathematics; including geometry, trigonometry, calculus, and so on. A good understanding of algebra is required to succeed at the other mathematics. Mathematics, itself, is the foundation of most other disciplines. This foundation is not just necessary for the sciences such as physics, electronics, chemistry, biology, astronomy, and so on. A mathematical foundation is necessary for many careers; including marketing, economics, architecture, and many, many others.

May all your calculations be prosperous ones!

Copyright and Historical Information

This article was originally published on 10/22/2010 at a site called "hubpages". The article remained there for a few years and was then transferred to a site called "owlcation". The article remained at the second site for a few years. Both sites had and have severe platform management problems, thus the article has been moved here to websitewithnoname.com. This article is copyrighted. If anyone comes across a stolen copy, please let me know in the Comments Section below. Meanwhile, the article has been updated throughout the years to make it be the best that it can be; the iterations will continue.

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MPH to FPS: Mental Math Calculation Conversion Formula for Miles per Hour to Feet per Second

Latest update: April 18, 2020

How to Instantly Convert MPH to FPS When Driving

- The Quick and Easy Math Trick Formula Equation -

Simply divide miles-per-hour by 2, then multiply the result by 3 to find feet-per-second.

This is the easy, quick math formula or equation to use and gives a usable, fairly accurate answer. Your answer will be accurate within 5%. As an example, 100 mph converts to 150 fps. If one does the more complicated method of math calculation (detailed further down the page), the resulting answer would be 147 fps.

"How many feet per second..." 
is a math question usually relating to cars and driving.

Easy Table Conversion from MPH to FPS Examples Chart

MPH
FPS
10
15
15
22.5
20
30
25
37.5
30
45
35
52.5
40
60
45
67.5
50
75
55
82.5
60
90
65
97.5
70
105
75
112.5
80
120
90
135
100
150
120
180
140
210
160
240


The More Difficult (and More Accurate) Answer

To prove the quick math shortcut conversion calculation works:
  1. First. convert MPH (miles-per-hour) to MPM (miles-per-minute) by dividing MPH by 60.
  2. Then convert miles to feet. There are 5,280 feet in one mile, so multiply MPM by 5,280 to get FPM (feet-per-minute).
  3. Lastly, convert FPM to FPS (feet-per-second) by dividing by 60.

Examples of the Longhand Math Converting Miles-per-Hour (MPH) to Feet-per-Second (FPS)


MPH to FPS Math Conversion Example One

You're going 25 mph. How many feet is that per second?
  1. Conversion from mph to fps is as follows: If we divide by 60, we get miles-per-minute: 25/60 = .416667.
  2. For accuracy's sake, now is a good time to convert miles to feet. There are 5,280 feet in a mile so we multiply mpm times 5,280 to get feet-per-minute: 416667 x 5280 = ~2199.99 feet-per-minute [note: "~" means approximate].
  3. To convert from feet-per-minute to feet-per-second, we divide the answer by 60: 2199.99/60 = ~36.667 feet-per second.
  4. This answer is within 5% of the one you get doing the easy math, which says 25 mph equals 37.5 fps.

MPH to FPS Math Conversion Example Two

65 mph. How many feet per second?
  1. Conversion from mph to fps is as follows: If we divide by 60, we get miles-per-minute: 65/60 = 1.0843333333333.This makes sense. After all, 60 miles an hour is the well-known mile-a-minute.
  2. For accuracy's sake, now is a good time to convert miles to feet. There are 5,280 feet in a mile, so we multiply the miles-per-minute times 5,280 to get feet-per-minute: 1.0843333333333 x 5280 = ~5725.28 feet-per-minute.
  3. To convert from feet-per-minute to feet-per-second, we divide the answer by 60: 5725.28/60 = ~95.421 feet-per second, which is the answer.
  4. This coincides within 5% of the table above, which says 65 mph equals 97.5 fps. 

MPH to FPS Math Conversion Example Three

120 mph is how many feet per second? Conversion from mph to fps is as follows.
  1. If we divide by 60, we get miles-per-minute: 120/60 = 2.000. This again makes sense; 60 miles an hour is the usual mile-a-minute. So 120 mph would be 2-miles-a-minute.
  2. To convert miles to feet. There are 5,280 feet in a mile, so we multiply the miles-per-minute times 5,280 to get feet-per-minute: 2 x 5280 = ~10560 feet-per-minute.
  3. To convert from feet-per-minute to feet-per-second, we divide the answer by 60: 10560/60 = ~176 feet-per second.
  4. This number is within 5% of the table above which says 120 mph equals 180 fps.

The Reverse FPS to MPH Math Conversion Examples

Quick way to find reverse, i.e., convert from feet-per-second to miles-per hour: divide fps by 3 and then multiply by 2 for mph.

Easy Table Conversion from FPS to MPH Examples Chart

FPS
MPH
10
6.67
20
13.33
25
16.67
40
26.68
50
33.33
75
50
100
66.67
150
100
200
133.33
500
333.33
1000
666.67

Doing 20 Miles-per-Hour? That Is 30 Feet-per-Second.

The laws of physics seldom take a vacation.
Most of these cars would not be able to stop in time.


And for Your Amusement,
the Most Ridiculous Car Chase Ever.


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How Much Does a Pint / Quart / Gallon Weigh in Pounds?

Latest update: October 21, 2020

How Many and How to Calculate Pints, Quarts, Gallons to and from Pounds

Alternate Titles or Questions

  • How to Convert Pints to Quarts to Gallons to and from Pounds?
  • Volume to Weight Conversions or Weight to Volume Conversions – 8:4:1:8.
  • How many pounds in X pints, quarts, or gallons?
  • How much does X gallons, quarts, or pints weigh?
A list of most-searched-for questions and answers is included. There is also a specific section relating to the calculating of the size/volume/weight of home water heaters.

Volume to Weight Conversions. Weight to Volume Conversions.
Simple math to convert or calculate pints, quarts, gallons, and pounds.

Be advised that this page is US-centric. As an example, this does not work in the UK where a pint is 1.25 pounds as opposed to US 1.044 pounds at room temperature (RT). There are also issues of temperature and density, both of which are addressed further down the page. The purpose of this page is for practical, everyday business-of-living uses only. For that, it will serve you well.

First – The Quick Answers to Volume Amounts and Ratios

Volume Definitions

  • 2 pints equals 1 quart.
  • 4 quarts equals 1 gallon.
  • 8 pints equals 1 gallon.

Or to Put It Another Way...

  • 1 quart equals 2 pints.
  • 1 gallon equals 8 pints.
  • 1 gallon equals 4 quarts.

Converting Volume to Weight

Converting volume to weight has everything to do with the density of the liquid. Fortunately, this question usually has to do with:
  • How much does the gasoline in your gas tank or a gas can weigh?
  • How much does a specific container of water or other mostly-water grocery items weigh?
  • Questions relating to home water heater size, calculated water volume and the resulting weight.
The rule of thumb, and the expression to remember is: "A pint's a pound the world around." The resulting estimates and extrapolations from this rule will serve you well for most everyday purposes.

Basic Formulas

  • A pint weighs a pound.
  • There are two pints in a quart, so a quart weighs 2 pounds.
  • There are four quarts in a gallon, so a gallon weighs 8 pounds. 
  • And the eight pints in a gallon, also weighing 8 pounds.
This pretty much answers the question. Here are some other typical examples:

Some Household Examples

  • A 1-quart bottle of Gatorade weighs 2 pounds. Note: there are 2 pints in a quart and 4 quarts in a gallon. As a side note: there are 16 fluid ounces in a pint, 32 fluid ounces in a quart; 1 and 2 pounds respectively.
  • A 2-liter bottle of Pepsi would convert to a weight of a little over 4 pounds. Note: A liter is slightly more than a quart.

Some gallon examples

  • A 1-gallon container would convert to a weight of 8 pounds.
  • A 5-gallon container weighs 40 pounds.
  • A 10-gallon container weighs 80 pounds.
  • A full, 25-gallon SUV gas tank means you are hauling around 200 pounds of fuel.
  • A typical city water tower can hold anywhere from 300,00 to 600,000 gallons of water, which converts to a weight of 2,400,000 to 4,800,000 pounds of water sitting on those "stilts".

The Density of the Liquid Significantly Affects the Rules Concerning Volume Conversion Calculations to Weight


A major component of converting fluid volume to a weight measurement is the density of the fluid. For gasoline, water, and most grocery items; the rule of a-pint's-a-pound will serve you just fine. However, as an example, the rule probably wouldn't work too well with any significant volume of engine oil. As an extreme example, the liquid metal/element mercury would totally throw the pint's-a-pound rule out the window. So of course would any molten metal or alloy.

Fluid density is also affected by temperature. This is why many people fill their gas tank first thing in the morning. There is more gas per gallon at 50 degrees Fahrenheit than at 90 degrees Fahrenheit. It should be noted the percentage difference is in the low single-digits.

List of Frequent Volume-to-Weight Conversion Q&A


The Most-Searched-for Questions and Answers for How Many Pounds

  • How much does 1.5 quarts weigh? Answer is 3 pounds.
  • How much does 2 quarts weigh? Answer is 4 pounds.
  • How much does 3 quarts weigh? Answer is 6 pounds.
  • How much does 5 quarts weigh? Answer is 10 pounds.
  • How much does 6 quarts weigh? Answer is 12 pounds.
  • How much does 10 quarts weigh? Answer is 20 pounds.
  • How much does 16 quarts weigh? Answer is 32 pounds.
  • How much does 5 gallons weigh? Answer is 40 pounds.
  • How much does 10 gallons weigh? Answer is 80 pounds.
  • How much does 15 gallons weigh? Answer is 120 pounds.
  • How much does 20 gallons weigh? Answer is 160 pounds.
  • How much does 50 gallons weigh? Answer is 400 pounds.
  • How much does 55 gallons weigh? Answer is 440 pounds.
  • How many pints is a pound? Answer is 1.0 pint.
  • How many quarts is a pound? Answer is 0.5 quarts.
  • How many gallons is a pound? Answer is 0.125 gallons.

The formulas for the volume of a sphere, the volume of a cube,
the volume of a cylinder, the volume of a rectangular prism.

For folks who are interested, here is a beginner's Algebra Tutorial.

More Water and Gasoline Volume-to-Weight Examples

Depending on what unit of measurement you use, volume will equal cubic English or cubic Metric; examples being cubic inches or cubic centimeters.

Side note: the tilde (~) is the mathematical symbol for approximate.

English

  • 29 cubic inches equals ~1 pint, which equals ~1 pound.
  • 58 cubic inches equals ~2 pints, which equals ~1 quart, which equals ~1/4 of a gallon, which equals ~2 pounds.
  • 231 cubic inches equals ~4 quarts, which equals 1 gallon, which equals 8 pounds.

Metric

Most of the world uses Metric. There is a reason for that. As an example, 1000 cubic centimeters equals one liter, 1000 grams equals one kilogram, etc.; all nice, neat, and tidy. The United States and others are trying to get with the program; Metric is already included with English measurement on most U.S. consumer items. It's only a matter of time.

How Much Does the Water Weigh in a Full Water Heater?
– Water Tank Size Volume Formula and Answers –

Serendipitous energy.gov page on everything about water heaters, including how to buy one.

How to Calculate the Weight of the Water in a Home Water Heater

How much does the total amount of water in a water heater weigh, volume to weight conversion.

From the above NASA chart we see the volume formula for a cylinder is V = (πd2h)/4.

Water tank heaters come in all sizes. For the purposes of this example, we will say the water tank heater has a measured height of approximately 54 inches; what with this, that, and the other, the water part is probably around 48". The diameter measured as 18"; what with insulation, etc., 16 probably works.

So,
d = 16
h = 48

Thus (" * " meaning to multiply),
V = (3.14 * 16 * 16 * 48) divided by 4.

Since all numbers are inches, the answer will be in cubic inches. We reduce the formula as follows:
  1. V = (3.14 * 256 * 48) divided by 4.
  2. V = (3.14 * 12288)/4
  3. V = 38514/4
  4. V = 9646 cubic inches
231 cubic inches is equal to a gallon, so we divide 9646 by 231.
9646/231 = 41.76 gallons.

What with the inner measurements being estimates, looks like it is a 40 gallon water heater.
  1. A gallon weighs 8 pounds.
  2. So multiplying 40 times 8 gives 320.
  3. A 40-gallon water heater contains 320 pounds of cold water.
Knowing the volume and weight of a 40-gallon water tank heater makes it easy to extrapolate the volume and weight of other water heaters.
  • 10 gallon water heater is 2310 cubic inches and the water weighs 80 pounds.
  • 20 gallon water heater is 4620 cubic inches and the water weighs 160 pounds.
  • 30 gallon water heater is 6930 cubic inches and the water weighs 240 pounds.
  • 40 gallon water heater is 9240 cubic inches and the water weighs 320 pounds.
  • 50 gallon water heater is 11550 cubic inches and the water weighs 400 pounds.
  • 80 gallon water heater is 18480 cubic inches and the water weighs 640 pounds.
  • 100 gallon water heater is 23100 cubic inches and the water weighs 800 pounds.
Do keep in mind the temperature versus density considerations and the expansion of water when heated, i.e., a fully hot water heater tank weighs slightly less than a cold or warm water tank heater that's been freshly refilled after water usage. The difference between hot and cold water density versus volume is significant enough that most water heaters have a temperature pressure relief valve and a drainage tube or pipe to compensate for this.

Converting Cubic Inches to Cubic Feet...

...and the corresponding volume and weight ratios. It is surprising how much just one cubic foot of water weighs.

One cubic foot of water weighs ~60 pounds (59.844 pounds at room temperature).

A cubic foot is 12 inches times 12 inches times 12 inches. So 1728 cubic inches equals 1 cubic foot. To convert cubic inches to cubic feet, simply divide the cubic inches by 1728. So using the above examples, we would have:
  • 2310 cubic inches equals 1.34 cubic feet equaling 10 gallons equaling 80 lb.
  • 4620 cubic inches equals 2.68 cubic feet equaling 20 gallons equaling 160 lb.
  • 6930 cubic inches equals 4.02 cubic feet equaling 30 gallons equaling 240 lb.
  • 9240 cubic inches equals 5.36 cubic feet equaling 40 gallons equaling 320 lb.
  • 11550 cubic inches equals 6.70 cubic feet equaling 50 gallons equaling 400 lb.
  • 18480 cubic inches equals 10.72 cubic feet equaling 80 gallons equaling 640 lb.
  • 23100 cubic inches equals 13.40 cubic feet equaling 100 gallons equaling 800 lb.
* I happened to stumble across this on Wikipedia: "Pound (mass), a unit of mass often abbreviated incorrectly as 'lbs' in the plural. Abbreviations of units of measure do not use an 's' on the end for plural."

A side note. For anyone interested in doing their own quick calculations here or for anything and anywhere else and not wanting to bother with calling up a spreadsheet, etc., here's a handy Google Calculator. Opens in a separate tab or window. After arriving and before entering numbers, you will need to click its rectangular number-entry box first to get its attention.

May all your calculations be prosperous ones.

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Mental Calculation of State Sales Tax From Total to Stop Being Overcharged

Latest update: July 7, 2020
Have you been overcharged on sales tax? Here is a way on how to mentally calculate state sales taxes on the spot and stop being cheated, catch errors, and prevent fraud attempts.

Sales Tax - The regressive taxing of the poor

Business, Vendor, or Store Overcharging on State Sales Tax?

When it comes to sales taxes, fraud is not that rare of an occurrence. Many times, smaller stores do deliberately overcharge state sales tax. In fact, I’ve seen news reports where even the larger, national chain stores have been caught overcharging sales taxes. And employees in all stores have also been known to make price and thus state sales tax mistakes as well.

Mentally calculating state sales tax to prevent being overcharged is easy. It all has to do with rounding, no degree in rocket surgery required. You are simply doing a quick approximation to prevent yourself from being a victim of sales tax fraud or simply to prevent being mistakenly overcharged.

[Be advised that this page is US-centric. Canada and most European countries have sales taxes or a value added tax (VAT) far exceeding 10%. However, if the VAT tax is close to another round number, one can still make this method work.]

Here are the four main premises of this page:
  • Most combined state, country, city sales taxes do not exceed 10 percent, but most sales tax totals are reasonably close to 10 percent.
  • Most thieves are greedy and will exceed the 10 percent amount.
  • Even my dog can mentally calculate 10% of something.
  • Even my dog can mentally add 10% of something to something.

You do not need any of these...

How to Mentally Calculate State Sales Tax – Some Examples

How do I calculate sales tax from a total?

The best way for this tutorial to demonstrate mentally calculating state total sales taxes is by giving lots of examples. In reality, you already know how to do this. You just don't know you know yet. So let's begin. You walk up to the counter and engage in a purchase which sells for...

$49.99
  1. You round the price to $50.
  2. You calculate the 10% as $5.
  3. You add the $50 plus $5 to get $55.
  4. If the counter person wants more than $55, welcome to the world of sales tax fraud and overcharges.

Other Examples...


$29.99
  1. You round it to $30.
  2. 10% is $3.
  3. Total is $33.
  4. If the final price is over $33, welcome to the world of sales tax fraud and overcharges.
$5.99
  1. Round to $6.
  2. 10% is $.60.
  3. Total is $6.60.
  4. Anything over $6.60, welcome to the world of sales tax fraud and overcharges.
$79.98
  1. $80.
  2. $8.
  3. $88.
  4. Over $88, cheated.
It should be noted that honest mistakes do happen. You will find out soon enough if the overcharge was deliberate or accidental.

Is It Sales Tax Fraud?


What to Do When the Person at the Counter is Overcharging You on total State Sales Tax

This depends on your mood, time constraints, the amount of money involved, the store and neighborhood, etc. Below are some typical scenarios and what one can do in each situation; followed by what you can do after the fact.

You Don't Care About the Amount Involved

  1. Say nothing.
  2. Pay it.
  3. Say nothing. Or say the routine "Thanks."
  4. [Optional] Locate and take one of the business cards offered on the counter.
  5. Leave.
  6. Once outside, note the date and time.
  7. Never go back.
  8. Maybe tell everyone you know.

You Do Care About the Amount Involved (Option One)

  1. Don't pay it.
  2. Say nothing.
  3. [Optional] Locate and take one of the business cards offered on the counter.
  4. Leave. Be advised however, the counter person (probably the owner) will immediately know that you know he was trying to cheat you. And you took one of his cards... And state sales tax fraud is a very serious offense...
  5. Once outside, note the date and time.
  6. Never go back.
  7. Maybe tell everyone you know.

You Do Care About the Amount Involved (Option Two)

  • Politely point out the total is incorrect and explain why you think so.

  • If the counter person reviews and corrects the error...
  1. Pay it.
  2. Call it a day.
  3. Maybe or maybe not give the place another chance in the future.
  • If the counter person denies, disputes, or otherwise argues with your statement...
  1. [Optional] Locate and take one of the business cards offered on the counter.
  2. Leave.
  3. Once outside, note the date and time.
  4. Never go back.
  5. Tell everyone you know.

Reward for Reporting State Sales Tax Fraud?


How to Report Stores and Other Businesses Who Overcharge State Sales Taxes

Not only are you doing a good deed for society, you might even make some money in the process.
  1. Find your state's website dealing with all things sales tax.
  2. Find where to report what you experienced. As an example, in California the California State Board of Equalization would be where to go. California does not pay a reward the last time I checked. However, reporting the fraud is still a good idea; wouldn't you like the thief (employee or owner) removed, so you can have an honest, local place to shop? Reports can be made anonymously and will still be investigated.
  3. For other states, determine if you might get a reward. Tell them your experience in detail, including date and time. Give them all the information on the business card. If you don't have the store's business card, that is ok; just be sure the store name and address you are reporting is correct. And don't worry; they're not going to just take your word for it. They will probably send the equivalent of a few "mystery shoppers" to the store to confirm. When they have absolutely verified and proven it is not an isolated incident; only then will the hammer fall on the deserving thief.
More than likely the store location is leased. With any luck, the thieving employee or owner will soon be gone; hopefully replaced with a new, honest employee or business.

A separate note as to state tax billing errors by business vendors. There it is in writing; it's a pretty good bet that this is an honest mistake and a simple phone call will fix it. While you're at it, you might want to review the previous invoices from this vendor.

A July 2020 Update
If you do online shopping, you will want to read this site: DarkPatterns.

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How to Calculate Miles-per-Gallon (MPG) and Cost-per-Mile (CPM) Formula and Savings

Latest update: July 22, 2020

Calculate Miles-per-Gallon and Cost-per-Mile Using Formula Templates

For folks who want accurate miles-per-gallon and cost-per-mile answers and savings.

For quick, easy answers; simply use the MPG and CPM formula templates below and you are done.

Not pretty.

This page will serve you well if your gas gauge is broken, inaccurate, or otherwise giving you problems. Or if you just want to know how well your car is doing. Can also be used for possibly figuring out ways to improve your mileage.

Needless to say, one needs to know the miles driven, how much gas was used, and the price of the gas before the templates will be of any use to you. If you do not already have these numbers, Section I below has everything you need to know on how to get started.

" Ω " Handy Google calculator. Opens in a separate tab or window.
  • Both " / " and " ÷ " means divide.
  • After arriving at the calculator and before entering numbers, you will need to click its numbers box first to get its attention.

Distance Traveled Precalculation Template


New Odometer When Tank Refill  -  Old Odometer From Previous Refill  =  Miles Driven

As previously mentioned, if you are just looking for approximate answers, then you can simply use the templates and call it a day. If you are looking for the most real, accurate results possible and savings, see Section I.

How to Calculate Miles per Gallon Formula Template


Miles Driven  /  Gallons Used  =  Miles-per-Gallon (MPG)

Example

  1. You drove 100 miles and used 5 gallons of gas.
  2. Your intuitive answer would be 20 miles-per-gallon. Your intuitive answer would be correct.
  3. 100 miles traveled, divided by 5 gallons of gas used, gives you 20 miles per gallon.
  4. 100/5 = 20 mpg.

How to Calculate Gas Cost per Mile Formula Template


Price per Gallon  /  Miles per Gallon  =  Cost-per-Mile (CPM)

Example

  1. You paid $5 for a gallon of gas, and you get 10 miles-per-gallon (mpg).
  2. Your intuitive answer would be $.50 a mile. Your intuitive answer would be correct.
  3. $5 paid, divided by 10 miles traveled per gallon (mpg), gives you $.50 cost per mile.
  4. 5/10 = $.50 cost-per-mile

About Your Miles per Gallon Results...

So what does the miles-per-gallon answer actually tell you? It tells you that...

  • This particular vehicle,
  • being mechanically maintained at a given level of efficiency,
  • using a specific brand and grade of gasoline,
  • being filled at a particular time of day,
  • from a particular gas station and a particular pump at a particular  fill speed,
  • and being driven a certain commute route,
  • by a specific driver...
...gets so many miles per gallon.

Real Miles-per Gallon Savings

If the numbers used to make the calculations are accurate, this can lead to some interesting experimentation. What if...
  • A different gas station or pump was used? Not all stations and pumps are the same.
  • A different pump fill speed was used? 
  • The tank was filled at a different time of day? Temperature affects fluid density, first thing in the morning is best; that is when the gas is coldest and most dense.
  • A different brand and/or grade of gasoline was used? See Section II.
  • A different commute route was tried?
  • Deficiencies were found as to the vehicle's maintenance?
  • The vehicle's ignition timing was experimented with (but staying within smog emission specifications)? See Section II.
  • The driver notices and alters a particular driving habit?
Probably other ideas might also come to mind over time. 

Section I - Using a Reasonably Scientific Method and Mistakes to Avoid

If you are looking to get the most real, accurate results possible; this procedure will help you do that. If you are just looking for an approximation, then you can skip it all and fill in the templates with your existing numbers.
  1. Pick a week, or other time period, when you will be doing your most typical driving pattern.
  2. Have two pens and paper in the car.
  3. Use the gas station you normally use. Fill the gas tank at your usual time. Note the pump number you are using. Note the pump speed you normally use. Do not top off. While waiting, write down your odometer reading, include tenths. If you have a trip-odometer, reset it to zero. Remember to not be distracted by all this to the point you forget to put back the gas cap.
  4. Commence with your week; the usual work commute, errands, etc. Combining your work commute with errands will increase your gas mileage, but only do so if it is what you intend to usually do. Continue your routine until you have less than a quarter-tank. Don't strive for a gas-gauge reading of empty unless it is what you normally do.
  5. Make sure you still have the pens and paper in your car.
  6. At the same time of day as before, return to your previous gas station.
  7. Attempt to use the same pump number you used before. Set to the same pump speed as before. While waiting: write down your odometer reading; write down your trip-odometer reading; include tenths from both. When the pump-handle clicks: write down how many gallons; and very definitely include tenths. Write down the price you paid per gallon. Save the receipt; if the gas station is at least half-coordinated, some or all of this information will be printed there for you. Does it match what's showing on the pump? Do not top off. If so inclined, reset trip-odometer to zero. And the gas cap thing again...
  8. Proceed with your normal routine. You'll do the calculations with the templates at your leisure.

Section II - List of Notes About Fuel Economy, Improving Gas Mileage and Saving Money


Tune-ups and tire pressure: These are The Big Two as to getting the best mileage.
  • Over-inflating tires increases gas mileage, but causes an immediate and significant increase in tire wear; so don't do that. Under-inflated tires reduces your mileage; and it doesn't do your sidewalls any good either.
  • As for tune ups, spark plugs are especially important. A fouled or carbon-built-up plug reduces mileage drastically, not to mention it will probably cause you to flunk a smog check. A personal note: Two different mechanics quoted me a price of over $100 to change a set of 6 spark plugs, plus the inflated cost of the plugs. In both cases, I departed the premises immediately. I ended up changing the plugs myself, it's not that hard to do. Buy yourself a Chilton or Haynes manual for your particular make and model of car, they have all sorts of useful information. Some auto parts stores even have tool-loaner programs if you don't want to buy your own. Depending on how the plugs are positioned, changing spark plugs can be an obnoxious task. However, there is no law that says one has to do them all at once. I just did one or two at a time when sufficiently motivated. As a side note, disconnecting more than one plug at a time is not a good idea; reconnection time can be a disaster waiting to happen.

Looking ahead and coasting to stop lights: Is a close third.

Speed: Once you are above 40 mph or so; the faster you go, the lower your mileage.

Weight: If you are carrying excess, unnecessary weight in the trunk, it will:
  • Reduce mileage
  • Increase engine wear and tear
  • Wear out your brakes faster

Ethanol: Do you have ethanol-times-of-year versus non-ethanol-times-of-year? It can be interesting to make mileage comparisons between the two. You probably won't be happy with the ethanol results.

Gasoline Grade: Putting premium in a car that takes regular will do absolutely nothing for your mileage. However, if your car is in the midgrade octane category and what with there being some octane rating overlap, it might be worth experimenting with trying both the lower and higher octanes; especially if you are also experimenting with the ignition timing.

Temperature and humidity: Mileage is better during cooler times of the year than during heatwaves. And the higher the humidity, the better the mileage. Yes, one does get better mileage on rainy days.

Air filter: When is the last time you replaced the air filter? A clogged air filter does reduce mileage and can cause smog test problems.

Fuel density and time of day: As mentioned earlier, always fill your tank first thing in the morning. Fluid density is affected by temperature. The colder it is, the more gas you get per gallon.

Logbook: If so inclined, this is as good a time as any to start one, especially if you want to try any of the aforementioned experiments..

A Couple of Relevant Federal Websites

I thought I'd include a couple of useful federal websites for future reference. Both are worth browsing the next time you have some time to kill.

If only urban freeway traffic looked like this...
From www.epa.gov/air-pollution-transportation. Has all sorts of links regarding vehicles and fuel efficiency and saving gas in general.

For when planning your next road trip...
From www.fueleconomy.gov/trip. This goes directly to their trip calculator page. What makes this calculator unique is you can specify the make and model of car you are using or are curious about. In addition to the total fuel cost calculation, they throw in a map and text directions as well. The rest of the site is also worth browsing.

On a final note, here's a quick article on how to mentally convert miles-per-hour to feet-per-second when driving: MPH to FPS.

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