Elementary Math Topic: Fractions, Percents, Rates, Ratios and Decimal Equivalences

After answering some questions on the WyzAnt Resources Answers message board, I feel there is a need to explain all the equivalent ways we can interpret the above laundry list of notions that repeatedly come up in Elementary Mathematics...
This equivalence is a key point in the COMMON CORE CURRICULUM for mathematics.
Understanding equivalences and conceptual relationships in mathematics is the major goal of the common core!
Let's start with percents, because these perhaps cause the most confusion...what does per-cent actually mean??
Well, 'centi' is a prefix used to represent 100...a centennial happens every 100 years...a centimeter is 1/100th of a meter.
So if we talk about 50%=50per-cent, that is the equivalent of 50 per 100...a ratio of 50:100, or a fraction 50/100=1/2.
It can also be thought of as 50 parts per whole 'cent'...or 50 parts per 100...
Think of this as analogous to 'parts per million' as is sometimes a measurement used for molecules in a fluid, only we've replaced million with 'cent', meaning 100=10^2 instead of 1million=1 and 6 zeros after it=10^6.
This leads us into the discussion of decimal do we rewrite a percent as a decimal??
42% means 42 parts per 100, or 42/100...
since our decimal ('deca'=10) system is a base ten system, it means that each place k spaces to the left of the decimal tells us how many whole parts of 10^k we have...the places k spaces to the right of the decimal represent how many whole parts of 10^{-k} we have.  Consider the decimal 3.5, this is 3 whole parts(10^0=1's), plus a half a part...or 5 many 1/10th pieces (10^{-1}=1/10).
Now consider what happens when we look at the number 5.0 and divide it by 10...what happens?
We get 5/10=1/2=50% or the amount of fractional parts we had before in our example of 3.5...that is to say if we remove the 3 whole parts we are left with 3.5-3=0.5
The decimal point moved from the right side of the 5, to the left side of the 5!
That is what dividing by 10 does in our base-10 (decimal) number system.
Some similar examples can convince you that multiplying by 10 moves the decimal in the other direction (5.0*10=50.0).
So how do we write 42% as a decimal?  Well, take 42 and divide by 100=10^2...divide by 10, twice!
42/10=4.2 and 4.2/10=0.42=.42
there is always assumed to be as many extra zeros to the left and right as we want ... the decimal point is either explicitly written to the left of some numbers, or omitted and assumed to be immediately following the last written number (which represents the whole parts of 10^0=1=the units digit).
We can think of this as an expansion of our number line concept...the decimal point is always where we have anchored the line at the ORIGIN.  
To the left the digits of our number represent quantities of powers of 10^(number of spaces to the left of the point)...
To the right the digits are inverse (reciprocal) quantities of powers of 10^(number of spaces to the right of the decimal).
ASIDE: we read/write numbers left to right, so the numbers/places/powers of ten get bigger to the left and smaller to the right...even thought we normally think of the positive direction of the number line to be to the right, we can still see a similarity to the number line...especially if we think in terms of a logarithmic scale (but that leads to a more complex discussion on logarithms, better left for another day).


Casey W.

Mathematics (and Science) Instruction by a Mathematician!

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