Fractions : Part Two

This is a longer post where we play with fractions and the intent is to take you to something you will see later when you learn about vectors, covectors and tensor calculus.

Think about what you see on an inch ruler and then look at…

4 * \dfrac {1}{4} = 1

…and with you imagination, tell yourself you are seeing…

\dfrac {4}{1} * \dfrac {1}{4} = \dfrac {1}{1}

Now try the following…

3 * 4 = 12

…and change it to…

3 * \dfrac {1}{1} * 4 = 12

…and then change that to…

3 * \dfrac {1}{a} * \dfrac {a}{1} * 4 = 12

…and you can let a be any real value that you want!

We want to put one more twist on this story. You know that 4 quarters is one dollar and that 20 nickels is also one dollar. You know that we measure quarters and nickels in cents and that one dollar is 100 cents.

With all this in mind, start with this…

4 * quarters = 100

…and then work through the following…

\dfrac {1}{1} * 4 * quarters = 100 \dfrac {1}{5} * \dfrac {5}{1} * 4 * quarters = 100

…switch some numbers around…

\dfrac {5}{1} * 4 * \dfrac {1}{5} * quarters = 100

…and then add parenthesis in two places…

(\dfrac {5}{1} * 4)*( \dfrac {1}{5} * quarters ) = 100

…then multiply two numbers to get a number and also take advantage of “a nickel is one-fifth of a quarter”…

(20)*(nickels ) = 100

You should be aware that you will often see things line “2.54 cm” and it is really “2.54 * centimeter”. It is very common to not type the * symbol.

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