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Density Equation Family

Date: 03/27/2002 at 20:07:04
From: Mary
Subject: Mass = ?

I know that density = mass/volume, but I need to know how to solve for 
mass using volume and density.

Date: 03/28/2002 at 09:12:15
From: Doctor Rick
Subject: Re: Mass = ?

Hi, Mary.

This sort of formula occurs a lot. You have probably seen the Rate 

  Rate = Distance / Time

The Rate Equation has two other equivalent forms:

  Distance = Rate * Time

  Time = Distance / Rate

One way to think of this is just like "fact families" (if you learned 
multiplication and division this way). Every multiplication or 
division problem is part of a "family" that uses the same numbers, for 

  2 * 5 = 10
  5 * 2 = 10
  10 / 2 = 5
  10 / 5 = 2

The three forms of the Rate Equation are just three members of the 
same fact family, except that we have names instead of particular 
numbers. (We leave out one because Distance = Time * Rate isn't any 
better than Distance = Rate * Time.)

The other way to think about it is the "algebraic" way. If two 
quantities are the same, and you multiply them by the same amount, 
then the new quantities are also the same. If Rate is equal to 
Distance divided by Time, then when I multiply both by Time, they are 
still the same:

  Distance * Time = Rate / Time * Time

But this can be rearranged:

  Distance * Time = Rate * (Time / Time)

Whatever Time is, Time divided by Time will always be 1, and 1 times 
Rate will always be Rate. Thus our new formula is

  Distance * Time = Rate

That's another member of the "Rate Equation Family."

Now, I hope you see that your formula, Density = Mass / Volume, is 
just the same as the Rate Equation, Rate = Distance / Time, except 
that the names have been changed. Can you see how to write two other 
members of the Density Equation Family?

- Doctor Rick, The Math Forum   
Associated Topics:
Middle School Word Problems

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